Subgroups — C₂×C₇×#3215 #13238000

Order 1792.

2242 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#124	C₂	1	1	yes	maximal, characteristic
3	C₇×#3215	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#497	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#464	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×#451	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#450	C₂	1	2	yes	maximal
9	C₂×C₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₇×Q₁₆	C₂²	1	4	yes	complemented
13	C₂²×C₇×SD₁₆	C₂²	1	4	yes	—
14	C₇×#533	—	2	4	no	—
15	C₇×#533	C₂²	1	8	yes	complemented
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₇×#125	—	2	1	no	—
19	C₂×C₇×#124	—	2	1	no	—
20	C₂×C₇×#124	—	2	1	no	—
21	C₂×C₇×#124	—	2	4	no	—
22	C₂×C₇×#124	C₂²	1	8	yes	—
23	C₂×C₇×#124	C₂²	1	2	yes	—
24	C₂×C₇×#124	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
26	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₄×C₇×Q₈	—	2	1	no	—
28	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	complemented
29	C₂×C₇×#117	—	2	1	no	—
30	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
31	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
32	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
33	C₇×#497	C₂²	1	16	yes	complemented
34	C₇×#489	C₂²	1	16	yes	complemented
35	C₇×#464	C₂²	1	32	yes	complemented
36	C₇×#451	C₂²	1	16	yes	complemented
37	C₇×#450	C₂²	1	16	yes	—
38	C₇×#432	C₂²	1	8	yes	complemented
39	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₇×#103	C₂²	1	4	yes	complemented
41	C₂×C₇×#102	C₂²	1	4	yes	—
42	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₇×#97	C₂²	1	2	yes	complemented
44	C₂×#3215	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁵×C₇	—	2	1	no	—
46	C₂²×C₇×Q₈	—	2	2	no	—
47	C₂²×C₇×Q₈	—	2	1	no	—
48	C₂²×C₇×Q₈	—	2	1	no	—
49	C₂²×C₇×Q₈	—	4	4	no	—
50	C₂²×C₇×Q₈	C₂³	1	2	yes	—
51	C₂²×C₇×Q₈	C₂³	1	2	yes	—
52	C₂²×C₇×Q₈	C₂³	1	4	yes	—
53	C₂²×C₇×D₄	—	2	2	no	—
54	C₂²×C₇×D₄	—	2	1	no	—
55	C₂²×C₇×D₄	—	4	4	no	—
56	C₂²×C₇×D₄	C₂³	1	2	yes	—
57	C₂²×C₇×D₄	C₂³	1	4	yes	—
58	C₂³×C₄×C₇	—	2	1	no	—
59	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
62	C₂×C₇×Q₁₆	—	4	8	no	—
63	C₂×C₇×Q₁₆	C₂³	1	16	yes	complemented
64	C₂×C₇×SD₁₆	—	4	8	no	—
65	C₂×C₇×SD₁₆	C₂³	1	16	yes	—
66	C₂²×C₇×C₈	C₂³	1	4	yes	—
67	C₇×C₄⋊Q₈	—	2	2	no	—
68	C₇×C₄⋊Q₈	—	2	2	no	—
69	C₇×C₄⋊Q₈	C₂³	1	4	yes	complemented
70	C₇×C₄⋊Q₈	C₂³	1	4	yes	complemented
71	C₇×#125	—	2	8	no	—
72	C₇×#124	—	2	8	no	—
73	C₇×#124	—	2	4	no	—
74	C₇×#124	—	2	8	no	—
75	C₇×#124	—	2	4	no	—
76	C₇×#124	C₂³	1	8	yes	—
77	C₇×#124	C₂³	1	8	yes	—
78	C₇×C₂²⋊Q₈	—	2	8	no	—
79	C₇×C₂²⋊Q₈	—	2	4	no	—
80	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
81	C₄×C₇×Q₈	—	2	8	no	—
82	C₄×C₇×Q₈	—	2	4	no	—
83	C₄×C₇×Q₈	C₂³	1	8	yes	complemented
84	C₇×#117	—	2	8	no	—
85	C₂×C₇×C₄⋊C₄	—	2	1	no	—
86	C₂×C₇×C₄⋊C₄	—	2	1	no	—
87	C₂×C₇×C₄⋊C₄	—	2	1	no	—
88	C₂×C₇×C₄⋊C₄	—	2	2	no	—
89	C₂×C₇×C₄⋊C₄	—	2	2	no	—
90	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
91	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
93	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
