Subgroups — C₂×C₇×#3222 #13238008

Order 1792.

2214 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#535	C₂	1	1	yes	maximal, characteristic
3	C₇×#3222	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#460	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×#452	C₂	1	2	yes	maximal, complemented
11	C₂×C₇×#450	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×#41	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	—
18	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
19	C₇×#541	—	2	4	no	—
20	C₇×#541	C₂²	1	8	yes	complemented
21	C₇×#535	—	2	4	no	—
22	C₇×#535	C₂²	1	8	yes	—
23	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₇×#127	C₂²	1	1	yes	characteristic
25	C₂×C₇×#127	C₂²	1	1	yes	characteristic
26	C₂×C₇×#126	—	2	2	no	—
27	C₂×C₇×#125	—	2	2	no	—
28	C₂×C₇×#125	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₇×#124	C₂²	1	1	yes	characteristic
30	C₂×C₇×#124	C₂²	1	1	yes	characteristic
31	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₂×C₇×#121	—	2	2	no	—
33	C₂×C₇×#120	—	2	2	no	—
34	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	complemented
35	C₂×C₇×#117	C₂²	1	1	yes	characteristic
36	C₇×#497	C₂²	1	8	yes	complemented
37	C₇×#492	C₂²	1	8	yes	complemented
38	C₇×#490	C₂²	1	8	yes	complemented
39	C₇×#488	C₂²	1	8	yes	complemented
40	C₇×#464	C₂²	1	16	yes	complemented
41	C₇×#460	C₂²	1	16	yes	complemented
42	C₇×#452	C₂²	1	16	yes	complemented
43	C₇×#450	C₂²	1	16	yes	complemented
44	C₇×#433	C₂²	1	8	yes	complemented
45	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
46	C₂×C₇×#103	C₂²	1	2	yes	complemented
47	C₂×C₇×#102	C₂²	1	2	yes	—
48	C₂×C₇×#102	C₂²	1	2	yes	—
49	C₂×C₇×#102	C₂²	1	2	yes	complemented
50	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
51	C₂×C₇×#97	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
53	C₂×#3222	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁵×C₇	—	2	1	no	—
55	C₂×C₇×#41	—	2	2	no	—
56	C₂×C₇×#41	—	4	4	no	—
57	C₂×C₇×#41	C₂³	1	4	yes	—
58	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
59	C₂²×C₇×Q₈	C₂³	1	2	yes	—
60	C₂²×C₇×D₄	—	2	1	no	—
61	C₂²×C₇×D₄	—	2	1	no	—
62	C₂²×C₇×D₄	—	2	2	no	—
63	C₂²×C₇×D₄	—	2	2	no	—
64	C₂²×C₇×D₄	—	2	2	no	—
65	C₂²×C₇×D₄	—	4	4	no	—
66	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
68	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₇×D₄	C₂³	1	4	yes	—
70	C₂³×C₄×C₇	—	2	1	no	—
71	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
72	C₂×C₇×Q₁₆	—	4	4	no	—
73	C₂×C₇×Q₁₆	C₂³	1	8	yes	complemented
74	C₂×C₇×SD₁₆	—	4	4	no	—
75	C₂×C₇×SD₁₆	—	4	4	no	—
76	C₂×C₇×SD₁₆	—	4	4	no	—
77	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
78	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
79	C₂×C₇×SD₁₆	C₂³	1	8	yes	complemented
80	C₂²×C₇×C₈	C₂³	1	2	yes	—
81	C₂²×C₇×C₈	C₂³	1	2	yes	—
82	C₇×C₄⋊Q₈	—	2	2	no	—
83	C₇×C₄⋊Q₈	C₂³	1	4	yes	complemented
84	C₇×#127	—	2	2	no	—
85	C₇×#127	—	2	2	no	—
86	C₇×#127	C₂³	1	4	yes	—
87	C₇×#127	C₂³	1	4	yes	—
88	C₇×#126	—	2	16	no	—
89	C₇×#125	—	2	16	no	—
90	C₇×#125	—	2	2	no	—
