Subgroups — C₇×#27070 #13204286

Order 1792.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2514	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2367	C₂	1	1	yes	maximal, characteristic
3	C₇×#2092	C₂	1	1	yes	maximal, characteristic
4	C₇×#2055	C₂	1	1	yes	maximal, characteristic
5	C₇×#2042	C₂	1	1	yes	maximal, characteristic
6	C₇×#2015	C₂	1	1	yes	maximal, characteristic
7	C₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1927	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1927	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1921	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1878	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1877	C₂	1	1	yes	maximal, characteristic
14	C₇×#1805	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
16	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₇×#124	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×#107	C₂²	1	1	yes	characteristic
22	C₂×C₇×#106	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×#103	C₂²	1	1	yes	characteristic
25	C₂×C₇×#103	C₂²	1	1	yes	characteristic
26	C₂×C₇×#103	C₂²	1	1	yes	characteristic
27	C₂×C₇×#103	C₂²	1	1	yes	characteristic
28	C₂×C₇×#103	C₂²	1	1	yes	characteristic
29	C₂×C₇×#102	C₂²	1	1	yes	characteristic
30	C₂×C₇×#102	C₂²	1	1	yes	characteristic
31	C₂×C₇×#102	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
33	C₇×#401	—	2	1	no	—
34	C₇×#400	—	2	1	no	—
35	C₇×#391	C₂²	1	1	yes	characteristic
36	C₇×#391	C₂²	1	1	yes	characteristic
37	C₇×#391	C₂²	1	1	yes	characteristic
38	C₇×#391	C₂²	1	1	yes	characteristic
39	C₇×#390	C₂²	1	1	yes	characteristic
40	C₇×#389	C₂²	1	1	yes	characteristic
41	C₇×#388	—	2	1	no	—
42	C₇×#388	—	2	1	no	—
43	C₇×#387	—	2	1	no	—
44	C₇×#385	—	2	1	no	—
45	C₇×#383	—	2	1	no	—
46	C₇×#382	C₂²	1	1	yes	characteristic
47	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
50	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
51	C₇×#340	C₂²	1	1	yes	characteristic
52	C₇×#340	C₂²	1	1	yes	characteristic
53	C₇×#340	C₂²	1	1	yes	characteristic
54	C₇×#340	C₂²	1	1	yes	characteristic
55	C₇×#336	C₂²	1	1	yes	characteristic
56	C₇×#336	C₂²	1	1	yes	characteristic
57	C₇×#336	C₂²	1	1	yes	characteristic
58	C₇×#336	C₂²	1	1	yes	characteristic
59	#27070	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
61	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₇	—	2	1	no	—
64	C₂×C₇×Q₁₆	—	2	2	no	—
65	C₂×C₇×Q₁₆	—	2	2	no	—
66	C₂×C₇×Q₁₆	—	4	1	no	—
67	C₂×C₇×SD₁₆	—	2	2	no	—
68	C₂×C₇×SD₁₆	—	4	1	no	—
69	C₂×C₇×SD₁₆	—	4	1	no	—
70	C₂²×C₇×C₈	C₂³	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	C₇×#124	—	2	2	no	—
75	C₇×#124	—	2	2	no	—
76	C₄×C₇×Q₈	—	2	2	no	—
77	C₄×C₇×Q₈	—	4	1	no	—
78	C₄×C₇×D₄	—	2	2	no	—
79	C₄×C₇×D₄	—	4	1	no	—
80	C₂×C₇×C₄⋊C₄	—	2	1	no	—
81	C₂×C₇×C₄⋊C₄	—	2	1	no	—
