Subgroups — C₂×C₇×#3211 #13237992

Order 1792.

2294 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3211	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#495	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#488	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#464	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#463	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#448	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#450	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#447	C₂	1	1	yes	maximal, characteristic
12	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₇×Q₁₆	C₂²	1	2	yes	complemented
16	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
17	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
18	C₂²×C₇×D₈	C₂²	1	2	yes	—
19	C₇×#529	—	2	4	no	—
20	C₇×#529	C₂²	1	8	yes	complemented
21	C₂×C₇×#126	—	2	1	no	—
22	C₂×C₇×#125	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₇×#124	—	2	1	no	—
24	C₂×C₇×#124	—	2	1	no	—
25	C₂×C₇×#124	—	2	4	no	—
26	C₂×C₇×#124	C₂²	1	2	yes	—
27	C₂×C₇×#124	C₂²	1	2	yes	—
28	C₂×C₇×#124	C₂²	1	8	yes	complemented
29	C₂×C₇×#124	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₇×#123	—	2	1	no	—
31	C₂×C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₇×#121	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×C₇×D₄	—	2	1	no	—
35	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₇×#117	—	2	1	no	—
37	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
38	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
39	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
40	C₇×#495	C₂²	1	16	yes	complemented
41	C₇×#488	C₂²	1	16	yes	complemented
42	C₇×#464	C₂²	1	16	yes	complemented
43	C₇×#463	C₂²	1	16	yes	complemented
44	C₇×#451	C₂²	1	8	yes	complemented
45	C₇×#450	C₂²	1	8	yes	complemented
46	C₇×#448	C₂²	1	8	yes	complemented
47	C₇×#447	C₂²	1	8	yes	—
48	C₇×#431	C₂²	1	8	yes	complemented
49	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
50	C₂×C₇×#103	C₂²	1	2	yes	complemented
51	C₂×C₇×#103	C₂²	1	2	yes	complemented
52	C₂×C₇×#102	C₂²	1	2	yes	—
53	C₂×C₇×#102	C₂²	1	2	yes	—
54	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
55	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
57	C₂×#3211	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁵×C₇	—	2	1	no	—
59	C₂²×C₇×Q₈	—	2	2	no	—
60	C₂²×C₇×Q₈	—	2	1	no	—
61	C₂²×C₇×Q₈	—	4	4	no	—
62	C₂²×C₇×Q₈	C₂³	1	2	yes	—
63	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
64	C₂²×C₇×Q₈	C₂³	1	4	yes	—
65	C₂²×C₇×D₄	—	2	1	no	—
66	C₂²×C₇×D₄	—	2	2	no	—
67	C₂²×C₇×D₄	—	2	1	no	—
68	C₂²×C₇×D₄	—	4	4	no	—
69	C₂²×C₇×D₄	C₂³	1	2	yes	—
70	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
71	C₂²×C₇×D₄	C₂³	1	4	yes	—
72	C₂³×C₄×C₇	—	2	1	no	—
73	C₂³×C₄×C₇	—	2	1	no	—
74	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
75	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
77	C₂×C₇×Q₁₆	—	4	4	no	—
78	C₂×C₇×Q₁₆	C₂³	1	8	yes	complemented
79	C₂×C₇×SD₁₆	—	4	4	no	—
80	C₂×C₇×SD₁₆	—	4	4	no	—