91	C₇×#125	C₂³	1	4	yes	complemented
92	C₇×#124	—	2	2	no	—
93	C₇×#124	—	2	2	no	—
94	C₇×#124	C₂³	1	4	yes	—
95	C₇×#124	C₂³	1	4	yes	—
96	C₇×C₂²⋊Q₈	—	2	4	no	—
97	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
98	C₇×#121	—	2	16	no	—
99	C₇×#120	—	2	16	no	—
100	C₄×C₇×Q₈	—	2	4	no	—
101	C₄×C₇×Q₈	C₂³	1	8	yes	complemented
102	C₇×#117	D₄	1	4	yes	—
103	C₇×#117	C₂³	1	4	yes	—
104	C₂×C₇×C₄⋊C₄	—	2	2	no	—
105	C₂×C₇×C₄⋊C₄	—	2	2	no	—
106	C₂×C₇×C₄⋊C₄	—	2	2	no	—
107	C₂×C₇×C₄⋊C₄	—	2	2	no	—
108	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
109	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
110	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
111	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
112	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
114	C₂×C₄²×C₇	—	2	2	no	—
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
117	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
118	C₇×#103	C₂³	1	8	yes	complemented
119	C₇×#102	C₂³	1	8	yes	—
120	C₇×#102	C₂³	1	8	yes	—
121	C₇×#102	C₂³	1	8	yes	complemented
122	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
123	C₇×#97	C₂³	1	4	yes	complemented
124	C₄×C₇×C₈	C₂³	1	4	yes	complemented
125	C₂×#541	C₂×C₇	1	1	yes	characteristic, complemented
126	C₂×#535	C₂×C₇	1	1	yes	characteristic
127	#3222	C₂×C₇	1	16	yes	direct product factor
128	C₂×#497	C₂×C₇	1	1	yes	characteristic, complemented
129	C₂×#492	C₂×C₇	1	1	yes	characteristic, complemented
130	C₂×#490	C₂×C₇	1	1	yes	characteristic, complemented
131	C₂×#488	C₂×C₇	1	1	yes	characteristic, complemented
132	C₂×#464	C₂×C₇	1	2	yes	complemented
133	C₂×#460	C₂×C₇	1	2	yes	complemented
134	C₂×#452	C₂×C₇	1	2	yes	complemented
135	C₂×#450	C₂×C₇	1	2	yes	complemented
136	C₂×#433	C₂×C₇	1	1	yes	characteristic, complemented
137	C₂⁴×C₇	—	2	1	no	—
138	C₂⁴×C₇	—	2	1	no	—
139	C₂⁴×C₇	—	2	1	no	—
140	C₂⁴×C₇	—	2	4	no	—
141	C₂⁴×C₇	—	4	4	no	—
142	C₂⁴×C₇	—	8	4	no	—
143	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
144	C₇×#41	—	4	8	no	—
145	C₇×#41	—	4	8	no	—
146	C₂×C₇×Q₈	—	2	2	no	—
147	C₂×C₇×Q₈	—	2	1	no	—
148	C₂×C₇×Q₈	—	2	4	no	—
149	C₂×C₇×Q₈	—	2	4	no	—
150	C₂×C₇×Q₈	—	2	4	no	—
151	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
152	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
153	C₂×C₇×D₄	—	2	4	no	—
154	C₂×C₇×D₄	—	2	4	no	—
155	C₂×C₇×D₄	—	2	1	no	—
156	C₂×C₇×D₄	—	2	1	no	—
157	C₂×C₇×D₄	—	2	1	no	—
158	C₂×C₇×D₄	—	2	8	no	—
159	C₂×C₇×D₄	—	2	8	no	—
160	C₂×C₇×D₄	—	2	4	no	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	—	2	4	no	—
163	C₂×C₇×D₄	—	4	8	no	—
164	C₂×C₇×D₄	—	4	4	no	—
165	C₂×C₇×D₄	—	4	8	no	—
166	C₂×C₇×D₄	—	4	4	no	—
167	C₂×C₇×D₄	—	8	4	no	—
168	C₂×C₇×D₄	—	8	4	no	—
169	C₂×C₇×D₄	C₂⁴	1	2	yes	—
170	C₂×C₇×D₄	C₂⁴	1	2	yes	—
171	C₂×C₇×D₄	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₇	—	2	4	no	—
173	C₂²×C₄×C₇	—	2	2	no	—
174	C₂²×C₄×C₇	—	2	2	no	—
175	C₂²×C₄×C₇	—	2	1	no	—
176	C₂²×C₄×C₇	—	2	4	no	—
177	C₂²×C₄×C₇	—	4	4	no	—
178	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
184	C₇×Q₁₆	—	4	8	no	—
185	C₇×SD₁₆	—	4	8	no	—