82	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
83	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
84	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
86	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
90	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
92	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
93	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
94	C₂×C₄²×C₇	—	2	1	no	—
95	C₂×C₄²×C₇	—	2	1	no	—
96	C₂×C₄²×C₇	—	2	1	no	—
97	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
100	C₇×#107	—	2	2	no	—
101	C₇×#106	—	4	1	no	—
102	C₇×C₄⋊C₈	—	4	1	no	—
103	C₇×#103	—	2	2	no	—
104	C₇×#103	—	2	2	no	—
105	C₇×#103	—	4	1	no	—
106	C₇×#103	—	4	1	no	—
107	C₇×#103	—	4	1	no	—
108	C₇×#102	—	2	2	no	—
109	C₇×#102	—	2	2	no	—
110	C₇×#102	—	4	1	no	—
111	C₄×C₇×C₈	—	4	1	no	—
112	C₇×#95	—	2	1	no	—
113	C₇×#95	—	2	1	no	—
114	C₇×#95	—	2	1	no	—
115	C₇×#95	—	2	1	no	—
116	C₇×#95	—	2	1	no	—
117	C₇×#95	—	2	1	no	—
118	C₇×#95	—	2	1	no	—
119	C₇×#95	—	2	1	no	—
120	C₇×#95	—	2	1	no	—
121	C₇×#95	—	2	1	no	—
122	C₇×#95	—	2	1	no	—
123	C₇×#95	—	2	1	no	—
124	C₇×#95	—	2	1	no	—
125	C₇×#95	D₄	1	1	yes	characteristic
126	C₇×#95	D₄	1	1	yes	characteristic
127	#2514	C₂×C₇	1	1	yes	characteristic, complemented
128	#2367	C₂×C₇	1	1	yes	characteristic
129	#2092	C₂×C₇	1	1	yes	characteristic
130	#2055	C₂×C₇	1	1	yes	characteristic
131	#2042	C₂×C₇	1	1	yes	characteristic
132	#2015	C₂×C₇	1	1	yes	characteristic
133	#1969	C₂×C₇	1	1	yes	characteristic, complemented
134	#1927	C₂×C₇	1	1	yes	characteristic, complemented
135	#1927	C₂×C₇	1	1	yes	characteristic, complemented
136	#1921	C₂×C₇	1	1	yes	characteristic, complemented
137	#1916	C₂×C₇	1	1	yes	characteristic, complemented
138	#1878	C₂×C₇	1	1	yes	characteristic, complemented
139	#1877	C₂×C₇	1	1	yes	characteristic
140	#1805	C₂×C₇	1	1	yes	characteristic, complemented
141	C₄×#102	C₂×C₇	1	1	yes	characteristic
142	C₂⁴×C₇	—	2	1	no	—
143	C₂×C₇×Q₈	—	2	1	no	—
144	C₂×C₇×Q₈	—	2	1	no	—
145	C₂×C₇×Q₈	—	2	1	no	—
146	C₂×C₇×Q₈	—	2	1	no	—
147	C₂×C₇×Q₈	—	2	1	no	—
148	C₂×C₇×Q₈	—	2	1	no	—
149	C₂×C₇×Q₈	—	2	1	no	—
150	C₂×C₇×Q₈	—	2	1	no	—
151	C₂×C₇×Q₈	—	2	1	no	—
152	C₂×C₇×Q₈	—	2	1	no	—
153	C₂×C₇×Q₈	—	2	1	no	—
154	C₂×C₇×Q₈	#41	1	2	yes	—
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	1	no	—
159	C₂×C₇×D₄	—	2	1	no	—
160	C₂×C₇×D₄	#41	1	2	yes	—
161	C₂²×C₄×C₇	—	2	1	no	—
162	C₂²×C₄×C₇	—	2	1	no	—
163	C₂²×C₄×C₇	—	2	1	no	—
164	C₂²×C₄×C₇	—	2	1	no	—
165	C₂²×C₄×C₇	—	2	1	no	—
166	C₂²×C₄×C₇	—	2	1	no	—
167	C₂²×C₄×C₇	—	2	1	no	—
168	C₂²×C₄×C₇	—	4	2	no	—
169	C₂²×C₄×C₇	—	4	2	no	—
170	C₂²×C₄×C₇	—	8	1	no	—
171	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
178	C₇×Q₁₆	—	4	4	no	—
179	C₇×SD₁₆	—	4	4	no	—
180	C₂×C₇×C₈	—	2	1	no	—
181	C₂×C₇×C₈	—	2	1	no	—