94	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
96	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
97	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
98	C₂×C₄²×C₇	—	2	2	no	—
99	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₇	D₄	1	4	yes	—
102	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₇	C₂³	1	4	yes	—
105	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
106	C₇×#103	C₂³	1	16	yes	complemented
107	C₇×#102	C₂³	1	16	yes	—
108	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
109	C₇×#97	C₂³	1	8	yes	complemented
110	C₂×#533	C₂×C₇	1	1	yes	characteristic, complemented
111	C₂²×#124	C₂×C₇	1	1	yes	characteristic
112	#3215	C₂×C₇	1	16	yes	direct product factor
113	C₂×#497	C₂×C₇	1	2	yes	complemented
114	C₂×#489	C₂×C₇	1	2	yes	complemented
115	C₂×#464	C₂×C₇	1	4	yes	complemented
116	C₂×#451	C₂×C₇	1	2	yes	complemented
117	C₂×#450	C₂×C₇	1	2	yes	—
118	C₂×#432	C₂×C₇	1	1	yes	characteristic, complemented
119	C₂⁴×C₇	—	2	4	no	—
120	C₂⁴×C₇	—	2	2	no	—
121	C₂⁴×C₇	—	4	4	no	—
122	C₂⁴×C₇	—	4	4	no	—
123	C₂⁴×C₇	—	4	4	no	—
124	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₇×Q₈	—	2	2	no	—
126	C₂×C₇×Q₈	—	2	4	no	—
127	C₂×C₇×Q₈	—	2	4	no	—
128	C₂×C₇×Q₈	—	2	2	no	—
129	C₂×C₇×Q₈	—	2	8	no	—
130	C₂×C₇×Q₈	—	2	8	no	—
131	C₂×C₇×Q₈	—	4	8	no	—
132	C₂×C₇×Q₈	—	4	4	no	—
133	C₂×C₇×Q₈	—	4	4	no	—
134	C₂×C₇×Q₈	—	4	8	no	—
135	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
136	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
137	C₂×C₇×D₄	—	2	4	no	—
138	C₂×C₇×D₄	—	2	2	no	—
139	C₂×C₇×D₄	—	2	8	no	—
140	C₂×C₇×D₄	—	4	8	no	—
141	C₂×C₇×D₄	—	4	4	no	—
142	C₂×C₇×D₄	—	4	8	no	—
143	C₂×C₇×D₄	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₇	—	2	4	no	—
145	C₂²×C₄×C₇	—	2	4	no	—
146	C₂²×C₄×C₇	—	2	4	no	—
147	C₂²×C₄×C₇	—	2	2	no	—
148	C₂²×C₄×C₇	—	2	2	no	—
149	C₂²×C₄×C₇	—	2	2	no	—
150	C₂²×C₄×C₇	—	4	2	no	—
151	C₂²×C₄×C₇	—	4	4	no	—
152	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
162	C₇×Q₁₆	—	4	16	no	—
163	C₇×SD₁₆	—	4	16	no	—
164	C₂×C₇×C₈	—	2	8	no	—
165	C₂×C₇×C₈	C₂⁴	1	8	yes	—
166	C₇×C₄⋊C₄	—	2	4	no	—
167	C₇×C₄⋊C₄	—	2	4	no	—
168	C₇×C₄⋊C₄	—	2	4	no	—
169	C₇×C₄⋊C₄	—	2	8	no	—
170	C₇×C₄⋊C₄	—	2	8	no	—
171	C₇×C₄⋊C₄	—	2	2	no	—
172	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
173	C₇×C₂²⋊C₄	—	2	8	no	—
174	C₇×C₂²⋊C₄	—	2	8	no	—
175	C₇×C₂²⋊C₄	—	2	8	no	—
176	C₇×C₂²⋊C₄	—	2	8	no	—
177	C₇×C₂²⋊C₄	—	2	8	no	—
178	C₄²×C₇	—	2	8	no	—
179	C₄²×C₇	C₂×D₄	1	4	yes	—
180	C₄²×C₇	C₂×D₄	1	2	yes	—
181	C₄²×C₇	C₂×D₄	1	4	yes	—
182	C₄²×C₇	C₂×D₄	1	2	yes	—
183	C₄²×C₇	C₂⁴	1	2	yes	—
184	C₄²×C₇	C₂⁴	1	2	yes	—
185	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
186	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
187	C₂²×Q₁₆	C₂²×C₇	1	4	yes	complemented
188	C₂²×SD₁₆	C₂²×C₇	1	4	yes	—
189	#533	—	2	4	no	—
190	#533	C₂²×C₇	1	8	yes	complemented
191	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
192	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
193	C₂×#125	—	2	1	no	—
194	C₂×#124	—	2	1	no	—
195	C₂×#124	—	2	1	no	—
196	C₂×#124	—	2	4	no	—
197	C₂×#124	C₂²×C₇	1	2	yes	—
198	C₂×#124	C₂²×C₇	1	8	yes	—
199	C₂×#124	C₂²×C₇	1	2	yes	—
200	C₂×C₂²⋊Q₈	—	2	1	no	—