81	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
82	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
83	C₂×C₇×D₈	—	4	4	no	—
84	C₂×C₇×D₈	C₂³	1	8	yes	—
85	C₂²×C₇×C₈	C₂³	1	2	yes	—
86	C₂²×C₇×C₈	C₂³	1	2	yes	—
87	C₇×#126	—	2	8	no	—
88	C₇×#125	—	2	2	no	—
89	C₇×#125	C₂³	1	4	yes	complemented
90	C₇×#124	—	2	4	no	—
91	C₇×#124	—	2	4	no	—
92	C₇×#124	—	2	8	no	—
93	C₇×#124	—	2	8	no	—
94	C₇×#124	—	2	2	no	—
95	C₇×#124	C₂³	1	8	yes	—
96	C₇×#124	C₂³	1	8	yes	—
97	C₇×#124	C₂³	1	4	yes	complemented
98	C₇×#123	—	2	8	no	—
99	C₇×C₂²⋊Q₈	—	2	2	no	—
100	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
101	C₇×#121	—	2	2	no	—
102	C₇×#121	C₂³	1	4	yes	—
103	C₄×C₇×Q₈	—	2	2	no	—
104	C₄×C₇×Q₈	C₂³	1	4	yes	complemented
105	C₄×C₇×D₄	—	2	8	no	—
106	C₄×C₇×D₄	—	2	2	no	—
107	C₄×C₇×D₄	C₂³	1	4	yes	complemented
108	C₇×#117	—	2	8	no	—
109	C₂×C₇×C₄⋊C₄	—	2	1	no	—
110	C₂×C₇×C₄⋊C₄	—	2	1	no	—
111	C₂×C₇×C₄⋊C₄	—	2	1	no	—
112	C₂×C₇×C₄⋊C₄	—	2	1	no	—
113	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
115	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
116	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
117	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
118	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
119	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
121	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
122	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
123	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
126	C₂×C₄²×C₇	—	2	1	no	—
127	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
129	C₂×C₄²×C₇	D₄	1	4	yes	—
130	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₇	C₂³	1	4	yes	—
133	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
134	C₇×#103	C₂³	1	8	yes	complemented
135	C₇×#103	C₂³	1	8	yes	complemented
136	C₇×#102	C₂³	1	8	yes	—
137	C₇×#102	C₂³	1	8	yes	—
138	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
139	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
140	C₄×C₇×C₈	C₂³	1	8	yes	complemented
141	C₂×#529	C₂×C₇	1	1	yes	characteristic, complemented
142	C₂²×#124	C₂×C₇	1	1	yes	characteristic, complemented
143	#3211	C₂×C₇	1	16	yes	direct product factor
144	C₂×#495	C₂×C₇	1	2	yes	complemented
145	C₂×#488	C₂×C₇	1	2	yes	complemented
146	C₂×#464	C₂×C₇	1	2	yes	complemented
147	C₂×#463	C₂×C₇	1	2	yes	complemented
148	C₂×#451	C₂×C₇	1	1	yes	characteristic, complemented
149	C₂×#448	C₂×C₇	1	1	yes	characteristic, complemented
150	C₂×#450	C₂×C₇	1	1	yes	characteristic, complemented
151	C₂×#447	C₂×C₇	1	1	yes	characteristic
152	C₂×#431	C₂×C₇	1	1	yes	characteristic, complemented
153	C₂⁴×C₇	—	2	4	no	—
154	C₂⁴×C₇	—	2	1	no	—
155	C₂⁴×C₇	—	2	2	no	—
156	C₂⁴×C₇	—	4	4	no	—
157	C₂⁴×C₇	—	4	4	no	—
158	C₂⁴×C₇	—	4	4	no	—
159	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
160	C₂×C₇×Q₈	—	2	1	no	—
161	C₂×C₇×Q₈	—	2	4	no	—
162	C₂×C₇×Q₈	—	2	2	no	—
163	C₂×C₇×Q₈	—	2	4	no	—
164	C₂×C₇×Q₈	—	2	4	no	—
165	C₂×C₇×Q₈	—	2	4	no	—
166	C₂×C₇×Q₈	—	4	8	no	—
167	C₂×C₇×Q₈	—	4	4	no	—
168	C₂×C₇×Q₈	—	4	8	no	—