182	C₂×C₇×C₈	—	2	1	no	—
183	C₂×C₇×C₈	—	2	1	no	—
184	C₂×C₇×C₈	—	2	1	no	—
185	C₂×C₇×C₈	—	2	1	no	—
186	C₂×C₇×C₈	—	2	1	no	—
187	C₂×C₇×C₈	—	2	1	no	—
188	C₂×C₇×C₈	—	2	1	no	—
189	C₂×C₇×C₈	—	2	1	no	—
190	C₂×C₇×C₈	—	2	1	no	—
191	C₂×C₇×C₈	#41	1	2	yes	—
192	C₇×C₄⋊C₄	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	1	no	—
194	C₇×C₄⋊C₄	—	2	1	no	—
195	C₇×C₄⋊C₄	—	2	1	no	—
196	C₇×C₄⋊C₄	—	2	1	no	—
197	C₇×C₄⋊C₄	—	2	1	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	1	no	—
200	C₇×C₄⋊C₄	—	2	1	no	—
201	C₇×C₄⋊C₄	—	2	1	no	—
202	C₇×C₄⋊C₄	—	4	1	no	—
203	C₇×C₄⋊C₄	—	4	2	no	—
204	C₇×C₄⋊C₄	—	4	2	no	—
205	C₇×C₄⋊C₄	#41	1	2	yes	—
206	C₇×C₂²⋊C₄	—	2	4	no	—
207	C₇×C₂²⋊C₄	—	2	4	no	—
208	C₇×C₂²⋊C₄	—	4	2	no	—
209	C₇×C₂²⋊C₄	—	4	2	no	—
210	C₇×C₂²⋊C₄	—	8	1	no	—
211	C₄²×C₇	—	2	1	no	—
212	C₄²×C₇	—	2	1	no	—
213	C₄²×C₇	—	2	1	no	—
214	C₄²×C₇	—	2	1	no	—
215	C₄²×C₇	—	2	1	no	—
216	C₄²×C₇	—	2	1	no	—
217	C₄²×C₇	—	4	2	no	—
218	C₄²×C₇	—	4	2	no	—
219	C₄²×C₇	—	8	1	no	—
220	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
221	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
222	C₂×#124	C₂²×C₇	1	1	yes	characteristic
223	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
224	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
225	C₂×#107	C₂²×C₇	1	1	yes	characteristic
226	C₂×#106	C₂²×C₇	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
228	C₂×#103	C₂²×C₇	1	1	yes	characteristic
229	C₂×#103	C₂²×C₇	1	1	yes	characteristic
230	C₂×#103	C₂²×C₇	1	1	yes	characteristic
231	C₂×#103	C₂²×C₇	1	1	yes	characteristic
232	C₂×#103	C₂²×C₇	1	1	yes	characteristic
233	C₂×#102	C₂²×C₇	1	1	yes	characteristic
234	C₂×#102	C₂²×C₇	1	1	yes	characteristic
235	C₂×#102	C₂²×C₇	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
237	#401	—	2	1	no	—
238	#400	—	2	1	no	—
239	#391	C₂²×C₇	1	1	yes	characteristic
240	#391	C₂²×C₇	1	1	yes	characteristic
241	#391	C₂²×C₇	1	1	yes	characteristic
242	#391	C₂²×C₇	1	1	yes	characteristic
243	#390	C₂²×C₇	1	1	yes	characteristic
244	#389	C₂²×C₇	1	1	yes	characteristic
245	#388	—	2	1	no	—
246	#388	—	2	1	no	—
247	#387	—	2	1	no	—
248	#385	—	2	1	no	—
249	#383	—	2	1	no	—
250	#382	C₂²×C₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
252	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
253	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
254	C₄×C₂²⋊C₄	—	2	1	no	—
255	#340	C₂²×C₇	1	1	yes	characteristic
256	#340	C₂²×C₇	1	1	yes	characteristic
257	#340	C₂²×C₇	1	1	yes	characteristic
258	#340	C₂²×C₇	1	1	yes	characteristic
259	#336	C₂²×C₇	1	1	yes	characteristic
260	#336	C₂²×C₇	1	1	yes	characteristic
261	#336	C₂²×C₇	1	1	yes	characteristic
262	#336	C₂²×C₇	1	1	yes	characteristic
263	C₂³×C₇	—	2	2	no	—
264	C₂³×C₇	—	4	1	no	—
265	C₂³×C₇	—	4	1	no	—
266	C₂³×C₇	—	4	1	no	—
267	C₂³×C₇	—	4	1	no	—
