Subgroups — C₇×#22131 #13199347

Order 1792.

636 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3076	C₂	1	1	yes	maximal, characteristic
2	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2922	C₂	1	1	yes	maximal, characteristic
4	C₇×#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1946	C₂	1	2	yes	maximal, complemented
7	C₇×#1938	C₂	1	2	yes	maximal
8	C₇×#1929	C₂	1	2	yes	maximal
9	C₇×#1922	C₂	1	2	yes	maximal, complemented
10	C₇×#1806	C₂	1	1	yes	maximal, characteristic
11	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#533	C₂²	1	1	yes	characteristic
13	C₄×C₇×#41	C₂²	1	1	yes	characteristic
14	C₇×#479	C₄	1	2	yes	complemented
15	C₇×#478	C₄	1	2	yes	complemented
16	C₇×#477	C₄	1	1	yes	characteristic, complemented
17	C₇×#476	C₄	1	1	yes	characteristic, complemented
18	C₇×#475	C₄	1	1	yes	characteristic, complemented
19	C₇×#474	C₄	1	1	yes	characteristic, complemented
20	C₇×#446	—	2	1	no	—
21	C₇×#446	—	2	1	no	—
22	C₇×C₈×Q₈	—	2	1	no	—
23	C₇×C₈×Q₈	—	2	1	no	—
24	C₇×#433	—	2	1	no	—
25	C₇×#433	—	2	1	no	—
26	C₇×#431	—	2	1	no	—
27	C₇×#431	—	2	1	no	—
28	C₂×C₇×#108	C₂²	1	2	yes	—
29	C₇×#427	C₂²	1	2	yes	—
30	C₇×#424	C₂²	1	1	yes	characteristic
31	C₇×#424	C₂²	1	1	yes	characteristic
32	C₇×#424	C₂²	1	2	yes	—
33	C₇×#423	C₂²	1	1	yes	characteristic
34	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₇×#104	C₂²	1	2	yes	—
36	C₇×#416	C₂²	1	2	yes	—
37	C₇×#408	C₂²	1	1	yes	characteristic
38	C₇×#408	C₂²	1	1	yes	characteristic
39	C₇×#405	C₂²	1	2	yes	—
40	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₇×#97	C₂²	1	2	yes	—
42	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
43	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	C₇×#341	C₂²	1	2	yes	—
45	C₇×#339	C₂²	1	2	yes	—
46	C₇×#339	C₂²	1	2	yes	—
47	C₇×#339	C₂²	1	2	yes	—
48	C₇×#336	C₂²	1	2	yes	—
49	#22131	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₇×#41	C₂³	1	1	yes	characteristic
51	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₇×M₁₆	C₂³	1	2	yes	—
54	C₂²×C₇×C₈	C₂³	1	2	yes	—
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
59	C₇×#125	C₂×C₄	1	2	yes	—
60	C₇×C₂²⋊Q₈	—	2	1	no	—
61	C₇×C₂²⋊Q₈	—	2	1	no	—
62	C₄×C₇×Q₈	—	2	1	no	—
63	C₄×C₇×Q₈	—	2	1	no	—
64	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₇×D₄	—	2	1	no	—
67	C₄×C₇×D₄	—	2	1	no	—
68	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
69	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
70	C₇×#117	—	2	1	no	—
71	C₇×#117	C₂³	1	1	yes	characteristic
72	C₇×#117	C₂³	1	2	yes	—
73	C₂×C₇×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄²×C₇	—	2	1	no	—
75	C₂×C₄²×C₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
77	C₇×#108	—	4	2	no	—
78	C₇×#107	C₂×C₄	1	2	yes	—
79	C₇×#107	C₂×C₄	1	2	yes	—
80	C₇×#106	C₂×C₄	1	2	yes	—
81	C₇×#106	C₂×C₄	1	2	yes	—
82	C₇×C₄⋊C₈	—	2	1	no	—
83	C₇×C₄⋊C₈	—	2	1	no	—
84	C₇×C₄⋊C₈	—	2	1	no	—
85	C₇×C₄⋊C₈	—	2	1	no	—
86	C₇×C₄⋊C₈	—	2	2	no	—
87	C₇×C₄⋊C₈	—	2	2	no	—
88	C₇×C₄⋊C₈	—	2	1	no	—
89	C₇×C₄⋊C₈	—	2	1	no	—
90	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
91	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
92	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
93	C₇×#104	—	4	2	no	—
94	C₇×#103	C₂×C₄	1	2	yes	—
95	C₇×#103	C₂×C₄	1	2	yes	—
96	C₇×#102	C₂×C₄	1	2	yes	—
97	C₇×#102	C₂×C₄	1	2	yes	—
98	C₇×C₂²⋊C₈	—	2	1	no	—
99	C₇×C₂²⋊C₈	—	2	1	no	—
100	C₇×C₂²⋊C₈	—	2	1	no	—
101	C₇×C₂²⋊C₈	—	2	1	no	—
102	C₇×#97	—	2	2	no	—
103	C₇×#97	—	2	1	no	—
104	C₇×#97	—	4	2	no	—
105	C₄×C₇×C₈	—	2	2	no	—
106	C₄×C₇×C₈	—	2	1	no	—
107	C₄×C₇×C₈	D₄	1	1	yes	characteristic
108	C₄×C₇×C₈	D₄	1	1	yes	characteristic
109	C₄×C₇×C₈	D₄	1	1	yes	characteristic
110	C₄×C₇×C₈	D₄	1	1	yes	characteristic
111	C₇×#95	—	2	1	no	—
112	#3076	C₂×C₇	1	1	yes	characteristic
113	#2960	C₂×C₇	1	1	yes	characteristic, complemented