169	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
170	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
171	C₂×C₇×D₄	—	2	4	no	—
172	C₂×C₇×D₄	—	2	1	no	—
173	C₂×C₇×D₄	—	2	4	no	—
174	C₂×C₇×D₄	—	2	2	no	—
175	C₂×C₇×D₄	—	2	4	no	—
176	C₂×C₇×D₄	—	2	4	no	—
177	C₂×C₇×D₄	—	2	4	no	—
178	C₂×C₇×D₄	—	4	8	no	—
179	C₂×C₇×D₄	—	4	4	no	—
180	C₂×C₇×D₄	—	4	8	no	—
181	C₂×C₇×D₄	—	4	4	no	—
182	C₂×C₇×D₄	C₂⁴	1	4	yes	—
183	C₂×C₇×D₄	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₇	—	2	4	no	—
185	C₂²×C₄×C₇	—	2	4	no	—
186	C₂²×C₄×C₇	—	2	4	no	—
187	C₂²×C₄×C₇	—	2	1	no	—
188	C₂²×C₄×C₇	—	2	1	no	—
189	C₂²×C₄×C₇	—	2	2	no	—
190	C₂²×C₄×C₇	—	2	1	no	—
191	C₂²×C₄×C₇	—	2	4	no	—
192	C₂²×C₄×C₇	—	4	2	no	—
193	C₂²×C₄×C₇	—	4	4	no	—
194	C₂²×C₄×C₇	—	4	4	no	—
195	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
197	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
205	C₇×Q₁₆	—	4	8	no	—
206	C₇×SD₁₆	—	4	8	no	—
207	C₇×SD₁₆	—	4	8	no	—
208	C₇×D₈	—	4	8	no	—
209	C₂×C₇×C₈	—	2	4	no	—
210	C₂×C₇×C₈	—	2	4	no	—
211	C₂×C₇×C₈	C₂⁴	1	4	yes	—
212	C₂×C₇×C₈	C₂⁴	1	4	yes	—
213	C₇×C₄⋊C₄	—	2	4	no	—
214	C₇×C₄⋊C₄	—	2	4	no	—
215	C₇×C₄⋊C₄	—	2	4	no	—
216	C₇×C₄⋊C₄	—	2	4	no	—
217	C₇×C₄⋊C₄	—	2	1	no	—
218	C₇×C₄⋊C₄	—	2	1	no	—
219	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
220	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
221	C₇×C₂²⋊C₄	—	2	4	no	—
222	C₇×C₂²⋊C₄	—	2	8	no	—
223	C₇×C₂²⋊C₄	—	2	8	no	—
224	C₇×C₂²⋊C₄	—	2	8	no	—
225	C₇×C₂²⋊C₄	—	2	8	no	—
226	C₇×C₂²⋊C₄	—	2	4	no	—
227	C₇×C₂²⋊C₄	—	2	4	no	—
228	C₇×C₂²⋊C₄	—	2	4	no	—
229	C₇×C₂²⋊C₄	—	2	4	no	—
230	C₄²×C₇	—	2	4	no	—
231	C₄²×C₇	C₂×D₄	1	4	yes	—
232	C₄²×C₇	C₂×D₄	1	2	yes	—
233	C₄²×C₇	C₂×D₄	1	2	yes	—
234	C₄²×C₇	C₂×D₄	1	4	yes	—
235	C₄²×C₇	C₂⁴	1	2	yes	—
236	C₄²×C₇	C₂⁴	1	2	yes	—
237	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
238	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
239	C₂²×Q₁₆	C₂²×C₇	1	2	yes	complemented
240	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
241	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
242	C₂²×D₈	C₂²×C₇	1	2	yes	—
243	#529	—	2	4	no	—
244	#529	C₂²×C₇	1	8	yes	complemented
245	C₂×#126	—	2	1	no	—
246	C₂×#125	C₂²×C₇	1	1	yes	characteristic, complemented
247	C₂×#124	—	2	1	no	—
248	C₂×#124	—	2	1	no	—
249	C₂×#124	—	2	4	no	—
250	C₂×#124	C₂²×C₇	1	2	yes	—
251	C₂×#124	C₂²×C₇	1	2	yes	—
252	C₂×#124	C₂²×C₇	1	8	yes	complemented
253	C₂×#124	C₂²×C₇	1	1	yes	characteristic, complemented
254	C₂×#123	—	2	1	no	—
255	C₂×C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
256	C₂×#121	C₂²×C₇	1	1	yes	characteristic
257	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
258	C₂×C₄×D₄	—	2	1	no	—
259	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
260	C₂×#117	—	2	1	no	—
261	C₂²×C₂²⋊C₄	—	2	1	no	—
262	C₂²×C₂²⋊C₄	—	2	1	no	—
263	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
264	#495	C₂²×C₇	1	16	yes	complemented
265	#488	C₂²×C₇	1	16	yes	complemented
266	#464	C₂²×C₇	1	16	yes	complemented
267	#463	C₂²×C₇	1	16	yes	complemented