186	C₇×SD₁₆	—	4	8	no	—
187	C₇×SD₁₆	—	4	8	no	—
188	C₂×C₇×C₈	—	2	4	no	—
189	C₂×C₇×C₈	—	2	4	no	—
190	C₂×C₇×C₈	C₂⁴	1	4	yes	—
191	C₂×C₇×C₈	C₂⁴	1	4	yes	—
192	C₇×C₄⋊C₄	—	2	8	no	—
193	C₇×C₄⋊C₄	—	2	8	no	—
194	C₇×C₄⋊C₄	—	2	8	no	—
195	C₇×C₄⋊C₄	—	2	8	no	—
196	C₇×C₄⋊C₄	—	2	2	no	—
197	C₇×C₄⋊C₄	C₂×D₄	1	8	yes	—
198	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
199	C₇×C₂²⋊C₄	—	2	8	no	—
200	C₇×C₂²⋊C₄	—	2	8	no	—
201	C₇×C₂²⋊C₄	—	2	8	no	—
202	C₇×C₂²⋊C₄	C₂×D₄	1	8	yes	—
203	C₄²×C₇	—	2	8	no	—
204	C₄²×C₇	C₂×D₄	1	2	yes	—
205	C₄²×C₇	C₂×D₄	1	2	yes	—
206	C₄²×C₇	C₂⁴	1	2	yes	—
207	C₄²×C₇	C₂⁴	1	2	yes	—
208	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
209	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
210	C₂²×Q₁₆	C₂²×C₇	1	2	yes	complemented
211	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
212	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
213	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
214	#541	—	2	4	no	—
215	#541	C₂²×C₇	1	8	yes	complemented
216	#535	—	2	4	no	—
217	#535	C₂²×C₇	1	8	yes	—
218	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
219	C₂×#127	C₂²×C₇	1	1	yes	characteristic
220	C₂×#127	C₂²×C₇	1	1	yes	characteristic
221	C₂×#126	—	2	2	no	—
222	C₂×#125	—	2	2	no	—
223	C₂×#125	C₂²×C₇	1	1	yes	characteristic, complemented
224	C₂×#124	C₂²×C₇	1	1	yes	characteristic
225	C₂×#124	C₂²×C₇	1	1	yes	characteristic
226	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
227	C₂×#121	—	2	2	no	—
228	C₂×#120	—	2	2	no	—
229	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	complemented
230	C₂×#117	C₂²×C₇	1	1	yes	characteristic
231	#497	C₂²×C₇	1	8	yes	complemented
232	#492	C₂²×C₇	1	8	yes	complemented
233	#490	C₂²×C₇	1	8	yes	complemented
234	#488	C₂²×C₇	1	8	yes	complemented
235	#464	C₂²×C₇	1	16	yes	complemented
236	#460	C₂²×C₇	1	16	yes	complemented
237	#452	C₂²×C₇	1	16	yes	complemented
238	#450	C₂²×C₇	1	16	yes	complemented
239	#433	C₂²×C₇	1	8	yes	complemented
240	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
241	C₂×#103	C₂²×C₇	1	2	yes	complemented
242	C₂×#102	C₂²×C₇	1	2	yes	—
243	C₂×#102	C₂²×C₇	1	2	yes	—
244	C₂×#102	C₂²×C₇	1	2	yes	complemented
245	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
246	C₂×#97	C₂²×C₇	1	1	yes	characteristic, complemented
247	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
248	C₂³×C₇	—	2	2	no	—
249	C₂³×C₇	—	2	4	no	—
250	C₂³×C₇	—	2	2	no	—
251	C₂³×C₇	—	2	2	no	—
252	C₂³×C₇	—	2	2	no	—
253	C₂³×C₇	—	4	8	no	—
254	C₂³×C₇	—	4	2	no	—
255	C₂³×C₇	—	4	2	no	—
256	C₂³×C₇	—	4	2	no	—
257	C₂³×C₇	—	4	4	no	—
258	C₂³×C₇	—	4	4	no	—
259	C₂³×C₇	—	4	4	no	—
260	C₂³×C₇	—	8	8	no	—
261	C₂³×C₇	—	8	16	no	—
262	C₂³×C₇	C₂²×D₄	1	2	yes	—
263	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
264	C₇×Q₈	—	2	4	no	—
265	C₇×Q₈	—	2	4	no	—
266	C₇×Q₈	—	2	4	no	—
267	C₇×Q₈	—	4	4	no	—
268	C₇×Q₈	—	4	2	no	—
269	C₇×D₄	—	2	4	no	—
270	C₇×D₄	—	2	4	no	—
271	C₇×D₄	—	2	4	no	—
272	C₇×D₄	—	4	8	no	—
273	C₇×D₄	—	4	8	no	—
274	C₇×D₄	—	4	2	no	—
275	C₇×D₄	—	4	2	no	—
276	C₇×D₄	—	4	2	no	—
277	C₇×D₄	—	8	8	no	—