114	#2922	C₂×C₇	1	1	yes	characteristic
115	#1986	C₂×C₇	1	1	yes	characteristic, complemented
116	#1950	C₂×C₇	1	1	yes	characteristic, complemented
117	#1946	C₂×C₇	1	2	yes	complemented
118	#1938	C₂×C₇	1	2	yes	—
119	#1929	C₂×C₇	1	2	yes	—
120	#1922	C₂×C₇	1	2	yes	complemented
121	#1806	C₂×C₇	1	1	yes	characteristic
122	#1770	C₂×C₇	1	1	yes	characteristic, complemented
123	C₇×#41	—	2	1	no	—
124	C₇×#41	—	2	1	no	—
125	C₇×#41	—	2	1	no	—
126	C₇×#41	—	2	1	no	—
127	C₂×C₇×Q₈	—	2	1	no	—
128	C₂×C₇×Q₈	—	2	1	no	—
129	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₇×D₄	—	2	1	no	—
131	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₇	—	2	1	no	—
133	C₂²×C₄×C₇	—	2	1	no	—
134	C₂²×C₄×C₇	—	2	1	no	—
135	C₂²×C₄×C₇	—	2	2	no	—
136	C₂²×C₄×C₇	#41	1	1	yes	characteristic
137	C₂²×C₄×C₇	#41	1	1	yes	characteristic
138	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
139	C₇×M₁₆	—	2	2	no	—
140	C₇×M₁₆	—	2	1	no	—
141	C₇×M₁₆	—	2	1	no	—
142	C₇×M₁₆	—	2	1	no	—
143	C₇×M₁₆	—	2	1	no	—
144	C₇×M₁₆	—	2	2	no	—
145	C₂×C₇×C₈	—	2	1	no	—
146	C₂×C₇×C₈	—	2	2	no	—
147	C₂×C₇×C₈	—	2	2	no	—
148	C₂×C₇×C₈	—	2	1	no	—
149	C₂×C₇×C₈	—	2	1	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	—	2	1	no	—
152	C₂×C₇×C₈	—	2	1	no	—
153	C₂×C₇×C₈	—	2	2	no	—
154	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
155	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
156	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
157	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
158	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
159	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
160	C₇×C₄⋊C₄	—	2	1	no	—
161	C₇×C₄⋊C₄	—	2	1	no	—
162	C₇×C₄⋊C₄	—	2	2	no	—
163	C₇×C₄⋊C₄	—	2	2	no	—
164	C₇×C₄⋊C₄	—	2	1	no	—
165	C₇×C₄⋊C₄	—	2	1	no	—
166	C₇×C₄⋊C₄	—	4	2	no	—
167	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
168	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
169	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
170	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
171	C₇×C₂²⋊C₄	—	2	1	no	—
172	C₇×C₂²⋊C₄	—	2	2	no	—
173	C₇×C₂²⋊C₄	—	2	1	no	—
174	C₇×C₂²⋊C₄	—	2	1	no	—
175	C₄²×C₇	—	2	1	no	—
176	C₄²×C₇	—	2	1	no	—
177	C₄²×C₇	—	2	2	no	—
178	C₄²×C₇	—	2	1	no	—
179	C₄²×C₇	—	2	2	no	—
180	C₄²×C₇	—	2	2	no	—
181	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
182	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
185	#533	C₂²×C₇	1	1	yes	characteristic
186	C₄×#41	C₂²×C₇	1	1	yes	characteristic
187	#479	C₄×C₇	1	2	yes	complemented
188	#478	C₄×C₇	1	2	yes	complemented
189	#477	C₄×C₇	1	1	yes	characteristic, complemented
190	#476	C₄×C₇	1	1	yes	characteristic, complemented
191	#475	C₄×C₇	1	1	yes	characteristic, complemented
192	#474	C₄×C₇	1	1	yes	characteristic, complemented
193	#446	—	2	1	no	—
194	#446	—	2	1	no	—
195	C₈×Q₈	—	2	1	no	—
196	C₈×Q₈	—	2	1	no	—
197	#433	—	2	1	no	—
198	#433	—	2	1	no	—
199	#431	—	2	1	no	—
200	#431	—	2	1	no	—
201	C₂×#108	C₂²×C₇	1	2	yes	—
202	#427	C₂²×C₇	1	2	yes	—
203	#424	C₂²×C₇	1	1	yes	characteristic
204	#424	C₂²×C₇	1	1	yes	characteristic
205	#424	C₂²×C₇	1	2	yes	—
206	#423	C₂²×C₇	1	1	yes	characteristic
207	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
208	C₂×#104	C₂²×C₇	1	2	yes	—
209	#416	C₂²×C₇	1	2	yes	—
210	#408	C₂²×C₇	1	1	yes	characteristic
211	#408	C₂²×C₇	1	1	yes	characteristic
212	#405	C₂²×C₇	1	2	yes	—
213	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
214	C₂×#97	C₂²×C₇	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
216	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
217	#341	C₂²×C₇	1	2	yes	—
218	#339	C₂²×C₇	1	2	yes	—
219	#339	C₂²×C₇	1	2	yes	—
220	#339	C₂²×C₇	1	2	yes	—
221	#336	C₂²×C₇	1	2	yes	—
222	C₂³×C₇	—	2	1	no	—
223	C₂³×C₇	C₂×#41	1	1	yes	characteristic
224	C₇×Q₈	—	2	1	no	—