268	#451	C₂²×C₇	1	8	yes	complemented
269	#450	C₂²×C₇	1	8	yes	complemented
270	#448	C₂²×C₇	1	8	yes	complemented
271	#447	C₂²×C₇	1	8	yes	—
272	#431	C₂²×C₇	1	8	yes	complemented
273	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
274	C₂×#103	C₂²×C₇	1	2	yes	complemented
275	C₂×#103	C₂²×C₇	1	2	yes	complemented
276	C₂×#102	C₂²×C₇	1	2	yes	—
277	C₂×#102	C₂²×C₇	1	2	yes	—
278	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
279	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
280	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
281	C₂³×C₇	—	2	2	no	—
282	C₂³×C₇	—	2	2	no	—
283	C₂³×C₇	—	2	2	no	—
284	C₂³×C₇	—	2	4	no	—
285	C₂³×C₇	—	4	8	no	—
286	C₂³×C₇	—	4	8	no	—
287	C₂³×C₇	—	4	4	no	—
288	C₂³×C₇	—	4	2	no	—
289	C₂³×C₇	—	4	2	no	—
290	C₂³×C₇	—	4	8	no	—
291	C₂³×C₇	—	4	4	no	—
292	C₂³×C₇	—	4	2	no	—
293	C₂³×C₇	—	4	4	no	—
294	C₂³×C₇	—	8	16	no	—
295	C₂³×C₇	#135	1	4	yes	—
296	C₂³×C₇	C₂²×D₄	1	2	yes	—
297	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
298	C₇×Q₈	—	2	4	no	—
299	C₇×Q₈	—	2	4	no	—
300	C₇×Q₈	—	2	4	no	—
301	C₇×Q₈	—	4	8	no	—
302	C₇×Q₈	—	4	2	no	—
303	C₇×Q₈	—	4	4	no	—
304	C₇×D₄	—	2	4	no	—
305	C₇×D₄	—	2	4	no	—
306	C₇×D₄	—	2	4	no	—
307	C₇×D₄	—	4	8	no	—
308	C₇×D₄	—	4	8	no	—
309	C₇×D₄	—	4	2	no	—
310	C₇×D₄	—	4	4	no	—
311	C₂×C₄×C₇	—	2	2	no	—
312	C₂×C₄×C₇	—	2	8	no	—
313	C₂×C₄×C₇	—	2	2	no	—
314	C₂×C₄×C₇	—	2	2	no	—
315	C₂×C₄×C₇	—	2	2	no	—
316	C₂×C₄×C₇	—	2	2	no	—
317	C₂×C₄×C₇	—	2	4	no	—
318	C₂×C₄×C₇	—	2	4	no	—
319	C₂×C₄×C₇	—	2	2	no	—
320	C₂×C₄×C₇	—	2	2	no	—
321	C₂×C₄×C₇	—	2	4	no	—
322	C₂×C₄×C₇	—	2	2	no	—
323	C₂×C₄×C₇	—	4	4	no	—
324	C₂×C₄×C₇	—	4	8	no	—
325	C₂×C₄×C₇	—	4	8	no	—
326	C₂×C₄×C₇	—	4	4	no	—
327	C₂×C₄×C₇	—	4	2	no	—
328	C₂×C₄×C₇	—	4	2	no	—
329	C₂×C₄×C₇	—	4	2	no	—
330	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
331	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
332	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
333	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
334	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
335	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
337	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
338	C₇×C₈	—	2	4	no	—
339	C₇×C₈	—	2	4	no	—
340	C₂⁵	—	2	1	no	—
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×Q₈	—	2	2	no	—
343	C₂²×Q₈	—	4	4	no	—
344	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
345	C₂²×Q₈	C₂³×C₇	1	2	yes	—
346	C₂²×Q₈	C₂³×C₇	1	4	yes	—
347	C₂²×D₄	—	2	1	no	—
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	4	4	no	—
351	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
352	C₂²×D₄	C₂³×C₇	1	2	yes	—
353	C₂²×D₄	C₂³×C₇	1	4	yes	—
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	—	2	1	no	—
356	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
357	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
359	C₂×Q₁₆	—	4	4	no	—
360	C₂×Q₁₆	C₂³×C₇	1	8	yes	complemented
361	C₂×SD₁₆	—	4	4	no	—
362	C₂×SD₁₆	—	4	4	no	—
363	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
364	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
