Subgroups — C₇×#23114 #13200330

Order 1792.

778 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3035	C₂	1	1	yes	maximal, characteristic
2	C₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1975	C₂	1	2	yes	maximal, complemented
5	C₇×#1962	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1929	C₂	1	2	yes	maximal, complemented
8	C₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1765	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	2	yes	—
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₇×#532	C₂²	1	1	yes	characteristic
15	C₄×C₇×#41	C₂²	1	1	yes	characteristic
16	C₇×#464	C₄	1	2	yes	complemented
17	C₇×#463	C₄	1	2	yes	complemented
18	C₇×#462	C₄	1	1	yes	characteristic, complemented
19	C₇×#461	C₄	1	1	yes	characteristic, complemented
20	C₇×#460	C₄	1	1	yes	characteristic, complemented
21	C₇×#459	C₄	1	1	yes	characteristic, complemented
22	C₇×#436	—	2	1	no	—
23	C₇×#435	—	2	2	no	—
24	C₇×C₈×D₄	—	2	1	no	—
25	C₇×#433	—	2	2	no	—
26	C₇×#432	—	2	1	no	—
27	C₇×#431	—	2	1	no	—
28	C₂×C₇×#108	C₂²	1	2	yes	—
29	C₇×#423	C₂²	1	1	yes	characteristic
30	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₇×#104	C₂²	1	1	yes	characteristic
33	C₂×C₇×#104	C₂²	1	2	yes	—
34	C₂×C₇×#104	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	2	yes	—
38	C₇×#408	C₂²	1	2	yes	—
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	C₄³×C₇	C₂²	1	1	yes	characteristic
43	C₇×#341	C₂²	1	2	yes	—
44	C₇×#339	C₂²	1	1	yes	characteristic
45	C₇×#339	C₂²	1	1	yes	characteristic
46	C₇×#336	C₂²	1	2	yes	—
47	#23114	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₇×#41	C₂³	1	1	yes	characteristic
49	C₂×C₇×#41	C₂³	1	2	yes	—
50	C₇×#135	—	4	2	no	—
51	C₇×#135	—	4	2	no	—
52	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
53	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
55	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
56	C₇×#131	—	2	4	no	—
57	C₇×#131	—	4	2	no	—
58	C₂×C₇×M₁₆	—	2	2	no	—
59	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₇×M₁₆	C₂³	1	2	yes	—
61	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
62	C₂²×C₇×C₈	—	2	2	no	—
63	C₂²×C₇×C₈	C₂³	1	2	yes	—
64	C₂²×C₇×C₈	C₂³	1	2	yes	—
65	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	C₇×#127	C₂×C₄	1	1	yes	characteristic
67	C₇×#124	C₂×C₄	1	2	yes	—
68	C₇×#121	—	2	1	no	—
69	C₇×#121	—	2	1	no	—
70	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₇×D₄	—	2	1	no	—
73	C₄×C₇×D₄	—	2	1	no	—
74	C₄×C₇×D₄	—	2	1	no	—
75	C₄×C₇×D₄	—	2	1	no	—
76	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
77	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
78	C₇×#117	—	2	1	no	—
79	C₇×#117	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₇	—	2	1	no	—
81	C₂×C₄²×C₇	—	2	1	no	—
82	C₂×C₄²×C₇	—	2	1	no	—
83	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
86	C₇×#108	—	4	2	no	—
87	C₇×C₄⋊C₈	—	2	1	no	—
88	C₇×C₄⋊C₈	—	2	1	no	—
89	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
90	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
91	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
92	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
93	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
94	C₇×C₄⋊C₈	D₄	1	2	yes	—
95	C₇×#104	—	2	2	no	—
