Subgroups — C₇×#23019 #13200235

Order 1792.

880 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2933	C₂	1	2	yes	maximal, complemented
4	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1929	C₂	1	2	yes	maximal, complemented
8	C₇×#1871	C₂	1	2	yes	maximal, complemented
9	C₇×#1866	C₂	1	2	yes	maximal, complemented
10	C₇×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	1	yes	characteristic
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
15	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×#525	C₂²	1	2	yes	—
17	C₇×#467	C₄	1	1	yes	characteristic, complemented
18	C₇×#467	C₄	1	1	yes	characteristic, complemented
19	C₇×#466	C₄	1	1	yes	characteristic, complemented
20	C₇×#466	C₄	1	1	yes	characteristic, complemented
21	C₇×#465	C₄	1	2	yes	complemented
22	C₇×#465	C₄	1	2	yes	complemented
23	C₇×#444	—	2	4	no	—
24	C₇×#436	—	2	2	no	—
25	C₇×#435	—	2	2	no	—
26	C₇×#435	—	2	2	no	—
27	C₇×C₈×D₄	—	2	2	no	—
28	C₂×C₇×#108	—	2	4	no	—
29	C₂×C₇×#108	C₂²	1	1	yes	characteristic
30	C₂×C₇×#108	C₂²	1	2	yes	—
31	C₂×C₇×#108	C₂²	1	1	yes	characteristic
32	C₇×#427	C₂²	1	1	yes	characteristic
33	C₇×#424	C₂²	1	2	yes	—
34	C₂×C₇×#104	C₂²	1	2	yes	—
35	C₂×C₇×#104	C₂²	1	2	yes	—
36	C₇×#416	C₂²	1	2	yes	—
37	C₇×#408	C₂²	1	2	yes	—
38	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₇×#405	C₂²	1	2	yes	—
40	C₂×C₇×#97	C₂²	1	1	yes	characteristic
41	C₂×C₇×#97	C₂²	1	1	yes	characteristic
42	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
43	C₇×#339	C₂²	1	2	yes	—
44	C₇×#339	C₂²	1	2	yes	—
45	C₇×#336	C₂²	1	1	yes	characteristic
46	C₇×#336	C₂²	1	1	yes	characteristic
47	#23019	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₇×#41	C₂³	1	2	yes	—
49	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₇×#135	—	2	2	no	—
51	C₇×#135	—	2	2	no	—
52	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
55	C₇×#131	—	2	4	no	—
56	C₇×#131	—	4	2	no	—
57	C₂×C₇×M₁₆	—	2	2	no	—
58	C₂×C₇×M₁₆	—	2	2	no	—
59	C₂×C₇×M₁₆	—	2	4	no	—
60	C₂×C₇×M₁₆	—	4	2	no	—
61	C₂×C₇×M₁₆	C₂³	1	2	yes	—
62	C₂×C₇×M₁₆	C₂³	1	2	yes	—
63	C₂²×C₇×C₈	—	2	2	no	—
64	C₂²×C₇×C₈	—	2	2	no	—
65	C₂²×C₇×C₈	—	2	2	no	—
66	C₂²×C₇×C₈	—	2	2	no	—
67	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
75	C₇×#123	—	2	2	no	—
76	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
77	C₇×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₇×#121	C₂×C₄	1	2	yes	—
79	C₇×#121	C₂×C₄	1	2	yes	—
80	C₇×#120	—	2	2	no	—
81	C₄×C₇×D₄	—	2	2	no	—
82	C₄×C₇×D₄	—	2	2	no	—
83	C₇×#117	C₂³	1	2	yes	—
84	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
85	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
87	C₇×#108	—	2	4	no	—
88	C₇×#108	—	2	2	no	—
89	C₇×#108	D₄	1	4	yes	complemented
90	C₇×#107	C₂×C₄	1	1	yes	characteristic
91	C₇×#107	C₂×C₄	1	1	yes	characteristic
92	C₇×#106	C₂×C₄	1	2	yes	—
93	C₇×C₄⋊C₈	—	2	2	no	—
94	C₇×C₄⋊C₈	—	2	2	no	—
95	C₇×#104	—	2	4	no	—
96	C₇×#104	—	4	2	no	—
97	C₇×#103	C₂×C₄	1	2	yes	—
98	C₇×#103	C₂×C₄	1	2	yes	—
99	C₇×#102	C₂×C₄	1	2	yes	—
100	C₇×#102	C₂×C₄	1	2	yes	—
101	C₇×C₂²⋊C₈	—	2	2	no	—