268	C₂³×C₇	—	4	1	no	—
269	C₂³×C₇	—	4	1	no	—
270	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
271	C₇×Q₈	—	2	2	no	—
272	C₇×Q₈	—	2	2	no	—
273	C₇×Q₈	—	4	1	no	—
274	C₇×Q₈	—	4	1	no	—
275	C₇×Q₈	—	4	1	no	—
276	C₇×Q₈	—	4	1	no	—
277	C₇×Q₈	—	4	1	no	—
278	C₇×Q₈	—	4	1	no	—
279	C₇×D₄	—	2	2	no	—
280	C₇×D₄	—	2	2	no	—
281	C₇×D₄	—	4	1	no	—
282	C₇×D₄	—	4	1	no	—
283	C₂×C₄×C₇	—	2	1	no	—
284	C₂×C₄×C₇	—	2	1	no	—
285	C₂×C₄×C₇	—	2	1	no	—
286	C₂×C₄×C₇	—	2	1	no	—
287	C₂×C₄×C₇	—	2	2	no	—
288	C₂×C₄×C₇	—	2	1	no	—
289	C₂×C₄×C₇	—	2	1	no	—
290	C₂×C₄×C₇	—	2	1	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	2	1	no	—
293	C₂×C₄×C₇	—	2	1	no	—
294	C₂×C₄×C₇	—	2	1	no	—
295	C₂×C₄×C₇	—	2	1	no	—
296	C₂×C₄×C₇	—	2	1	no	—
297	C₂×C₄×C₇	—	2	1	no	—
298	C₂×C₄×C₇	—	2	1	no	—
299	C₂×C₄×C₇	—	2	1	no	—
300	C₂×C₄×C₇	—	2	1	no	—
301	C₂×C₄×C₇	—	2	1	no	—
302	C₂×C₄×C₇	—	4	1	no	—
303	C₂×C₄×C₇	—	4	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	1	no	—
307	C₂×C₄×C₇	—	4	1	no	—
308	C₂×C₄×C₇	—	4	1	no	—
309	C₂×C₄×C₇	—	4	1	no	—
310	C₂×C₄×C₇	—	4	1	no	—
311	C₂×C₄×C₇	—	4	1	no	—
312	C₂×C₄×C₇	—	4	1	no	—
313	C₂×C₄×C₇	—	4	1	no	—
314	C₂×C₄×C₇	—	4	1	no	—
315	C₂×C₄×C₇	—	4	1	no	—
316	C₂×C₄×C₇	—	4	1	no	—
317	C₂×C₄×C₇	—	4	1	no	—
318	C₂×C₄×C₇	—	4	1	no	—
319	C₂×C₄×C₇	—	4	1	no	—
320	C₂×C₄×C₇	—	4	1	no	—
321	C₂×C₄×C₇	—	8	4	no	—
322	C₂×C₄×C₇	#135	1	2	yes	—
323	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
324	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
325	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
326	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
327	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
328	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
329	C₇×C₈	—	2	2	no	—
330	C₇×C₈	—	4	1	no	—
331	C₇×C₈	—	4	1	no	—
332	C₇×C₈	—	4	1	no	—
333	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
334	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
335	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
336	C₂³×C₄	—	2	1	no	—
337	C₂×Q₁₆	—	2	2	no	—
338	C₂×Q₁₆	—	2	2	no	—
339	C₂×Q₁₆	—	4	1	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	4	1	no	—
342	C₂×SD₁₆	—	4	1	no	—
343	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
347	#124	—	2	2	no	—
348	#124	—	2	2	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	4	1	no	—
351	C₄×D₄	—	2	2	no	—
352	C₄×D₄	—	4	1	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
361	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₂²⋊C₄	—	2	1	no	—
366	C₂×C₂²⋊C₄	—	2	1	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
372	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
373	#107	—	2	2	no	—
374	#106	—	4	1	no	—
375	C₄⋊C₈	—	4	1	no	—
376	#103	—	2	2	no	—
377	#103	—	2	2	no	—
378	#103	—	4	1	no	—
379	#103	—	4	1	no	—
380	#103	—	4	1	no	—
381	#102	—	2	2	no	—