96	C₇×#104	—	2	2	no	—
97	C₇×#104	—	2	4	no	—
98	C₇×#104	—	4	2	no	—
99	C₇×#103	C₂×C₄	1	2	yes	—
100	C₇×#103	C₂×C₄	1	2	yes	—
101	C₇×#102	C₂×C₄	1	2	yes	—
102	C₇×#102	C₂×C₄	1	2	yes	—
103	C₇×C₂²⋊C₈	—	2	2	no	—
104	C₇×C₂²⋊C₈	—	2	2	no	—
105	C₇×C₂²⋊C₈	—	2	2	no	—
106	C₇×C₂²⋊C₈	—	2	2	no	—
107	C₇×#97	—	2	2	no	—
108	C₇×#97	—	2	1	no	—
109	C₇×#97	—	4	2	no	—
110	C₄×C₇×C₈	—	2	2	no	—
111	C₄×C₇×C₈	—	2	1	no	—
112	#3035	C₂×C₇	1	1	yes	characteristic
113	#2935	C₂×C₇	1	1	yes	characteristic, complemented
114	#2925	C₂×C₇	1	1	yes	characteristic, complemented
115	#1975	C₂×C₇	1	2	yes	complemented
116	#1962	C₂×C₇	1	1	yes	characteristic, complemented
117	#1950	C₂×C₇	1	1	yes	characteristic, complemented
118	#1929	C₂×C₇	1	2	yes	complemented
119	#1876	C₂×C₇	1	2	yes	complemented
120	#1767	C₂×C₇	1	1	yes	characteristic, complemented
121	#1765	C₂×C₇	1	2	yes	complemented
122	C₄×#104	C₂×C₇	1	1	yes	characteristic, complemented
123	C₇×#41	—	2	1	no	—
124	C₇×#41	—	2	1	no	—
125	C₇×#41	—	2	2	no	—
126	C₇×#41	—	2	2	no	—
127	C₇×#41	—	2	2	no	—
128	C₇×#41	—	2	2	no	—
129	C₇×#41	—	2	1	no	—
130	C₇×#41	—	2	1	no	—
131	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
132	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₇×D₄	—	2	1	no	—
134	C₂×C₇×D₄	—	2	2	no	—
135	C₂×C₇×D₄	—	2	1	no	—
136	C₂×C₇×D₄	—	2	1	no	—
137	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
138	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₇	—	2	1	no	—
140	C₂²×C₄×C₇	—	2	1	no	—
141	C₂²×C₄×C₇	—	2	1	no	—
142	C₂²×C₄×C₇	—	2	1	no	—
143	C₂²×C₄×C₇	—	2	2	no	—
144	C₂²×C₄×C₇	—	2	1	no	—
145	C₂²×C₄×C₇	#41	1	1	yes	characteristic
146	C₂²×C₄×C₇	#41	1	1	yes	characteristic
147	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
148	C₇×Q₁₆	—	4	2	no	—
149	C₇×SD₁₆	—	4	2	no	—
150	C₇×SD₁₆	—	4	2	no	—
151	C₇×D₈	—	4	2	no	—
152	C₇×M₁₆	—	2	1	no	—
153	C₇×M₁₆	—	2	1	no	—
154	C₇×M₁₆	—	2	2	no	—
155	C₇×M₁₆	—	2	2	no	—
156	C₇×M₁₆	—	2	1	no	—
157	C₇×M₁₆	—	4	4	no	—
158	C₇×M₁₆	#41	1	2	yes	—
159	C₂×C₇×C₈	—	2	1	no	—
160	C₂×C₇×C₈	—	2	2	no	—
161	C₂×C₇×C₈	—	2	2	no	—
162	C₂×C₇×C₈	—	2	2	no	—
163	C₂×C₇×C₈	—	2	2	no	—
164	C₂×C₇×C₈	—	2	1	no	—
165	C₂×C₇×C₈	—	2	2	no	—
166	C₂×C₇×C₈	—	4	4	no	—
167	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
168	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
169	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
170	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
171	C₇×C₄⋊C₄	—	2	1	no	—
172	C₇×C₄⋊C₄	—	2	1	no	—
173	C₇×C₄⋊C₄	—	2	1	no	—
174	C₇×C₄⋊C₄	—	2	1	no	—
175	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
176	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
177	C₇×C₂²⋊C₄	—	2	1	no	—
178	C₇×C₂²⋊C₄	—	2	1	no	—
179	C₇×C₂²⋊C₄	—	2	1	no	—
180	C₇×C₂²⋊C₄	—	2	2	no	—
181	C₇×C₂²⋊C₄	—	2	2	no	—
182	C₄²×C₇	—	2	1	no	—
183	C₄²×C₇	—	2	1	no	—
184	C₄²×C₇	—	2	1	no	—
185	C₄²×C₇	—	2	1	no	—
186	C₄²×C₇	—	2	1	no	—
187	C₄²×C₇	—	2	1	no	—
188	C₄²×C₇	—	4	2	no	—
189	C₄²×C₇	—	4	2	no	—
190	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₇	#41	1	2	yes	—
195	C₂×#135	C₂²×C₇	1	2	yes	—
196	C₂×#131	C₂²×C₇	1	2	yes	—
197	#532	C₂²×C₇	1	1	yes	characteristic
198	C₄×#41	C₂²×C₇	1	1	yes	characteristic
199	#464	C₄×C₇	1	2	yes	complemented
200	#463	C₄×C₇	1	2	yes	complemented