102	C₇×C₂²⋊C₈	—	2	2	no	—
103	C₇×C₂²⋊C₈	D₄	1	2	yes	—
104	C₇×C₂²⋊C₈	D₄	1	2	yes	—
105	C₇×#97	—	2	2	no	—
106	C₇×#97	—	2	2	no	—
107	C₇×#97	—	2	2	no	—
108	C₄×C₇×C₈	—	2	2	no	—
109	C₇×#95	—	2	1	no	—
110	#3050	C₂×C₇	1	1	yes	characteristic
111	C₂×#444	C₂×C₇	1	1	yes	characteristic, complemented
112	#2933	C₂×C₇	1	2	yes	complemented
113	C₂²×#108	C₂×C₇	1	1	yes	characteristic, complemented
114	#1975	C₂×C₇	1	1	yes	characteristic, complemented
115	#1946	C₂×C₇	1	1	yes	characteristic, complemented
116	#1929	C₂×C₇	1	2	yes	complemented
117	#1871	C₂×C₇	1	2	yes	complemented
118	#1866	C₂×C₇	1	2	yes	complemented
119	#1773	C₂×C₇	1	1	yes	characteristic, complemented
120	#1752	C₂×C₇	1	1	yes	characteristic, complemented
121	C₂⁴×C₇	#41	1	1	yes	characteristic
122	C₇×#41	—	2	2	no	—
123	C₇×#41	—	2	2	no	—
124	C₇×#41	—	2	2	no	—
125	C₇×#41	—	2	2	no	—
126	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
127	C₂×C₇×D₄	—	2	2	no	—
128	C₂×C₇×D₄	—	2	2	no	—
129	C₂×C₇×D₄	—	2	2	no	—
130	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₇	—	2	1	no	—
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	—	2	1	no	—
137	C₂²×C₄×C₇	—	2	1	no	—
138	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₇	#41	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
144	C₇×Q₁₆	—	2	2	no	—
145	C₇×SD₁₆	—	2	4	no	—
146	C₇×D₈	—	2	2	no	—
147	C₇×M₁₆	—	2	2	no	—
148	C₇×M₁₆	—	2	2	no	—
149	C₇×M₁₆	—	2	2	no	—
150	C₇×M₁₆	—	4	4	no	—
151	C₇×M₁₆	—	4	4	no	—
152	C₇×M₁₆	C₂×D₄	1	4	yes	—
153	C₂×C₇×C₈	—	2	1	no	—
154	C₂×C₇×C₈	—	2	1	no	—
155	C₂×C₇×C₈	—	2	2	no	—
156	C₂×C₇×C₈	—	2	1	no	—
157	C₂×C₇×C₈	—	2	1	no	—
158	C₂×C₇×C₈	—	2	1	no	—
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	2	no	—
165	C₂×C₇×C₈	—	4	4	no	—
166	C₂×C₇×C₈	—	4	4	no	—
167	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
172	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
173	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
174	C₂×C₇×C₈	#41	1	2	yes	—
175	C₇×C₄⋊C₄	—	2	2	no	—
176	C₇×C₄⋊C₄	—	2	2	no	—
177	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
178	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
179	C₇×C₂²⋊C₄	—	2	2	no	—
180	C₇×C₂²⋊C₄	—	2	2	no	—
181	C₇×C₂²⋊C₄	—	2	2	no	—
182	C₇×C₂²⋊C₄	—	2	2	no	—
183	C₇×C₂²⋊C₄	—	2	2	no	—
184	C₇×C₂²⋊C₄	—	4	2	no	—
185	C₄²×C₇	—	2	2	no	—
186	C₄²×C₇	—	2	1	no	—
187	C₄²×C₇	—	2	1	no	—
188	C₄²×C₇	—	2	1	no	—
189	C₄²×C₇	—	2	1	no	—
190	C₂×#135	C₂²×C₇	1	1	yes	characteristic
191	C₂×#131	C₂²×C₇	1	2	yes	—
192	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
193	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
194	#525	C₂²×C₇	1	2	yes	—
195	#467	C₄×C₇	1	1	yes	characteristic, complemented
196	#467	C₄×C₇	1	1	yes	characteristic, complemented
197	#466	C₄×C₇	1	1	yes	characteristic, complemented
198	#466	C₄×C₇	1	1	yes	characteristic, complemented
199	#465	C₄×C₇	1	2	yes	complemented
200	#465	C₄×C₇	1	2	yes	complemented
201	#444	—	2	4	no	—
202	#436	—	2	2	no	—
203	#435	—	2	2	no	—
204	#435	—	2	2	no	—
205	C₈×D₄	—	2	2	no	—
206	C₂×#108	—	2	4	no	—
207	C₂×#108	C₂²×C₇	1	2	yes	—