225	C₇×Q₈	—	2	1	no	—
226	C₇×Q₈	—	4	2	no	—
227	C₇×Q₈	—	4	2	no	—
228	C₇×D₄	—	2	1	no	—
229	C₇×D₄	—	2	1	no	—
230	C₇×D₄	—	4	1	no	—
231	C₇×D₄	—	4	1	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	1	no	—
234	C₂×C₄×C₇	—	2	1	no	—
235	C₂×C₄×C₇	—	2	1	no	—
236	C₂×C₄×C₇	—	2	2	no	—
237	C₂×C₄×C₇	—	2	2	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	2	no	—
244	C₂×C₄×C₇	—	4	2	no	—
245	C₂×C₄×C₇	—	4	2	no	—
246	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
247	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
248	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
249	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
250	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
253	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
254	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
255	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
256	C₇×C₈	—	2	2	no	—
257	C₇×C₈	—	2	2	no	—
258	C₇×C₈	—	2	1	no	—
259	C₇×C₈	—	2	1	no	—
260	C₇×C₈	—	2	1	no	—
261	C₇×C₈	—	2	1	no	—
262	C₇×C₈	—	4	1	no	—
263	C₇×C₈	—	4	1	no	—
264	C₇×C₈	—	4	2	no	—
265	C₂×#41	C₂³×C₇	1	1	yes	characteristic
266	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
267	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
268	C₂×M₁₆	C₂³×C₇	1	2	yes	—
269	C₂²×C₈	C₂³×C₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
272	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
273	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
274	#125	C₂×C₄×C₇	1	2	yes	—
275	C₂²⋊Q₈	—	2	1	no	—
276	C₂²⋊Q₈	—	2	1	no	—
277	C₄×Q₈	—	2	1	no	—
278	C₄×Q₈	—	2	1	no	—
279	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
280	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	—	2	1	no	—
283	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
284	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
285	#117	—	2	1	no	—
286	#117	C₂³×C₇	1	1	yes	characteristic
287	#117	C₂³×C₇	1	2	yes	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	C₂³×C₇	1	2	yes	—
291	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
292	#108	—	4	2	no	—
293	#107	C₂×C₄×C₇	1	2	yes	—
294	#107	C₂×C₄×C₇	1	2	yes	—
295	#106	C₂×C₄×C₇	1	2	yes	—
296	#106	C₂×C₄×C₇	1	2	yes	—
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	—	2	1	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	2	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
306	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
307	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
308	#104	—	4	2	no	—
309	#103	C₂×C₄×C₇	1	2	yes	—
310	#103	C₂×C₄×C₇	1	2	yes	—
311	#102	C₂×C₄×C₇	1	2	yes	—
312	#102	C₂×C₄×C₇	1	2	yes	—
313	C₂²⋊C₈	—	2	1	no	—
314	C₂²⋊C₈	—	2	1	no	—
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	2	1	no	—
317	#97	—	2	2	no	—
318	#97	—	2	1	no	—
319	#97	—	4	2	no	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	C₇×D₄	1	1	yes	characteristic
323	C₄×C₈	C₇×D₄	1	1	yes	characteristic
324	C₄×C₈	C₇×D₄	1	1	yes	characteristic
325	C₄×C₈	C₇×D₄	1	1	yes	characteristic
326	#95	—	2	1	no	—
327	C₂²×C₇	—	2	1	no	—
328	C₂²×C₇	—	2	1	no	—
329	C₂²×C₇	—	4	1	no	—
330	C₂²×C₇	—	4	1	no	—
331	C₂²×C₇	—	4	1	no	—
332	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
333	C₂²×C₇	#541	1	1	yes	characteristic
334	C₂²×C₇	#542	1	1	yes	characteristic
335	C₄×C₇	—	2	1	no	—
336	C₄×C₇	—	2	1	no	—
337	C₄×C₇	—	2	1	no	—
338	C₄×C₇	—	4	1	no	—
339	C₄×C₇	—	4	1	no	—
340	C₄×C₇	—	4	2	no	—
341	C₄×C₇	—	4	1	no	—
342	C₄×C₇	—	4	2	no	—
343	C₄×C₇	—	8	2	no	—
344	C₄×C₇	#444	1	2	yes	—
345	C₄×C₇	#519	1	1	yes	characteristic
346	C₄×C₇	#520	1	1	yes	characteristic
347	C₄×C₇	#536	1	1	yes	characteristic
348	C₄×C₇	#537	1	1	yes	characteristic
349	#41	—	2	1	no	—
350	#41	—	2	1	no	—
351	#41	—	2	1	no	—
352	#41	—	2	1	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	C₇×#41	1	1	yes	characteristic