278	C₇×D₄	—	8	8	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	4	no	—
283	C₂×C₄×C₇	—	2	4	no	—
284	C₂×C₄×C₇	—	2	4	no	—
285	C₂×C₄×C₇	—	2	4	no	—
286	C₂×C₄×C₇	—	2	2	no	—
287	C₂×C₄×C₇	—	4	8	no	—
288	C₂×C₄×C₇	—	4	2	no	—
289	C₂×C₄×C₇	—	4	4	no	—
290	C₂×C₄×C₇	—	4	4	no	—
291	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
292	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
296	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
297	C₇×C₈	—	2	4	no	—
298	C₇×C₈	—	2	4	no	—
299	C₂⁵	—	2	1	no	—
300	C₂×#41	—	2	2	no	—
301	C₂×#41	—	4	4	no	—
302	C₂×#41	C₂³×C₇	1	4	yes	—
303	C₂²×Q₈	C₂³×C₇	1	2	yes	—
304	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	—	2	2	no	—
309	C₂²×D₄	—	2	2	no	—
310	C₂²×D₄	—	4	4	no	—
311	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
313	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₇	1	4	yes	—
315	C₂³×C₄	—	2	1	no	—
316	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
317	C₂×Q₁₆	—	4	4	no	—
318	C₂×Q₁₆	C₂³×C₇	1	8	yes	complemented
319	C₂×SD₁₆	—	4	4	no	—
320	C₂×SD₁₆	—	4	4	no	—
321	C₂×SD₁₆	—	4	4	no	—
322	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
323	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
324	C₂×SD₁₆	C₂³×C₇	1	8	yes	complemented
325	C₂²×C₈	C₂³×C₇	1	2	yes	—
326	C₂²×C₈	C₂³×C₇	1	2	yes	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	C₂³×C₇	1	4	yes	complemented
329	#127	—	2	2	no	—
330	#127	—	2	2	no	—
331	#127	C₂³×C₇	1	4	yes	—
332	#127	C₂³×C₇	1	4	yes	—
333	#126	—	2	16	no	—
334	#125	—	2	16	no	—
335	#125	—	2	2	no	—
336	#125	C₂³×C₇	1	4	yes	complemented
337	#124	—	2	2	no	—
338	#124	—	2	2	no	—
339	#124	C₂³×C₇	1	4	yes	—
340	#124	C₂³×C₇	1	4	yes	—
341	C₂²⋊Q₈	—	2	4	no	—
342	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
343	#121	—	2	16	no	—
344	#120	—	2	16	no	—
345	C₄×Q₈	—	2	4	no	—
346	C₄×Q₈	C₂³×C₇	1	8	yes	complemented
347	#117	C₇×D₄	1	4	yes	—
348	#117	C₂³×C₇	1	4	yes	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
359	C₂×C₄²	—	2	2	no	—
360	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
362	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
363	#103	C₂³×C₇	1	8	yes	complemented
364	#102	C₂³×C₇	1	8	yes	—
365	#102	C₂³×C₇	1	8	yes	—
366	#102	C₂³×C₇	1	8	yes	complemented
367	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
368	#97	C₂³×C₇	1	4	yes	complemented
369	C₄×C₈	C₂³×C₇	1	4	yes	complemented
370	C₂²×C₇	—	2	2	no	—
371	C₂²×C₇	—	2	4	no	—
372	C₂²×C₇	—	4	4	no	—
373	C₂²×C₇	—	4	2	no	—
374	C₂²×C₇	—	4	2	no	—
375	C₂²×C₇	—	4	2	no	—
376	C₂²×C₇	—	4	4	no	—
377	C₂²×C₇	—	4	4	no	—
378	C₂²×C₇	—	4	4	no	—
379	C₂²×C₇	—	8	16	no	—
380	C₂²×C₇	—	8	4	no	—
381	C₂²×C₇	—	8	4	no	—
382	C₂²×C₇	—	8	4	no	—
383	C₂²×C₇	#535	1	2	yes	—
384	C₂²×C₇	#577	1	4	yes	—
385	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
386	C₄×C₇	—	2	2	no	—
387	C₄×C₇	—	2	4	no	—
388	C₄×C₇	—	2	4	no	—
389	C₄×C₇	—	4	2	no	—
390	C₄×C₇	—	4	4	no	—
391	C₄×C₇	—	4	4	no	—
392	C₄×C₇	#535	1	2	yes	—
393	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
394	C₂⁴	—	2	1	no	—
395	C₂⁴	—	2	1	no	—