365	C₂×D₈	—	4	4	no	—
366	C₂×D₈	C₂³×C₇	1	8	yes	—
367	C₂²×C₈	C₂³×C₇	1	2	yes	—
368	C₂²×C₈	C₂³×C₇	1	2	yes	—
369	#126	—	2	8	no	—
370	#125	—	2	2	no	—
371	#125	C₂³×C₇	1	4	yes	complemented
372	#124	—	2	4	no	—
373	#124	—	2	4	no	—
374	#124	—	2	8	no	—
375	#124	—	2	8	no	—
376	#124	—	2	2	no	—
377	#124	C₂³×C₇	1	8	yes	—
378	#124	C₂³×C₇	1	8	yes	—
379	#124	C₂³×C₇	1	4	yes	complemented
380	#123	—	2	8	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
383	#121	—	2	2	no	—
384	#121	C₂³×C₇	1	4	yes	—
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	C₂³×C₇	1	4	yes	complemented
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	C₂³×C₇	1	4	yes	complemented
390	#117	—	2	8	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	—	2	2	no	—
400	C₂×C₂²⋊C₄	—	2	2	no	—
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	8	no	—
403	C₂×C₂²⋊C₄	—	2	8	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
410	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
411	C₂×C₄²	C₇×D₄	1	4	yes	—
412	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₇	1	4	yes	—
415	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
416	#103	C₂³×C₇	1	8	yes	complemented
417	#103	C₂³×C₇	1	8	yes	complemented
418	#102	C₂³×C₇	1	8	yes	—
419	#102	C₂³×C₇	1	8	yes	—
420	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
421	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
422	C₄×C₈	C₂³×C₇	1	8	yes	complemented
423	C₂²×C₇	—	2	2	no	—
424	C₂²×C₇	—	2	8	no	—
425	C₂²×C₇	—	2	2	no	—
426	C₂²×C₇	—	4	4	no	—
427	C₂²×C₇	—	4	2	no	—
428	C₂²×C₇	—	4	2	no	—
429	C₂²×C₇	—	4	4	no	—
430	C₂²×C₇	—	4	2	no	—
431	C₂²×C₇	—	4	4	no	—
432	C₂²×C₇	—	8	16	no	—
433	C₂²×C₇	—	8	4	no	—
434	C₂²×C₇	—	8	8	no	—
435	C₂²×C₇	#535	1	2	yes	—
436	C₂²×C₇	C₂×#135	1	4	yes	—
437	C₂²×C₇	C₂×#135	1	2	yes	—
438	C₂²×C₇	#575	1	2	yes	—
439	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
440	C₄×C₇	—	2	2	no	—
441	C₄×C₇	—	2	2	no	—
442	C₄×C₇	—	2	2	no	—
443	C₄×C₇	—	2	2	no	—
444	C₄×C₇	—	2	2	no	—
445	C₄×C₇	—	4	4	no	—
446	C₄×C₇	—	4	2	no	—
447	C₄×C₇	—	4	2	no	—
448	C₄×C₇	—	4	2	no	—
449	C₄×C₇	#535	1	2	yes	—
450	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
451	C₂⁴	—	2	1	no	—
452	C₂⁴	—	2	4	no	—
453	C₂⁴	—	2	2	no	—
454	C₂⁴	—	4	4	no	—
455	C₂⁴	—	4	4	no	—
456	C₂⁴	—	4	4	no	—
457	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	4	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×Q₈	—	2	4	no	—
464	C₂×Q₈	—	4	8	no	—
465	C₂×Q₈	—	4	4	no	—
466	C₂×Q₈	—	4	8	no	—
467	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
468	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	2	4	no	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	4	8	no	—
477	C₂×D₄	—	4	4	no	—
478	C₂×D₄	—	4	8	no	—
479	C₂×D₄	—	4	4	no	—
480	C₂×D₄	C₂⁴×C₇	1	4	yes	—
481	C₂×D₄	C₂⁴×C₇	1	2	yes	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	—	4	4	no	—
493	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
494	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
495	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