201	#462	C₄×C₇	1	1	yes	characteristic, complemented
202	#461	C₄×C₇	1	1	yes	characteristic, complemented
203	#460	C₄×C₇	1	1	yes	characteristic, complemented
204	#459	C₄×C₇	1	1	yes	characteristic, complemented
205	#436	—	2	1	no	—
206	#435	—	2	2	no	—
207	C₈×D₄	—	2	1	no	—
208	#433	—	2	2	no	—
209	#432	—	2	1	no	—
210	#431	—	2	1	no	—
211	C₂×#108	C₂²×C₇	1	2	yes	—
212	#423	C₂²×C₇	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
214	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
215	C₂×#104	C₂²×C₇	1	1	yes	characteristic
216	C₂×#104	C₂²×C₇	1	2	yes	—
217	C₂×#104	C₂²×C₇	1	1	yes	characteristic
218	C₂×#104	C₂²×C₇	1	2	yes	—
219	#416	C₂²×C₇	1	2	yes	—
220	#408	C₂²×C₇	1	2	yes	—
221	#408	C₂²×C₇	1	2	yes	—
222	#405	C₂²×C₇	1	2	yes	—
223	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
224	C₂×#97	C₂²×C₇	1	2	yes	—
225	C₄³	C₂²×C₇	1	1	yes	characteristic
226	#341	C₂²×C₇	1	2	yes	—
227	#339	C₂²×C₇	1	1	yes	characteristic
228	#339	C₂²×C₇	1	1	yes	characteristic
229	#336	C₂²×C₇	1	2	yes	—
230	C₂³×C₇	—	2	1	no	—
231	C₂³×C₇	—	2	2	no	—
232	C₂³×C₇	C₂×#41	1	1	yes	characteristic
233	C₇×Q₈	—	2	2	no	—
234	C₇×Q₈	—	2	1	no	—
235	C₇×Q₈	—	2	1	no	—
236	C₇×Q₈	—	2	2	no	—
237	C₇×D₄	—	2	2	no	—
238	C₇×D₄	—	2	1	no	—
239	C₇×D₄	—	2	1	no	—
240	C₇×D₄	—	2	2	no	—
241	C₇×D₄	—	4	1	no	—
242	C₇×D₄	—	4	2	no	—
243	C₇×D₄	—	4	1	no	—
244	C₇×D₄	—	4	2	no	—
245	C₇×D₄	—	4	2	no	—
246	C₇×D₄	—	8	1	no	—
247	C₂×C₄×C₇	—	2	2	no	—
248	C₂×C₄×C₇	—	2	2	no	—
249	C₂×C₄×C₇	—	2	1	no	—
250	C₂×C₄×C₇	—	2	1	no	—
251	C₂×C₄×C₇	—	2	1	no	—
252	C₂×C₄×C₇	—	2	2	no	—
253	C₂×C₄×C₇	—	2	2	no	—
254	C₂×C₄×C₇	—	4	1	no	—
255	C₂×C₄×C₇	—	4	1	no	—
256	C₂×C₄×C₇	—	4	1	no	—
257	C₂×C₄×C₇	—	4	1	no	—
258	C₂×C₄×C₇	—	4	1	no	—
259	C₂×C₄×C₇	—	4	2	no	—
260	C₂×C₄×C₇	—	4	1	no	—
261	C₂×C₄×C₇	—	4	1	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	—	4	2	no	—
264	C₂×C₄×C₇	—	8	1	no	—
265	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
266	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
267	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
268	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
269	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
270	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
275	C₇×C₈	—	2	2	no	—
276	C₇×C₈	—	2	2	no	—
277	C₇×C₈	—	2	2	no	—
278	C₇×C₈	—	2	2	no	—
279	C₇×C₈	—	2	2	no	—
280	C₇×C₈	—	4	1	no	—
281	C₇×C₈	—	4	2	no	—
282	C₂×#41	C₂³×C₇	1	1	yes	characteristic
283	C₂×#41	C₂³×C₇	1	2	yes	—
284	#135	—	4	2	no	—
285	#135	—	4	2	no	—
286	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
287	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
288	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
289	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
290	#131	—	2	4	no	—
291	#131	—	4	2	no	—
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₇	1	2	yes	—
295	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	C₂³×C₇	1	2	yes	—
298	C₂²×C₈	C₂³×C₇	1	2	yes	—
299	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
300	#127	C₂×C₄×C₇	1	1	yes	characteristic
301	#124	C₂×C₄×C₇	1	2	yes	—
302	#121	—	2	1	no	—
303	#121	—	2	1	no	—
304	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
305	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	—	2	1	no	—