208	C₂×#108	C₂²×C₇	1	1	yes	characteristic
209	C₂×#108	C₂²×C₇	1	1	yes	characteristic
210	#427	C₂²×C₇	1	1	yes	characteristic
211	#424	C₂²×C₇	1	2	yes	—
212	C₂×#104	C₂²×C₇	1	2	yes	—
213	C₂×#104	C₂²×C₇	1	2	yes	—
214	#416	C₂²×C₇	1	2	yes	—
215	#408	C₂²×C₇	1	2	yes	—
216	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
217	#405	C₂²×C₇	1	2	yes	—
218	C₂×#97	C₂²×C₇	1	1	yes	characteristic
219	C₂×#97	C₂²×C₇	1	1	yes	characteristic
220	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
221	#339	C₂²×C₇	1	2	yes	—
222	#339	C₂²×C₇	1	2	yes	—
223	#336	C₂²×C₇	1	1	yes	characteristic
224	#336	C₂²×C₇	1	1	yes	characteristic
225	C₂³×C₇	—	2	1	no	—
226	C₂³×C₇	—	2	1	no	—
227	C₂³×C₇	—	2	2	no	—
228	C₂³×C₇	—	2	2	no	—
229	C₂³×C₇	—	4	1	no	—
230	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
231	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
232	C₂³×C₇	C₂×#41	1	1	yes	characteristic
233	C₇×Q₈	—	2	2	no	—
234	C₇×Q₈	—	2	2	no	—
235	C₇×D₄	—	2	2	no	—
236	C₇×D₄	—	2	2	no	—
237	C₇×D₄	—	4	2	no	—
238	C₇×D₄	—	4	2	no	—
239	C₇×D₄	—	4	4	no	—
240	C₇×D₄	—	8	2	no	—
241	C₂×C₄×C₇	—	2	1	no	—
242	C₂×C₄×C₇	—	2	1	no	—
243	C₂×C₄×C₇	—	2	1	no	—
244	C₂×C₄×C₇	—	2	1	no	—
245	C₂×C₄×C₇	—	2	1	no	—
246	C₂×C₄×C₇	—	2	1	no	—
247	C₂×C₄×C₇	—	2	1	no	—
248	C₂×C₄×C₇	—	2	1	no	—
249	C₂×C₄×C₇	—	2	1	no	—
250	C₂×C₄×C₇	—	2	1	no	—
251	C₂×C₄×C₇	—	2	2	no	—
252	C₂×C₄×C₇	—	2	2	no	—
253	C₂×C₄×C₇	—	2	2	no	—
254	C₂×C₄×C₇	—	4	2	no	—
255	C₂×C₄×C₇	—	4	2	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	1	no	—
260	C₂×C₄×C₇	—	4	1	no	—
261	C₂×C₄×C₇	—	4	1	no	—
262	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
263	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
264	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
270	C₇×C₈	—	2	1	no	—
271	C₇×C₈	—	2	1	no	—
272	C₇×C₈	—	2	2	no	—
273	C₇×C₈	—	2	4	no	—
274	C₇×C₈	—	2	2	no	—
275	C₇×C₈	—	4	2	no	—
276	C₇×C₈	C₄×D₄	1	2	yes	—
277	C₇×C₈	C₄×D₄	1	2	yes	—
278	C₂×#41	C₂³×C₇	1	2	yes	—
279	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
280	#135	—	2	2	no	—
281	#135	—	2	2	no	—
282	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
283	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
284	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
285	#131	—	2	4	no	—
286	#131	—	4	2	no	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	4	2	no	—
291	C₂×M₁₆	C₂³×C₇	1	2	yes	—
292	C₂×M₁₆	C₂³×C₇	1	2	yes	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	—	2	2	no	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
298	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
299	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
300	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
305	#123	—	2	2	no	—
306	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
307	C₂²⋊Q₈	C₂×C₄×C₇	1	2	yes	—
308	#121	C₂×C₄×C₇	1	2	yes	—
309	#121	C₂×C₄×C₇	1	2	yes	—
310	#120	—	2	2	no	—
311	C₄×D₄	—	2	2	no	—
312	C₄×D₄	—	2	2	no	—
313	#117	C₂³×C₇	1	2	yes	—
314	C₂×C₂²⋊C₄	—	2	1	no	—
315	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