496	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
497	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
498	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
499	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
500	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
501	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
502	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
503	Q₁₆	—	4	8	no	—
504	SD₁₆	—	4	8	no	—
505	SD₁₆	—	4	8	no	—
506	D₈	—	4	8	no	—
507	C₂×C₈	—	2	4	no	—
508	C₂×C₈	—	2	4	no	—
509	C₂×C₈	C₂⁴×C₇	1	4	yes	—
510	C₂×C₈	C₂⁴×C₇	1	4	yes	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
518	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	—	2	8	no	—
523	C₂²⋊C₄	—	2	8	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₄²	—	2	4	no	—
529	C₄²	C₂×C₇×D₄	1	4	yes	—
530	C₄²	C₂×C₇×D₄	1	2	yes	—
531	C₄²	C₂×C₇×D₄	1	2	yes	—
532	C₄²	C₂×C₇×D₄	1	4	yes	—
533	C₄²	C₂⁴×C₇	1	2	yes	—
534	C₄²	C₂⁴×C₇	1	2	yes	—
535	C₂×C₇	—	2	4	no	—
536	C₂×C₇	—	8	2	no	—
537	C₂×C₇	—	8	4	no	—
538	C₂×C₇	#3211	1	4	yes	direct product factor
539	C₂×C₇	C₂×#535	1	1	yes	characteristic
540	C₂×C₇	C₂²×#135	1	1	yes	characteristic
541	C₂×C₇	C₂×#575	1	1	yes	characteristic
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	8	no	—
548	C₂³	—	4	4	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	2	no	—
551	C₂³	—	4	8	no	—
552	C₂³	—	4	4	no	—
553	C₂³	—	4	2	no	—
554	C₂³	—	4	4	no	—
555	C₂³	—	8	16	no	—
556	C₂³	C₇×#135	1	4	yes	—
557	C₂³	C₂²×C₇×D₄	1	2	yes	—
558	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
559	Q₈	—	2	4	no	—
560	Q₈	—	2	4	no	—
561	Q₈	—	2	4	no	—
562	Q₈	—	4	8	no	—
563	Q₈	—	4	2	no	—
564	Q₈	—	4	4	no	—
565	D₄	—	2	4	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	4	no	—
568	D₄	—	4	8	no	—
569	D₄	—	4	8	no	—
570	D₄	—	4	2	no	—
571	D₄	—	4	4	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
592	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
593	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
594	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
595	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
596	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
597	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
598	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
599	C₈	—	2	4	no	—
600	C₈	—	2	4	no	—
601	C₇	C₂×#3211	1	1	yes	characteristic, direct product factor, Sylow
602	C₂²	—	2	2	no	—
603	C₂²	—	2	8	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	4	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	8	16	no	—
612	C₂²	—	8	8	no	—
613	C₂²	—	8	4	no	—
614	C₂²	C₇×#535	1	2	yes	—
615	C₂²	C₂×C₇×#135	1	4	yes	—
616	C₂²	C₂×C₇×#135	1	2	yes	—
617	C₂²	C₇×#575	1	2	yes	—
618	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	4	no	—
627	C₄	—	4	2	no	—
628	C₄	C₇×#535	1	2	yes	—
629	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
630	C₂	—	2	4	no	—
631	C₂	—	8	4	no	—
632	C₂	—	8	2	no	—
633	C₂	C₇×#3211	1	4	yes	direct product factor
634	C₂	C₂×C₇×#535	1	1	yes	characteristic