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
311	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
312	#117	—	2	1	no	—
313	#117	C₂³×C₇	1	1	yes	characteristic
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
320	#108	—	4	2	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
324	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
325	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
326	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
327	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
328	C₄⋊C₈	C₇×D₄	1	2	yes	—
329	#104	—	2	2	no	—
330	#104	—	2	4	no	—
331	#104	—	2	2	no	—
332	#104	—	4	2	no	—
333	#103	C₂×C₄×C₇	1	2	yes	—
334	#103	C₂×C₄×C₇	1	2	yes	—
335	#102	C₂×C₄×C₇	1	2	yes	—
336	#102	C₂×C₄×C₇	1	2	yes	—
337	C₂²⋊C₈	—	2	2	no	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	—	2	2	no	—
340	C₂²⋊C₈	—	2	2	no	—
341	#97	—	2	2	no	—
342	#97	—	2	1	no	—
343	#97	—	4	2	no	—
344	C₄×C₈	—	2	2	no	—
345	C₄×C₈	—	2	1	no	—
346	C₂²×C₇	—	2	1	no	—
347	C₂²×C₇	—	2	1	no	—
348	C₂²×C₇	—	4	2	no	—
349	C₂²×C₇	—	4	1	no	—
350	C₂²×C₇	—	4	1	no	—
351	C₂²×C₇	—	4	2	no	—
352	C₂²×C₇	—	4	1	no	—
353	C₂²×C₇	—	4	2	no	—
354	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
355	C₂²×C₇	#553	1	1	yes	characteristic
356	C₂²×C₇	#555	1	1	yes	characteristic
357	C₄×C₇	—	2	1	no	—
358	C₄×C₇	—	2	1	no	—
359	C₄×C₇	—	2	1	no	—
360	C₄×C₇	—	4	2	no	—
361	C₄×C₇	—	4	2	no	—
362	C₄×C₇	—	4	1	no	—
363	C₄×C₇	—	4	2	no	—
364	C₄×C₇	—	4	1	no	—
365	C₄×C₇	—	4	1	no	—
366	C₄×C₇	#444	1	2	yes	—
367	C₄×C₇	C₄×#41	1	1	yes	characteristic
368	C₄×C₇	#518	1	1	yes	characteristic
369	C₄×C₇	#546	1	1	yes	characteristic
370	C₄×C₇	#550	1	1	yes	characteristic
371	#41	—	2	1	no	—
372	#41	—	2	1	no	—
373	#41	—	2	2	no	—
374	#41	—	2	2	no	—
375	#41	—	2	2	no	—
376	#41	—	2	2	no	—
377	#41	—	2	1	no	—
378	#41	—	2	1	no	—
379	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
380	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
386	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	C₇×#41	1	1	yes	characteristic
394	C₂²×C₄	C₇×#41	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
396	Q₁₆	—	4	2	no	—
397	SD₁₆	—	4	2	no	—
398	SD₁₆	—	4	2	no	—
399	D₈	—	4	2	no	—
400	M₁₆	—	2	1	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	1	no	—
405	M₁₆	—	4	4	no	—
406	M₁₆	C₇×#41	1	2	yes	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	4	4	no	—
415	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
416	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
417	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
418	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
424	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
425	C₂²⋊C₄	—	2	1	no	—
426	C₂²⋊C₄	—	2	1	no	—
427	C₂²⋊C₄	—	2	1	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₄²	—	2	1	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	4	2	no	—
437	C₄²	—	4	2	no	—
438	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
439	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
440	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
441	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
442	C₄²	C₇×#41	1	2	yes	—
443	C₂×C₇	—	2	1	no	—
444	C₂×C₇	—	2	1	no	—
445	C₂×C₇	—	8	1	no	—
446	C₂×C₇	—	8	2	no	—