382	#102	—	2	2	no	—
383	#102	—	4	1	no	—
384	C₄×C₈	—	4	1	no	—
385	#95	—	2	1	no	—
386	#95	—	2	1	no	—
387	#95	—	2	1	no	—
388	#95	—	2	1	no	—
389	#95	—	2	1	no	—
390	#95	—	2	1	no	—
391	#95	—	2	1	no	—
392	#95	—	2	1	no	—
393	#95	—	2	1	no	—
394	#95	—	2	1	no	—
395	#95	—	2	1	no	—
396	#95	—	2	1	no	—
397	#95	—	2	1	no	—
398	#95	C₇×D₄	1	1	yes	characteristic
399	#95	C₇×D₄	1	1	yes	characteristic
400	C₂²×C₇	—	4	2	no	—
401	C₂²×C₇	—	4	2	no	—
402	C₂²×C₇	—	4	2	no	—
403	C₂²×C₇	—	8	1	no	—
404	C₂²×C₇	—	8	1	no	—
405	C₂²×C₇	—	8	1	no	—
406	C₂²×C₇	—	8	1	no	—
407	C₂²×C₇	#525	1	1	yes	characteristic
408	C₂²×C₇	#546	1	1	yes	characteristic
409	C₂²×C₇	#550	1	1	yes	characteristic
410	C₂²×C₇	C₂×#135	1	1	yes	characteristic
411	C₂²×C₇	#575	1	1	yes	characteristic
412	C₂²×C₇	#577	1	1	yes	characteristic
413	C₂²×C₇	#578	1	1	yes	characteristic
414	C₄×C₇	—	2	2	no	—
415	C₄×C₇	—	4	1	no	—
416	C₄×C₇	—	4	1	no	—
417	C₄×C₇	—	4	2	no	—
418	C₄×C₇	—	4	1	no	—
419	C₄×C₇	—	4	2	no	—
420	C₄×C₇	—	4	1	no	—
421	C₄×C₇	—	4	1	no	—
422	C₄×C₇	—	8	1	no	—
423	C₄×C₇	—	8	1	no	—
424	C₄×C₇	—	8	1	no	—
425	C₄×C₇	—	8	1	no	—
426	C₄×C₇	—	8	1	no	—
427	C₄×C₇	#529	1	1	yes	characteristic
428	C₄×C₇	#538	1	1	yes	characteristic
429	C₄×C₇	#551	1	1	yes	characteristic
430	C₄×C₇	#553	1	1	yes	characteristic
431	C₂⁴	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	C₇×#41	1	2	yes	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	C₇×#41	1	2	yes	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	8	1	no	—
460	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
461	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
462	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
463	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
464	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
465	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
467	Q₁₆	—	4	4	no	—
468	SD₁₆	—	4	4	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	C₇×#41	1	2	yes	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	—	4	2	no	—
493	C₄⋊C₄	—	4	2	no	—
494	C₄⋊C₄	C₇×#41	1	2	yes	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	4	2	no	—
498	C₂²⋊C₄	—	4	2	no	—
499	C₂²⋊C₄	—	8	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	4	2	no	—
507	C₄²	—	4	2	no	—
508	C₄²	—	8	1	no	—
509	C₂×C₇	—	8	2	no	—
510	C₂×C₇	#2467	1	1	yes	characteristic
511	C₂×C₇	#3102	1	1	yes	characteristic
512	C₂×C₇	#3120	1	1	yes	characteristic
513	C₂×C₇	#3296	1	1	yes	characteristic
514	C₂×C₇	#3313	1	1	yes	characteristic
515	C₂×C₇	#3393	1	1	yes	characteristic
516	C₂×C₇	#3401	1	1	yes	characteristic
517	C₂³	—	2	2	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	1	no	—
520	C₂³	—	4	1	no	—
521	C₂³	—	4	1	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	4	1	no	—