317	#108	—	2	4	no	—
318	#108	—	2	2	no	—
319	#108	C₇×D₄	1	4	yes	complemented
320	#107	C₂×C₄×C₇	1	1	yes	characteristic
321	#107	C₂×C₄×C₇	1	1	yes	characteristic
322	#106	C₂×C₄×C₇	1	2	yes	—
323	C₄⋊C₈	—	2	2	no	—
324	C₄⋊C₈	—	2	2	no	—
325	#104	—	2	4	no	—
326	#104	—	4	2	no	—
327	#103	C₂×C₄×C₇	1	2	yes	—
328	#103	C₂×C₄×C₇	1	2	yes	—
329	#102	C₂×C₄×C₇	1	2	yes	—
330	#102	C₂×C₄×C₇	1	2	yes	—
331	C₂²⋊C₈	—	2	2	no	—
332	C₂²⋊C₈	—	2	2	no	—
333	C₂²⋊C₈	C₇×D₄	1	2	yes	—
334	C₂²⋊C₈	C₇×D₄	1	2	yes	—
335	#97	—	2	2	no	—
336	#97	—	2	2	no	—
337	#97	—	2	2	no	—
338	C₄×C₈	—	2	2	no	—
339	#95	—	2	1	no	—
340	C₂²×C₇	—	2	1	no	—
341	C₂²×C₇	—	2	1	no	—
342	C₂²×C₇	—	2	1	no	—
343	C₂²×C₇	—	2	1	no	—
344	C₂²×C₇	—	2	1	no	—
345	C₂²×C₇	—	2	1	no	—
346	C₂²×C₇	—	2	1	no	—
347	C₂²×C₇	—	4	2	no	—
348	C₂²×C₇	—	4	2	no	—
349	C₂²×C₇	—	4	2	no	—
350	C₂²×C₇	—	4	2	no	—
351	C₂²×C₇	—	4	2	no	—
352	C₂²×C₇	#444	1	2	yes	complemented
353	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
354	C₂²×C₇	#548	1	1	yes	characteristic
355	C₂²×C₇	#553	1	1	yes	characteristic
356	C₄×C₇	—	2	1	no	—
357	C₄×C₇	—	2	1	no	—
358	C₄×C₇	—	4	2	no	—
359	C₄×C₇	—	4	2	no	—
360	C₄×C₇	—	4	2	no	—
361	C₄×C₇	—	8	1	no	—
362	C₄×C₇	—	8	1	no	—
363	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
364	C₄×C₇	#520	1	1	yes	characteristic
365	C₄×C₇	D₄²	1	1	yes	characteristic
366	C₄×C₇	#547	1	1	yes	characteristic
367	C₂⁴	C₇×#41	1	1	yes	characteristic
368	#41	—	2	2	no	—
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	#41	—	2	2	no	—
372	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
385	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
386	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
387	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
388	C₂²×C₄	C₇×#41	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
390	Q₁₆	—	2	2	no	—
391	SD₁₆	—	2	4	no	—
392	D₈	—	2	2	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	4	4	no	—
397	M₁₆	—	4	4	no	—
398	M₁₆	C₂×C₇×D₄	1	4	yes	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	4	4	no	—
412	C₂×C₈	—	4	4	no	—
413	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
414	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
415	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
416	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
417	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
418	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
419	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
420	C₂×C₈	C₇×#41	1	2	yes	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
424	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	4	2	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₂×C₇	—	2	1	no	—
437	C₂×C₇	—	2	1	no	—
438	C₂×C₇	—	2	2	no	—
439	C₂×C₇	—	4	1	no	—
440	C₂×C₇	—	8	2	no	—
441	C₂×C₇	#2361	1	1	yes	characteristic
442	C₂×C₇	C₂×#444	1	1	yes	characteristic
443	C₂×C₇	#2957	1	1	yes	characteristic
444	C₂³	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	1	no	—
449	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