363	C₂²×C₄	C₇×#41	1	1	yes	characteristic
364	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
365	M₁₆	—	2	2	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	2	1	no	—
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
381	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
382	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
383	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
384	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
385	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
394	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
395	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
396	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
397	C₂²⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	2	1	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
408	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
409	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
410	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
411	C₂×C₇	—	2	1	no	—
412	C₂×C₇	—	2	1	no	—
413	C₂×C₇	—	8	1	no	—
414	C₂×C₇	#2323	1	1	yes	characteristic
415	C₂×C₇	C₂×#444	1	1	yes	characteristic
416	C₂×C₇	#2955	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	C₂×C₇×#41	1	1	yes	characteristic
419	Q₈	—	2	1	no	—
420	Q₈	—	2	1	no	—
421	Q₈	—	4	2	no	—
422	Q₈	—	4	2	no	—
423	D₄	—	2	1	no	—
424	D₄	—	2	1	no	—
425	D₄	—	4	1	no	—
426	D₄	—	4	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
442	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
443	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
444	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
445	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
446	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
447	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
448	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
449	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
450	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	2	2	no	—
453	C₈	—	2	1	no	—
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	4	2	no	—
460	C₇	#22131	1	1	yes	characteristic, direct product factor, Sylow
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
467	C₂²	C₇×#541	1	1	yes	characteristic
468	C₂²	C₇×#542	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	1	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	8	2	no	—
478	C₄	C₇×#444	1	2	yes	—
479	C₄	C₇×#519	1	1	yes	characteristic
480	C₄	C₇×#520	1	1	yes	characteristic
481	C₄	C₇×#536	1	1	yes	characteristic
482	C₄	C₇×#537	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	8	1	no	—
486	C₂	C₇×#2323	1	1	yes	characteristic
487	C₂	C₂×C₇×#444	1	1	yes	characteristic
488	C₂	C₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2960	C₂	semidirect
4. C₇×#1986	C₂	semidirect
5. C₇×#1950	C₂	semidirect
6. C₇×#1946	C₂	semidirect
9. C₇×#1922	C₂	semidirect
11. C₇×#1770	C₂	semidirect
14. C₇×#479	C₄	semidirect
15. C₇×#478	C₄	semidirect
16. C₇×#477	C₄	semidirect
17. C₇×#476	C₄	semidirect
18. C₇×#475	C₄	semidirect
19. C₇×#474	C₄	semidirect
49. #22131	C₇	direct (N × H)
113. #2960	C₂×C₇	semidirect
115. #1986	C₂×C₇	semidirect
116. #1950	C₂×C₇	semidirect
117. #1946	C₂×C₇	semidirect
120. #1922	C₂×C₇	semidirect
122. #1770	C₂×C₇	semidirect
187. #479	C₄×C₇	semidirect
188. #478	C₄×C₇	semidirect
189. #477	C₄×C₇	semidirect
190. #476	C₄×C₇	semidirect
191. #475	C₄×C₇	semidirect
192. #474	C₄×C₇	semidirect
460. C₇	#22131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3076	2
C₇×#2960	2
C₇×#2922	2
C₇×#1986	2
C₇×#1950	2
C₇×#1946	2
C₇×#1938	2
C₇×#1929	2
C₇×#1922	2
C₇×#1806	2
C₇×#1770	2
#22131	7

Subgroups — C7×#22131 #13199347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22131 #13199347