Subgroups — C₇×#27129 #13204345

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2513	C₂	1	1	yes	maximal, characteristic
2	C₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2065	C₂	1	1	yes	maximal, characteristic
5	C₇×#2058	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2023	C₂	1	1	yes	maximal, characteristic
7	C₇×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1934	C₂	1	1	yes	maximal, characteristic
9	C₇×#1926	C₂	1	1	yes	maximal, characteristic
10	C₇×#1926	C₂	1	1	yes	maximal, characteristic
11	C₇×#1925	C₂	1	1	yes	maximal, characteristic
12	C₇×#1921	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1919	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×Q₈	—	2	1	no	—
18	C₂×C₇×#107	C₂²	1	1	yes	characteristic
19	C₂×C₇×#106	C₂²	1	1	yes	characteristic
20	C₂×C₇×#106	C₂²	1	1	yes	characteristic
21	C₂×C₇×#106	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₇×#102	C₂²	1	1	yes	characteristic
24	C₂×C₇×#102	C₂²	1	1	yes	characteristic
25	C₂×C₇×#102	C₂²	1	1	yes	characteristic
26	C₂×C₇×#102	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#397	—	2	1	no	—
29	C₇×#397	—	2	1	no	—
30	C₇×#391	—	2	1	no	—
31	C₇×#390	C₂²	1	1	yes	characteristic
32	C₇×#390	C₂²	1	1	yes	characteristic
33	C₇×#390	C₂²	1	1	yes	characteristic
34	C₇×#389	C₂²	1	1	yes	characteristic
35	C₇×#389	C₂²	1	1	yes	characteristic
36	C₇×#389	C₂²	1	1	yes	characteristic
37	C₇×#388	—	2	1	no	—
38	C₇×#388	—	2	1	no	—
39	C₇×#387	—	2	1	no	—
40	C₇×#387	—	2	1	no	—
41	C₇×#382	C₂²	1	1	yes	characteristic
42	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
45	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
46	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
47	C₇×#340	C₂²	1	1	yes	characteristic
48	C₇×#340	C₂²	1	1	yes	characteristic
49	C₇×#340	C₂²	1	1	yes	characteristic
50	C₇×#340	C₂²	1	1	yes	characteristic
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	#27129	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₇×Q₈	—	2	1	no	—
61	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
62	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₇×C₄⋊Q₈	—	2	2	no	—
67	C₇×C₄⋊Q₈	—	2	2	no	—
68	C₄×C₇×Q₈	—	4	4	no	—
69	C₂×C₇×C₄⋊C₄	—	2	1	no	—
70	C₂×C₇×C₄⋊C₄	—	2	1	no	—
71	C₂×C₇×C₄⋊C₄	—	2	1	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
75	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
83	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
84	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
85	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
86	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
87	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
88	C₂×C₄²×C₇	—	2	1	no	—
89	C₂×C₄²×C₇	—	2	1	no	—
90	C₂×C₄²×C₇	—	2	1	no	—
91	C₂×C₄²×C₇	—	2	1	no	—
92	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
95	C₇×#107	—	2	2	no	—
96	C₇×#106	—	2	2	no	—
97	C₇×#106	—	2	2	no	—
98	C₇×#106	—	2	2	no	—
99	C₇×C₄⋊C₈	—	2	2	no	—
100	C₇×#102	—	2	2	no	—
101	C₇×#102	—	2	2	no	—
102	C₇×#102	—	2	2	no	—
103	C₇×#102	—	2	2	no	—