396	C₂⁴	—	2	1	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	—	4	4	no	—
399	C₂⁴	—	8	4	no	—
400	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
409	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	4	8	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	—	4	8	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	—	8	4	no	—
425	C₂×D₄	—	8	4	no	—
426	C₂×D₄	C₂⁴×C₇	1	2	yes	—
427	C₂×D₄	C₂⁴×C₇	1	2	yes	—
428	C₂×D₄	C₂⁴×C₇	1	2	yes	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
436	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
437	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
438	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
441	Q₁₆	—	4	8	no	—
442	SD₁₆	—	4	8	no	—
443	SD₁₆	—	4	8	no	—
444	SD₁₆	—	4	8	no	—
445	C₂×C₈	—	2	4	no	—
446	C₂×C₈	—	2	4	no	—
447	C₂×C₈	C₂⁴×C₇	1	4	yes	—
448	C₂×C₈	C₂⁴×C₇	1	4	yes	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	C₂×C₇×D₄	1	8	yes	—
455	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	C₂×C₇×D₄	1	8	yes	—
460	C₄²	—	2	8	no	—
461	C₄²	C₂×C₇×D₄	1	2	yes	—
462	C₄²	C₂×C₇×D₄	1	2	yes	—
463	C₄²	C₂⁴×C₇	1	2	yes	—
464	C₄²	C₂⁴×C₇	1	2	yes	—
465	C₂×C₇	—	4	2	no	—
466	C₂×C₇	—	8	2	no	—
467	C₂×C₇	—	8	2	no	—
468	C₂×C₇	—	8	2	no	—
469	C₂×C₇	#3222	1	4	yes	direct product factor
470	C₂×C₇	C₂×#535	1	1	yes	characteristic
471	C₂×C₇	C₂×#577	1	2	yes	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	4	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	4	8	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	4	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	8	8	no	—
485	C₂³	—	8	16	no	—
486	C₂³	C₂²×C₇×D₄	1	2	yes	—
487	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
488	Q₈	—	2	4	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	2	4	no	—
491	Q₈	—	4	4	no	—
492	Q₈	—	4	2	no	—
493	D₄	—	2	4	no	—
494	D₄	—	2	4	no	—
495	D₄	—	2	4	no	—
496	D₄	—	4	8	no	—
497	D₄	—	4	8	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	2	no	—
501	D₄	—	8	8	no	—
502	D₄	—	8	8	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
516	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
517	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
518	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
519	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
520	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
521	C₈	—	2	4	no	—
522	C₈	—	2	4	no	—
523	C₇	C₂×#3222	1	1	yes	characteristic, direct product factor, Sylow
524	C₂²	—	2	4	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	8	16	no	—
534	C₂²	—	8	4	no	—
535	C₂²	—	8	4	no	—
536	C₂²	—	8	4	no	—
537	C₂²	C₇×#535	1	2	yes	—
538	C₂²	C₇×#577	1	4	yes	—
539	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
540	C₄	—	2	4	no	—
541	C₄	—	2	4	no	—
542	C₄	—	2	2	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	4	no	—
546	C₄	C₇×#535	1	2	yes	—
547	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
548	C₂	—	4	2	no	—
549	C₂	—	8	2	no	—
550	C₂	—	8	2	no	—
551	C₂	—	8	2	no	—
552	C₂	C₇×#3222	1	4	yes	direct product factor