635	C₂	C₂²×C₇×#135	1	1	yes	characteristic
636	C₂	C₂×C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₇×#529	C₂	semidirect
2. C₂²×C₇×#124	C₂	semidirect
3. C₇×#3211	C₂	direct (N × H)
4. C₂×C₇×#495	C₂	semidirect
5. C₂×C₇×#488	C₂	semidirect
6. C₂×C₇×#464	C₂	semidirect
7. C₂×C₇×#463	C₂	semidirect
8. C₂×C₇×#451	C₂	semidirect
9. C₂×C₇×#448	C₂	semidirect
10. C₂×C₇×#450	C₂	semidirect
12. C₂×C₇×#431	C₂	semidirect
15. C₂²×C₇×Q₁₆	C₂²	semidirect
17. C₂²×C₇×SD₁₆	C₂²	semidirect
20. C₇×#529	C₂²	semidirect
22. C₂×C₇×#125	C₂²	semidirect
28. C₂×C₇×#124	C₂²	semidirect
29. C₂×C₇×#124	C₂²	semidirect
33. C₂×C₄×C₇×Q₈	C₂²	semidirect
35. C₂×C₄×C₇×D₄	C₂²	semidirect
40. C₇×#495	C₂²	semidirect
41. C₇×#488	C₂²	semidirect
42. C₇×#464	C₂²	semidirect
43. C₇×#463	C₂²	semidirect
44. C₇×#451	C₂²	semidirect
45. C₇×#450	C₂²	semidirect
46. C₇×#448	C₂²	semidirect
48. C₇×#431	C₂²	semidirect
49. C₂×C₇×C₄⋊C₈	C₂²	semidirect
50. C₂×C₇×#103	C₂²	semidirect
51. C₂×C₇×#103	C₂²	semidirect
56. C₂×C₄×C₇×C₈	C₂²	semidirect
57. C₂×#3211	C₇	direct (N × H)
78. C₂×C₇×Q₁₆	C₂³	semidirect
82. C₂×C₇×SD₁₆	C₂³	semidirect
89. C₇×#125	C₂³	semidirect
97. C₇×#124	C₂³	semidirect
104. C₄×C₇×Q₈	C₂³	semidirect
107. C₄×C₇×D₄	C₂³	semidirect
133. C₇×C₄⋊C₈	C₂³	semidirect
134. C₇×#103	C₂³	semidirect
135. C₇×#103	C₂³	semidirect
140. C₄×C₇×C₈	C₂³	semidirect
141. C₂×#529	C₂×C₇	semidirect
142. C₂²×#124	C₂×C₇	semidirect
143. #3211	C₂×C₇	direct (N × H)
144. C₂×#495	C₂×C₇	semidirect
145. C₂×#488	C₂×C₇	semidirect
146. C₂×#464	C₂×C₇	semidirect
147. C₂×#463	C₂×C₇	semidirect
148. C₂×#451	C₂×C₇	semidirect
149. C₂×#448	C₂×C₇	semidirect
150. C₂×#450	C₂×C₇	semidirect
152. C₂×#431	C₂×C₇	semidirect
239. C₂²×Q₁₆	C₂²×C₇	semidirect
241. C₂²×SD₁₆	C₂²×C₇	semidirect
244. #529	C₂²×C₇	semidirect
246. C₂×#125	C₂²×C₇	semidirect
252. C₂×#124	C₂²×C₇	semidirect
253. C₂×#124	C₂²×C₇	semidirect
257. C₂×C₄×Q₈	C₂²×C₇	semidirect
259. C₂×C₄×D₄	C₂²×C₇	semidirect
264. #495	C₂²×C₇	semidirect
265. #488	C₂²×C₇	semidirect
266. #464	C₂²×C₇	semidirect
267. #463	C₂²×C₇	semidirect
268. #451	C₂²×C₇	semidirect
269. #450	C₂²×C₇	semidirect
270. #448	C₂²×C₇	semidirect
272. #431	C₂²×C₇	semidirect
273. C₂×C₄⋊C₈	C₂²×C₇	semidirect
274. C₂×#103	C₂²×C₇	semidirect
275. C₂×#103	C₂²×C₇	semidirect
280. C₂×C₄×C₈	C₂²×C₇	semidirect
360. C₂×Q₁₆	C₂³×C₇	semidirect
364. C₂×SD₁₆	C₂³×C₇	semidirect
371. #125	C₂³×C₇	semidirect
379. #124	C₂³×C₇	semidirect
386. C₄×Q₈	C₂³×C₇	semidirect
389. C₄×D₄	C₂³×C₇	semidirect
415. C₄⋊C₈	C₂³×C₇	semidirect
416. #103	C₂³×C₇	semidirect
417. #103	C₂³×C₇	semidirect
422. C₄×C₈	C₂³×C₇	semidirect
538. C₂×C₇	#3211	direct (N × H)
601. C₇	C₂×#3211	direct (N × H)
633. C₂	C₇×#3211	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#529	2
C₂²×C₇×#124	2
C₇×#3211	2
C₂×C₇×#495	2
C₂×C₇×#488	2
C₂×C₇×#464	2
C₂×C₇×#463	2
C₂×C₇×#451	2
C₂×C₇×#448	2
C₂×C₇×#450	2
C₂×C₇×#447	2
C₂×C₇×#431	2
C₂×#3211	7

Subgroups — C2×C7×#3211 #13237992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3211 #13237992