104	C₄×C₇×C₈	—	2	2	no	—
105	C₇×#95	—	2	1	no	—
106	C₇×#95	—	2	1	no	—
107	C₇×#95	—	2	1	no	—
108	C₇×#95	—	2	1	no	—
109	C₇×#95	—	2	1	no	—
110	C₇×#95	—	2	1	no	—
111	C₇×#95	—	2	1	no	—
112	C₇×#95	—	2	1	no	—
113	C₇×#95	—	2	1	no	—
114	C₇×#95	D₄	1	1	yes	characteristic
115	C₇×#95	D₄	1	1	yes	characteristic
116	#2513	C₂×C₇	1	1	yes	characteristic
117	#2365	C₂×C₇	1	1	yes	characteristic, complemented
118	#2101	C₂×C₇	1	1	yes	characteristic, complemented
119	#2065	C₂×C₇	1	1	yes	characteristic
120	#2058	C₂×C₇	1	1	yes	characteristic, complemented
121	#2023	C₂×C₇	1	1	yes	characteristic
122	#1949	C₂×C₇	1	1	yes	characteristic, complemented
123	#1934	C₂×C₇	1	1	yes	characteristic
124	#1926	C₂×C₇	1	1	yes	characteristic
125	#1926	C₂×C₇	1	1	yes	characteristic
126	#1925	C₂×C₇	1	1	yes	characteristic
127	#1921	C₂×C₇	1	1	yes	characteristic, complemented
128	#1919	C₂×C₇	1	1	yes	characteristic, complemented
129	#1916	C₂×C₇	1	1	yes	characteristic, complemented
130	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
131	C₂⁴×C₇	—	2	1	no	—
132	C₂×C₇×Q₈	—	2	4	no	—
133	C₂×C₇×Q₈	—	4	2	no	—
134	C₂×C₇×Q₈	—	4	2	no	—
135	C₂×C₇×D₄	—	2	1	no	—
136	C₂×C₇×D₄	—	2	1	no	—
137	C₂×C₇×D₄	—	2	1	no	—
138	C₂×C₇×D₄	—	2	1	no	—
139	C₂×C₇×D₄	—	2	1	no	—
140	C₂×C₇×D₄	—	2	1	no	—
141	C₂²×C₄×C₇	—	2	1	no	—
142	C₂²×C₄×C₇	—	2	1	no	—
143	C₂²×C₄×C₇	—	2	1	no	—
144	C₂²×C₄×C₇	—	2	1	no	—
145	C₂²×C₄×C₇	—	2	1	no	—
146	C₂²×C₄×C₇	—	2	1	no	—
147	C₂²×C₄×C₇	—	2	1	no	—
148	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
155	C₂×C₇×C₈	—	2	1	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	1	no	—
161	C₂×C₇×C₈	—	2	1	no	—
162	C₂×C₇×C₈	—	2	1	no	—
163	C₂×C₇×C₈	—	2	1	no	—
164	C₂×C₇×C₈	—	2	1	no	—
165	C₂×C₇×C₈	—	2	1	no	—
166	C₂×C₇×C₈	#41	1	2	yes	—
167	C₇×C₄⋊C₄	—	2	4	no	—
168	C₇×C₄⋊C₄	—	2	2	no	—
169	C₇×C₄⋊C₄	—	2	2	no	—
170	C₇×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
176	C₇×C₄⋊C₄	—	2	1	no	—
177	C₇×C₄⋊C₄	—	2	1	no	—
178	C₇×C₄⋊C₄	—	2	1	no	—
179	C₇×C₄⋊C₄	—	2	1	no	—
180	C₇×C₄⋊C₄	—	2	1	no	—
181	C₇×C₄⋊C₄	—	4	2	no	—
182	C₇×C₄⋊C₄	—	4	2	no	—
183	C₇×C₄⋊C₄	—	8	1	no	—
184	C₇×C₄⋊C₄	#41	1	2	yes	—
185	C₇×C₄⋊C₄	#41	1	2	yes	—
186	C₇×C₄⋊C₄	#41	1	2	yes	—
187	C₇×C₂²⋊C₄	—	4	2	no	—
188	C₇×C₂²⋊C₄	—	4	2	no	—
189	C₇×C₂²⋊C₄	—	8	1	no	—
190	C₇×C₂²⋊C₄	—	8	1	no	—
191	C₇×C₂²⋊C₄	—	8	1	no	—
192	C₇×C₂²⋊C₄	—	8	1	no	—
193	C₄²×C₇	—	2	1	no	—
194	C₄²×C₇	—	2	1	no	—
195	C₄²×C₇	—	2	1	no	—
196	C₄²×C₇	—	2	1	no	—
197	C₄²×C₇	—	2	1	no	—
198	C₄²×C₇	—	2	1	no	—
199	C₄²×C₇	—	4	2	no	—
200	C₄²×C₇	—	4	2	no	—
201	C₄²×C₇	—	4	2	no	—
202	C₄²×C₇	—	8	1	no	—
203	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
204	C₂×C₄×Q₈	—	2	1	no	—
205	C₂×#107	C₂²×C₇	1	1	yes	characteristic
206	C₂×#106	C₂²×C₇	1	1	yes	characteristic
207	C₂×#106	C₂²×C₇	1	1	yes	characteristic
208	C₂×#106	C₂²×C₇	1	1	yes	characteristic
209	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
210	C₂×#102	C₂²×C₇	1	1	yes	characteristic
211	C₂×#102	C₂²×C₇	1	1	yes	characteristic
212	C₂×#102	C₂²×C₇	1	1	yes	characteristic