553	C₂	C₂×C₇×#535	1	1	yes	characteristic
554	C₂	C₂×C₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#541	C₂	semidirect
3. C₇×#3222	C₂	direct (N × H)
4. C₂×C₇×#497	C₂	semidirect
5. C₂×C₇×#492	C₂	semidirect
6. C₂×C₇×#490	C₂	semidirect
7. C₂×C₇×#488	C₂	semidirect
8. C₂×C₇×#464	C₂	semidirect
9. C₂×C₇×#460	C₂	semidirect
10. C₂×C₇×#452	C₂	semidirect
11. C₂×C₇×#450	C₂	semidirect
12. C₂×C₇×#433	C₂	semidirect
15. C₂²×C₇×Q₁₆	C₂²	semidirect
18. C₂²×C₇×SD₁₆	C₂²	semidirect
20. C₇×#541	C₂²	semidirect
23. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
28. C₂×C₇×#125	C₂²	semidirect
34. C₂×C₄×C₇×Q₈	C₂²	semidirect
36. C₇×#497	C₂²	semidirect
37. C₇×#492	C₂²	semidirect
38. C₇×#490	C₂²	semidirect
39. C₇×#488	C₂²	semidirect
40. C₇×#464	C₂²	semidirect
41. C₇×#460	C₂²	semidirect
42. C₇×#452	C₂²	semidirect
43. C₇×#450	C₂²	semidirect
44. C₇×#433	C₂²	semidirect
45. C₂×C₇×C₄⋊C₈	C₂²	semidirect
46. C₂×C₇×#103	C₂²	semidirect
49. C₂×C₇×#102	C₂²	semidirect
51. C₂×C₇×#97	C₂²	semidirect
52. C₂×C₄×C₇×C₈	C₂²	semidirect
53. C₂×#3222	C₇	direct (N × H)
73. C₂×C₇×Q₁₆	C₂³	semidirect
79. C₂×C₇×SD₁₆	C₂³	semidirect
83. C₇×C₄⋊Q₈	C₂³	semidirect
91. C₇×#125	C₂³	semidirect
101. C₄×C₇×Q₈	C₂³	semidirect
117. C₇×C₄⋊C₈	C₂³	semidirect
118. C₇×#103	C₂³	semidirect
121. C₇×#102	C₂³	semidirect
123. C₇×#97	C₂³	semidirect
124. C₄×C₇×C₈	C₂³	semidirect
125. C₂×#541	C₂×C₇	semidirect
127. #3222	C₂×C₇	direct (N × H)
128. C₂×#497	C₂×C₇	semidirect
129. C₂×#492	C₂×C₇	semidirect
130. C₂×#490	C₂×C₇	semidirect
131. C₂×#488	C₂×C₇	semidirect
132. C₂×#464	C₂×C₇	semidirect
133. C₂×#460	C₂×C₇	semidirect
134. C₂×#452	C₂×C₇	semidirect
135. C₂×#450	C₂×C₇	semidirect
136. C₂×#433	C₂×C₇	semidirect
210. C₂²×Q₁₆	C₂²×C₇	semidirect
213. C₂²×SD₁₆	C₂²×C₇	semidirect
215. #541	C₂²×C₇	semidirect
218. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
223. C₂×#125	C₂²×C₇	semidirect
229. C₂×C₄×Q₈	C₂²×C₇	semidirect
231. #497	C₂²×C₇	semidirect
232. #492	C₂²×C₇	semidirect
233. #490	C₂²×C₇	semidirect
234. #488	C₂²×C₇	semidirect
235. #464	C₂²×C₇	semidirect
236. #460	C₂²×C₇	semidirect
237. #452	C₂²×C₇	semidirect
238. #450	C₂²×C₇	semidirect
239. #433	C₂²×C₇	semidirect
240. C₂×C₄⋊C₈	C₂²×C₇	semidirect
241. C₂×#103	C₂²×C₇	semidirect
244. C₂×#102	C₂²×C₇	semidirect
246. C₂×#97	C₂²×C₇	semidirect
247. C₂×C₄×C₈	C₂²×C₇	semidirect
318. C₂×Q₁₆	C₂³×C₇	semidirect
324. C₂×SD₁₆	C₂³×C₇	semidirect
328. C₄⋊Q₈	C₂³×C₇	semidirect
336. #125	C₂³×C₇	semidirect
346. C₄×Q₈	C₂³×C₇	semidirect
362. C₄⋊C₈	C₂³×C₇	semidirect
363. #103	C₂³×C₇	semidirect
366. #102	C₂³×C₇	semidirect
368. #97	C₂³×C₇	semidirect
369. C₄×C₈	C₂³×C₇	semidirect
469. C₂×C₇	#3222	direct (N × H)
523. C₇	C₂×#3222	direct (N × H)
552. C₂	C₇×#3222	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×#541	2
C₂×C₇×#535	2
C₇×#3222	2
C₂×C₇×#497	2
C₂×C₇×#492	2
C₂×C₇×#490	2
C₂×C₇×#488	2
C₂×C₇×#464	2
C₂×C₇×#460	2
C₂×C₇×#452	2
C₂×C₇×#450	2
C₂×C₇×#433	2
C₂×#3222	7

Subgroups — C2×C7×#3222 #13238008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3222 #13238008