447	C₂×C₇	#2363	1	1	yes	characteristic
448	C₂×C₇	C₂×#444	1	1	yes	characteristic
449	C₂×C₇	#2956	1	1	yes	characteristic
450	C₂³	—	2	1	no	—
451	C₂³	—	2	2	no	—
452	C₂³	C₂×C₇×#41	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	1	no	—
455	Q₈	—	2	1	no	—
456	Q₈	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	1	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	4	2	no	—
466	D₄	—	8	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	8	1	no	—
485	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
486	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
487	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
488	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
489	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
490	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
491	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
492	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
493	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
494	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
495	C₈	—	2	2	no	—
496	C₈	—	2	2	no	—
497	C₈	—	2	2	no	—
498	C₈	—	2	2	no	—
499	C₈	—	2	2	no	—
500	C₈	—	4	1	no	—
501	C₈	—	4	2	no	—
502	C₇	#23114	1	1	yes	characteristic, direct product factor, Sylow
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
512	C₂²	C₇×#553	1	1	yes	characteristic
513	C₂²	C₇×#555	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	2	1	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	1	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	2	no	—
523	C₄	C₇×#444	1	2	yes	—
524	C₄	C₄×C₇×#41	1	1	yes	characteristic
525	C₄	C₇×#518	1	1	yes	characteristic
526	C₄	C₇×#546	1	1	yes	characteristic
527	C₄	C₇×#550	1	1	yes	characteristic
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	8	1	no	—
531	C₂	—	8	2	no	—
532	C₂	C₇×#2363	1	1	yes	characteristic
533	C₂	C₂×C₇×#444	1	1	yes	characteristic
534	C₂	C₇×#2956	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	C₇×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2935	C₂	semidirect
3. C₇×#2925	C₂	semidirect
4. C₇×#1975	C₂	semidirect
5. C₇×#1962	C₂	semidirect
6. C₇×#1950	C₂	semidirect
7. C₇×#1929	C₂	semidirect
8. C₇×#1876	C₂	semidirect
9. C₇×#1767	C₂	semidirect
10. C₇×#1765	C₂	semidirect
11. C₄×C₇×#104	C₂	semidirect
16. C₇×#464	C₄	semidirect
17. C₇×#463	C₄	semidirect
18. C₇×#462	C₄	semidirect
19. C₇×#461	C₄	semidirect
20. C₇×#460	C₄	semidirect
21. C₇×#459	C₄	semidirect
47. #23114	C₇	direct (N × H)
113. #2935	C₂×C₇	semidirect
114. #2925	C₂×C₇	semidirect
115. #1975	C₂×C₇	semidirect
116. #1962	C₂×C₇	semidirect
117. #1950	C₂×C₇	semidirect
118. #1929	C₂×C₇	semidirect
119. #1876	C₂×C₇	semidirect
120. #1767	C₂×C₇	semidirect
121. #1765	C₂×C₇	semidirect
122. C₄×#104	C₂×C₇	semidirect
199. #464	C₄×C₇	semidirect
200. #463	C₄×C₇	semidirect
201. #462	C₄×C₇	semidirect
202. #461	C₄×C₇	semidirect
203. #460	C₄×C₇	semidirect
204. #459	C₄×C₇	semidirect
502. C₇	#23114	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×#3035	2
C₇×#2935	2
C₇×#2925	2
C₇×#1975	2
C₇×#1962	2
C₇×#1950	2
C₇×#1929	2
C₇×#1876	2
C₇×#1767	2
C₇×#1765	2
C₄×C₇×#104	2
#23114	7

Subgroups — C7×#23114 #13200330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23114 #13200330