450	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
451	C₂³	C₂×C₇×#41	1	1	yes	characteristic
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	4	no	—
459	D₄	—	8	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	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	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
482	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
483	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
484	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
486	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
487	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
488	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
489	C₈	—	2	1	no	—
490	C₈	—	2	1	no	—
491	C₈	—	2	2	no	—
492	C₈	—	2	4	no	—
493	C₈	—	2	2	no	—
494	C₈	—	4	2	no	—
495	C₈	C₄×C₇×D₄	1	2	yes	—
496	C₈	C₄×C₇×D₄	1	2	yes	—
497	C₇	#23019	1	1	yes	characteristic, direct product factor, Sylow
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	C₇×#444	1	2	yes	complemented
511	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
512	C₂²	C₇×#548	1	1	yes	characteristic
513	C₂²	C₇×#553	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	8	1	no	—
520	C₄	—	8	1	no	—
521	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
522	C₄	C₇×#520	1	1	yes	characteristic
523	C₄	C₇×D₄²	1	1	yes	characteristic
524	C₄	C₇×#547	1	1	yes	characteristic
525	C₂	—	2	1	no	—
526	C₂	—	2	1	no	—
527	C₂	—	2	2	no	—
528	C₂	—	4	1	no	—
529	C₂	—	8	2	no	—
530	C₂	C₇×#2361	1	1	yes	characteristic
531	C₂	C₂×C₇×#444	1	1	yes	characteristic
532	C₂	C₇×#2957	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×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#444	C₂	semidirect
3. C₇×#2933	C₂	semidirect
4. C₂²×C₇×#108	C₂	semidirect
5. C₇×#1975	C₂	semidirect
6. C₇×#1946	C₂	semidirect
7. C₇×#1929	C₂	semidirect
8. C₇×#1871	C₂	semidirect
9. C₇×#1866	C₂	semidirect
10. C₇×#1773	C₂	semidirect
11. C₇×#1752	C₂	semidirect
17. C₇×#467	C₄	semidirect
18. C₇×#467	C₄	semidirect
19. C₇×#466	C₄	semidirect
20. C₇×#466	C₄	semidirect
21. C₇×#465	C₄	semidirect
22. C₇×#465	C₄	semidirect
47. #23019	C₇	direct (N × H)
89. C₇×#108	D₄	semidirect
111. C₂×#444	C₂×C₇	semidirect
112. #2933	C₂×C₇	semidirect
113. C₂²×#108	C₂×C₇	semidirect
114. #1975	C₂×C₇	semidirect
115. #1946	C₂×C₇	semidirect
116. #1929	C₂×C₇	semidirect
117. #1871	C₂×C₇	semidirect
118. #1866	C₂×C₇	semidirect
119. #1773	C₂×C₇	semidirect
120. #1752	C₂×C₇	semidirect
195. #467	C₄×C₇	semidirect
196. #467	C₄×C₇	semidirect
197. #466	C₄×C₇	semidirect
198. #466	C₄×C₇	semidirect
199. #465	C₄×C₇	semidirect
200. #465	C₄×C₇	semidirect
319. #108	C₇×D₄	semidirect
352. C₂²×C₇	#444	semidirect
497. C₇	#23019	direct (N × H)
510. C₂²	C₇×#444	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3050	2
C₂×C₇×#444	2
C₇×#2933	2
C₂²×C₇×#108	2
C₇×#1975	2
C₇×#1946	2
C₇×#1929	2
C₇×#1871	2
C₇×#1866	2
C₇×#1773	2
C₇×#1752	2
#23019	7

Subgroups — C7×#23019 #13200235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23019 #13200235