201	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
202	C₂×C₄×Q₈	—	2	1	no	—
203	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	complemented
204	C₂×#117	—	2	1	no	—
205	C₂²×C₂²⋊C₄	—	2	1	no	—
206	C₂²×C₂²⋊C₄	—	2	1	no	—
207	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
208	#497	C₂²×C₇	1	16	yes	complemented
209	#489	C₂²×C₇	1	16	yes	complemented
210	#464	C₂²×C₇	1	32	yes	complemented
211	#451	C₂²×C₇	1	16	yes	complemented
212	#450	C₂²×C₇	1	16	yes	—
213	#432	C₂²×C₇	1	8	yes	complemented
214	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
215	C₂×#103	C₂²×C₇	1	4	yes	complemented
216	C₂×#102	C₂²×C₇	1	4	yes	—
217	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
218	C₂×#97	C₂²×C₇	1	2	yes	complemented
219	C₂³×C₇	—	2	2	no	—
220	C₂³×C₇	—	2	2	no	—
221	C₂³×C₇	—	2	4	no	—
222	C₂³×C₇	—	4	8	no	—
223	C₂³×C₇	—	4	8	no	—
224	C₂³×C₇	—	4	4	no	—
225	C₂³×C₇	—	4	8	no	—
226	C₂³×C₇	—	4	4	no	—
227	C₂³×C₇	—	4	4	no	—
228	C₂³×C₇	—	8	16	no	—
229	C₂³×C₇	#137	1	4	yes	—
230	C₂³×C₇	C₂²×D₄	1	2	yes	—
231	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
232	C₇×Q₈	—	2	8	no	—
233	C₇×Q₈	—	2	8	no	—
234	C₇×Q₈	—	4	8	no	—
235	C₇×Q₈	—	4	8	no	—
236	C₇×Q₈	—	4	4	no	—
237	C₇×Q₈	—	4	4	no	—
238	C₇×D₄	—	2	8	no	—
239	C₇×D₄	—	4	8	no	—
240	C₇×D₄	—	4	4	no	—
241	C₂×C₄×C₇	—	2	2	no	—
242	C₂×C₄×C₇	—	2	8	no	—
243	C₂×C₄×C₇	—	2	2	no	—
244	C₂×C₄×C₇	—	2	2	no	—
245	C₂×C₄×C₇	—	2	2	no	—
246	C₂×C₄×C₇	—	2	2	no	—
247	C₂×C₄×C₇	—	2	4	no	—
248	C₂×C₄×C₇	—	2	4	no	—
249	C₂×C₄×C₇	—	2	4	no	—
250	C₂×C₄×C₇	—	2	4	no	—
251	C₂×C₄×C₇	—	2	4	no	—
252	C₂×C₄×C₇	—	4	4	no	—
253	C₂×C₄×C₇	—	4	8	no	—
254	C₂×C₄×C₇	—	4	4	no	—
255	C₂×C₄×C₇	—	4	4	no	—
256	C₂×C₄×C₇	—	4	4	no	—
257	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
264	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
265	C₇×C₈	—	2	8	no	—
266	C₂⁵	—	2	1	no	—
267	C₂²×Q₈	—	2	1	no	—
268	C₂²×Q₈	—	2	1	no	—
269	C₂²×Q₈	—	2	2	no	—
270	C₂²×Q₈	—	4	4	no	—
271	C₂²×Q₈	C₂³×C₇	1	2	yes	—
272	C₂²×Q₈	C₂³×C₇	1	2	yes	—
273	C₂²×Q₈	C₂³×C₇	1	4	yes	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	4	4	no	—
277	C₂²×D₄	C₂³×C₇	1	2	yes	—
278	C₂²×D₄	C₂³×C₇	1	4	yes	—
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
281	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
282	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
283	C₂×Q₁₆	—	4	8	no	—
284	C₂×Q₁₆	C₂³×C₇	1	16	yes	complemented
285	C₂×SD₁₆	—	4	8	no	—
286	C₂×SD₁₆	C₂³×C₇	1	16	yes	—
287	C₂²×C₈	C₂³×C₇	1	4	yes	—
288	C₄⋊Q₈	—	2	2	no	—
289	C₄⋊Q₈	—	2	2	no	—
290	C₄⋊Q₈	C₂³×C₇	1	4	yes	complemented
291	C₄⋊Q₈	C₂³×C₇	1	4	yes	complemented
292	#125	—	2	8	no	—
293	#124	—	2	8	no	—
294	#124	—	2	4	no	—
295	#124	—	2	8	no	—
296	#124	—	2	4	no	—
297	#124	C₂³×C₇	1	8	yes	—
298	#124	C₂³×C₇	1	8	yes	—
299	C₂²⋊Q₈	—	2	8	no	—
300	C₂²⋊Q₈	—	2	4	no	—
301	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
302	C₄×Q₈	—	2	8	no	—
303	C₄×Q₈	—	2	4	no	—
304	C₄×Q₈	C₂³×C₇	1	8	yes	complemented
305	#117	—	2	8	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	8	no	—
318	C₂×C₂²⋊C₄	—	2	8	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
321	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