213	C₂×#102	C₂²×C₇	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
215	#397	—	2	1	no	—
216	#397	—	2	1	no	—
217	#391	—	2	1	no	—
218	#390	C₂²×C₇	1	1	yes	characteristic
219	#390	C₂²×C₇	1	1	yes	characteristic
220	#390	C₂²×C₇	1	1	yes	characteristic
221	#389	C₂²×C₇	1	1	yes	characteristic
222	#389	C₂²×C₇	1	1	yes	characteristic
223	#389	C₂²×C₇	1	1	yes	characteristic
224	#388	—	2	1	no	—
225	#388	—	2	1	no	—
226	#387	—	2	1	no	—
227	#387	—	2	1	no	—
228	#382	C₂²×C₇	1	1	yes	characteristic
229	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
230	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
231	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
232	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
233	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
234	#340	C₂²×C₇	1	1	yes	characteristic
235	#340	C₂²×C₇	1	1	yes	characteristic
236	#340	C₂²×C₇	1	1	yes	characteristic
237	#340	C₂²×C₇	1	1	yes	characteristic
238	#340	C₂²×C₇	1	1	yes	characteristic
239	#340	C₂²×C₇	1	1	yes	characteristic
240	#340	C₂²×C₇	1	1	yes	characteristic
241	#340	C₂²×C₇	1	1	yes	characteristic
242	#336	C₂²×C₇	1	1	yes	characteristic
243	#336	C₂²×C₇	1	1	yes	characteristic
244	#336	C₂²×C₇	1	1	yes	characteristic
245	#336	C₂²×C₇	1	1	yes	characteristic
246	C₂³×C₇	—	4	1	no	—
247	C₂³×C₇	—	4	1	no	—
248	C₂³×C₇	—	4	1	no	—
249	C₂³×C₇	—	4	1	no	—
250	C₂³×C₇	—	4	1	no	—
251	C₂³×C₇	—	4	1	no	—
252	C₂³×C₇	—	4	1	no	—
253	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
254	C₇×Q₈	—	4	8	no	—
255	C₇×D₄	—	4	1	no	—
256	C₇×D₄	—	4	1	no	—
257	C₇×D₄	—	4	1	no	—
258	C₇×D₄	—	4	1	no	—
259	C₂×C₄×C₇	—	2	1	no	—
260	C₂×C₄×C₇	—	2	1	no	—
261	C₂×C₄×C₇	—	2	1	no	—
262	C₂×C₄×C₇	—	2	1	no	—
263	C₂×C₄×C₇	—	2	1	no	—
264	C₂×C₄×C₇	—	2	2	no	—
265	C₂×C₄×C₇	—	2	2	no	—
266	C₂×C₄×C₇	—	2	1	no	—
267	C₂×C₄×C₇	—	2	1	no	—
268	C₂×C₄×C₇	—	2	1	no	—
269	C₂×C₄×C₇	—	2	2	no	—
270	C₂×C₄×C₇	—	2	1	no	—
271	C₂×C₄×C₇	—	2	1	no	—
272	C₂×C₄×C₇	—	2	1	no	—
273	C₂×C₄×C₇	—	2	1	no	—
274	C₂×C₄×C₇	—	2	1	no	—
275	C₂×C₄×C₇	—	2	1	no	—
276	C₂×C₄×C₇	—	2	1	no	—
277	C₂×C₄×C₇	—	2	1	no	—
278	C₂×C₄×C₇	—	4	1	no	—
279	C₂×C₄×C₇	—	4	1	no	—
280	C₂×C₄×C₇	—	4	1	no	—
281	C₂×C₄×C₇	—	4	1	no	—
282	C₂×C₄×C₇	—	4	1	no	—
283	C₂×C₄×C₇	—	4	1	no	—
284	C₂×C₄×C₇	—	4	1	no	—
285	C₂×C₄×C₇	—	4	1	no	—
286	C₂×C₄×C₇	—	4	1	no	—
287	C₂×C₄×C₇	—	4	1	no	—
288	C₂×C₄×C₇	—	4	1	no	—
289	C₂×C₄×C₇	—	4	1	no	—
290	C₂×C₄×C₇	—	4	1	no	—
291	C₂×C₄×C₇	—	4	1	no	—
292	C₂×C₄×C₇	—	4	1	no	—
293	C₂×C₄×C₇	—	4	1	no	—
294	C₂×C₄×C₇	—	4	1	no	—
295	C₂×C₄×C₇	—	4	1	no	—
296	C₂×C₄×C₇	#135	1	2	yes	—
297	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
298	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
299	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
300	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
301	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
303	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
304	C₇×C₈	—	2	2	no	—
305	C₇×C₈	—	4	1	no	—
306	C₇×C₈	—	4	1	no	—
307	C₇×C₈	—	4	1	no	—
308	C₂²×Q₈	—	2	1	no	—