524	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
525	Q₈	—	2	2	no	—
526	Q₈	—	2	2	no	—
527	Q₈	—	4	1	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	4	1	no	—
530	Q₈	—	4	1	no	—
531	Q₈	—	4	1	no	—
532	Q₈	—	4	1	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	2	no	—
535	D₄	—	4	1	no	—
536	D₄	—	4	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	8	4	no	—
576	C₂×C₄	C₇×#135	1	2	yes	—
577	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
578	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
579	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
580	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
581	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
582	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
583	C₈	—	2	2	no	—
584	C₈	—	4	1	no	—
585	C₈	—	4	1	no	—
586	C₈	—	4	1	no	—
587	C₇	#27070	1	1	yes	characteristic, direct product factor, Sylow
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	8	1	no	—
592	C₂²	—	8	1	no	—
593	C₂²	—	8	1	no	—
594	C₂²	—	8	1	no	—
595	C₂²	C₇×#525	1	1	yes	characteristic
596	C₂²	C₇×#546	1	1	yes	characteristic
597	C₂²	C₇×#550	1	1	yes	characteristic
598	C₂²	C₂×C₇×#135	1	1	yes	characteristic
599	C₂²	C₇×#575	1	1	yes	characteristic
600	C₂²	C₇×#577	1	1	yes	characteristic
601	C₂²	C₇×#578	1	1	yes	characteristic
602	C₄	—	2	2	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	1	no	—
609	C₄	—	4	1	no	—
610	C₄	—	8	1	no	—
611	C₄	—	8	1	no	—
612	C₄	—	8	1	no	—
613	C₄	—	8	1	no	—
614	C₄	—	8	1	no	—
615	C₄	C₇×#529	1	1	yes	characteristic
616	C₄	C₇×#538	1	1	yes	characteristic
617	C₄	C₇×#551	1	1	yes	characteristic
618	C₄	C₇×#553	1	1	yes	characteristic
619	C₂	—	8	2	no	—
620	C₂	C₇×#2467	1	1	yes	characteristic
621	C₂	C₇×#3102	1	1	yes	characteristic
622	C₂	C₇×#3120	1	1	yes	characteristic
623	C₂	C₇×#3296	1	1	yes	characteristic
624	C₂	C₇×#3313	1	1	yes	characteristic
625	C₂	C₇×#3393	1	1	yes	characteristic
626	C₂	C₇×#3401	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₇×#2514	C₂	semidirect
7. C₇×#1969	C₂	semidirect
8. C₇×#1927	C₂	semidirect
9. C₇×#1927	C₂	semidirect
10. C₇×#1921	C₂	semidirect
11. C₇×#1916	C₂	semidirect
12. C₇×#1878	C₂	semidirect
14. C₇×#1805	C₂	semidirect
59. #27070	C₇	direct (N × H)
127. #2514	C₂×C₇	semidirect
133. #1969	C₂×C₇	semidirect
134. #1927	C₂×C₇	semidirect
135. #1927	C₂×C₇	semidirect
136. #1921	C₂×C₇	semidirect
137. #1916	C₂×C₇	semidirect
138. #1878	C₂×C₇	semidirect
140. #1805	C₂×C₇	semidirect
587. C₇	#27070	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×#2514	2
C₇×#2367	2
C₇×#2092	2
C₇×#2055	2
C₇×#2042	2
C₇×#2015	2
C₇×#1969	2
C₇×#1927	2
C₇×#1921	2
C₇×#1916	2
C₇×#1878	2
C₇×#1877	2
C₇×#1805	2
C₄×C₇×#102	2
#27070	7

Subgroups — C7×#27070 #13204286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#27070 #13204286