322	C₂×C₄²	C₇×D₄	1	4	yes	—
323	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₇	1	4	yes	—
326	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
327	#103	C₂³×C₇	1	16	yes	complemented
328	#102	C₂³×C₇	1	16	yes	—
329	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
330	#97	C₂³×C₇	1	8	yes	complemented
331	C₂²×C₇	—	2	2	no	—
332	C₂²×C₇	—	2	8	no	—
333	C₂²×C₇	—	2	2	no	—
334	C₂²×C₇	—	4	4	no	—
335	C₂²×C₇	—	4	4	no	—
336	C₂²×C₇	—	4	4	no	—
337	C₂²×C₇	—	8	16	no	—
338	C₂²×C₇	—	8	8	no	—
339	C₂²×C₇	#535	1	2	yes	—
340	C₂²×C₇	C₂×#137	1	4	yes	—
341	C₂²×C₇	C₂×#137	1	2	yes	—
342	C₂²×C₇	#575	1	2	yes	—
343	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
344	C₄×C₇	—	2	2	no	—
345	C₄×C₇	—	2	2	no	—
346	C₄×C₇	—	2	2	no	—
347	C₄×C₇	—	2	2	no	—
348	C₄×C₇	—	2	2	no	—
349	C₄×C₇	—	4	4	no	—
350	C₄×C₇	—	4	4	no	—
351	C₄×C₇	—	4	4	no	—
352	C₄×C₇	#535	1	2	yes	—
353	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
354	C₂⁴	—	2	4	no	—
355	C₂⁴	—	2	2	no	—
356	C₂⁴	—	4	4	no	—
357	C₂⁴	—	4	4	no	—
358	C₂⁴	—	4	4	no	—
359	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	8	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	—	4	8	no	—
367	C₂×Q₈	—	4	4	no	—
368	C₂×Q₈	—	4	8	no	—
369	C₂×Q₈	—	4	4	no	—
370	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
371	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	—	4	8	no	—
378	C₂×D₄	C₂⁴×C₇	1	4	yes	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	4	no	—
387	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
388	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
389	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
390	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
391	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
392	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
393	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
394	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
397	Q₁₆	—	4	16	no	—
398	SD₁₆	—	4	16	no	—
399	C₂×C₈	—	2	8	no	—
400	C₂×C₈	C₂⁴×C₇	1	8	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₄²	—	2	8	no	—
414	C₄²	C₂×C₇×D₄	1	4	yes	—
415	C₄²	C₂×C₇×D₄	1	2	yes	—
416	C₄²	C₂×C₇×D₄	1	4	yes	—
417	C₄²	C₂×C₇×D₄	1	2	yes	—
418	C₄²	C₂⁴×C₇	1	2	yes	—
419	C₄²	C₂⁴×C₇	1	2	yes	—
420	C₂×C₇	—	2	4	no	—
421	C₂×C₇	—	8	4	no	—
422	C₂×C₇	#3215	1	4	yes	direct product factor
423	C₂×C₇	C₂×#535	1	1	yes	characteristic
424	C₂×C₇	C₂²×#137	1	1	yes	characteristic
425	C₂×C₇	C₂×#575	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	4	8	no	—
430	C₂³	—	4	8	no	—
431	C₂³	—	4	4	no	—
432	C₂³	—	4	8	no	—
433	C₂³	—	4	4	no	—
434	C₂³	—	4	4	no	—
435	C₂³	—	8	16	no	—
436	C₂³	C₇×#137	1	4	yes	—
437	C₂³	C₂²×C₇×D₄	1	2	yes	—
438	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
439	Q₈	—	2	8	no	—
440	Q₈	—	2	8	no	—
441	Q₈	—	4	8	no	—
442	Q₈	—	4	8	no	—
443	Q₈	—	4	4	no	—
444	Q₈	—	4	4	no	—
445	D₄	—	2	8	no	—
446	D₄	—	4	8	no	—
447	D₄	—	4	4	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
465	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
466	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
467	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
468	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
469	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
470	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
471	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
472	C₈	—	2	8	no	—
473	C₇	C₂×#3215	1	1	yes	characteristic, direct product factor, Sylow
474	C₂²	—	2	2	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	8	16	no	—
481	C₂²	—	8	8	no	—
482	C₂²	C₇×#535	1	2	yes	—
483	C₂²	C₂×C₇×#137	1	4	yes	—
484	C₂²	C₂×C₇×#137	1	2	yes	—
485	C₂²	C₇×#575	1	2	yes	—
486	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	4	4	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	4	no	—
495	C₄	C₇×#535	1	2	yes	—
496	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
497	C₂	—	2	4	no	—
498	C₂	—	8	4	no	—
499	C₂	C₇×#3215	1	4	yes	direct product factor
500	C₂	C₂×C₇×#535	1	1	yes	characteristic
501	C₂	C₂²×C₇×#137	1	1	yes	characteristic
502	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	C₂³×C₇×D₄	C₂²
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×#533	C₂	semidirect
3. C₇×#3215	C₂	direct (N × H)
4. C₂×C₇×#497	C₂	semidirect
5. C₂×C₇×#489	C₂	semidirect
6. C₂×C₇×#464	C₂	semidirect
7. C₂×C₇×#451	C₂	semidirect
9. C₂×C₇×#432	C₂	semidirect
12. C₂²×C₇×Q₁₆	C₂²	semidirect
15. C₇×#533	C₂²	semidirect
16. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
28. C₂×C₄×C₇×Q₈	C₂²	semidirect
33. C₇×#497	C₂²	semidirect
34. C₇×#489	C₂²	semidirect
35. C₇×#464	C₂²	semidirect
36. C₇×#451	C₂²	semidirect
38. C₇×#432	C₂²	semidirect
39. C₂×C₇×C₄⋊C₈	C₂²	semidirect
40. C₂×C₇×#103	C₂²	semidirect
43. C₂×C₇×#97	C₂²	semidirect
44. C₂×#3215	C₇	direct (N × H)
63. C₂×C₇×Q₁₆	C₂³	semidirect
69. C₇×C₄⋊Q₈	C₂³	semidirect
70. C₇×C₄⋊Q₈	C₂³	semidirect
83. C₄×C₇×Q₈	C₂³	semidirect
105. C₇×C₄⋊C₈	C₂³	semidirect
106. C₇×#103	C₂³	semidirect
109. C₇×#97	C₂³	semidirect
110. C₂×#533	C₂×C₇	semidirect
112. #3215	C₂×C₇	direct (N × H)
113. C₂×#497	C₂×C₇	semidirect
114. C₂×#489	C₂×C₇	semidirect
115. C₂×#464	C₂×C₇	semidirect
116. C₂×#451	C₂×C₇	semidirect
118. C₂×#432	C₂×C₇	semidirect
187. C₂²×Q₁₆	C₂²×C₇	semidirect
190. #533	C₂²×C₇	semidirect
191. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
192. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
203. C₂×C₄×Q₈	C₂²×C₇	semidirect
208. #497	C₂²×C₇	semidirect
209. #489	C₂²×C₇	semidirect
210. #464	C₂²×C₇	semidirect
211. #451	C₂²×C₇	semidirect
213. #432	C₂²×C₇	semidirect
214. C₂×C₄⋊C₈	C₂²×C₇	semidirect
215. C₂×#103	C₂²×C₇	semidirect
218. C₂×#97	C₂²×C₇	semidirect
284. C₂×Q₁₆	C₂³×C₇	semidirect
290. C₄⋊Q₈	C₂³×C₇	semidirect
291. C₄⋊Q₈	C₂³×C₇	semidirect
304. C₄×Q₈	C₂³×C₇	semidirect
326. C₄⋊C₈	C₂³×C₇	semidirect
327. #103	C₂³×C₇	semidirect
330. #97	C₂³×C₇	semidirect
422. C₂×C₇	#3215	direct (N × H)
473. C₇	C₂×#3215	direct (N × H)
499. C₂	C₇×#3215	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×#533	2
C₂²×C₇×#124	2
C₇×#3215	2
C₂×C₇×#497	2
C₂×C₇×#489	2
C₂×C₇×#464	2
C₂×C₇×#451	2
C₂×C₇×#450	2
C₂×C₇×#432	2
C₂×#3215	7

Subgroups — C2×C7×#3215 #13238000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3215 #13238000