309	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄×Q₈	—	4	4	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₄²	—	2	1	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
343	#107	—	2	2	no	—
344	#106	—	2	2	no	—
345	#106	—	2	2	no	—
346	#106	—	2	2	no	—
347	C₄⋊C₈	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	2	2	no	—
350	#102	—	2	2	no	—
351	#102	—	2	2	no	—
352	C₄×C₈	—	2	2	no	—
353	#95	—	2	1	no	—
354	#95	—	2	1	no	—
355	#95	—	2	1	no	—
356	#95	—	2	1	no	—
357	#95	—	2	1	no	—
358	#95	—	2	1	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	#95	—	2	1	no	—
362	#95	C₇×D₄	1	1	yes	characteristic
363	#95	C₇×D₄	1	1	yes	characteristic
364	C₂²×C₇	—	8	1	no	—
365	C₂²×C₇	—	8	1	no	—
366	C₂²×C₇	—	8	1	no	—
367	C₂²×C₇	—	8	1	no	—
368	C₂²×C₇	—	8	1	no	—
369	C₂²×C₇	—	8	1	no	—
370	C₂²×C₇	—	8	1	no	—
371	C₂²×C₇	#525	1	1	yes	characteristic
372	C₂²×C₇	#547	1	1	yes	characteristic
373	C₂²×C₇	#548	1	1	yes	characteristic
374	C₂²×C₇	C₂×#135	1	1	yes	characteristic
375	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
376	C₂²×C₇	#577	1	1	yes	characteristic
377	C₂²×C₇	#577	1	1	yes	characteristic
378	C₄×C₇	—	2	2	no	—
379	C₄×C₇	—	2	2	no	—
380	C₄×C₇	—	4	1	no	—
381	C₄×C₇	—	4	1	no	—
382	C₄×C₇	—	4	1	no	—
383	C₄×C₇	—	4	2	no	—
384	C₄×C₇	—	4	2	no	—
385	C₄×C₇	—	4	2	no	—
386	C₄×C₇	—	4	1	no	—
387	C₄×C₇	—	8	1	no	—
388	C₄×C₇	—	8	1	no	—
389	C₄×C₇	—	8	1	no	—
390	C₄×C₇	—	8	1	no	—
391	C₄×C₇	#529	1	1	yes	characteristic
392	C₄×C₇	#541	1	1	yes	characteristic
393	C₄×C₇	#556	1	1	yes	characteristic
394	C₄×C₇	#556	1	1	yes	characteristic
395	C₂⁴	—	2	1	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	4	2	no	—
398	C₂×Q₈	—	4	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
413	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
414	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
415	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
416	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
417	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
418	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	C₇×#41	1	2	yes	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	4	2	no	—
446	C₄⋊C₄	—	4	2	no	—
447	C₄⋊C₄	—	8	1	no	—
448	C₄⋊C₄	C₇×#41	1	2	yes	—
449	C₄⋊C₄	C₇×#41	1	2	yes	—
450	C₄⋊C₄	C₇×#41	1	2	yes	—
451	C₂²⋊C₄	—	4	2	no	—
452	C₂²⋊C₄	—	4	2	no	—
453	C₂²⋊C₄	—	8	1	no	—
454	C₂²⋊C₄	—	8	1	no	—
455	C₂²⋊C₄	—	8	1	no	—
456	C₂²⋊C₄	—	8	1	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₄²	—	2	1	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	2	1	no	—
462	C₄²	—	2	1	no	—
463	C₄²	—	4	2	no	—
464	C₄²	—	4	2	no	—
465	C₄²	—	4	2	no	—
466	C₄²	—	8	1	no	—
467	C₂×C₇	—	16	1	no	—
468	C₂×C₇	#2469	1	1	yes	characteristic
469	C₂×C₇	#3105	1	1	yes	characteristic
470	C₂×C₇	#3126	1	1	yes	characteristic
471	C₂×C₇	#3340	1	1	yes	characteristic
472	C₂×C₇	#3345	1	1	yes	characteristic
473	C₂×C₇	#3365	1	1	yes	characteristic
474	C₂×C₇	#3373	1	1	yes	characteristic
475	C₂³	—	4	1	no	—
476	C₂³	—	4	1	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	4	1	no	—
480	C₂³	—	4	1	no	—
481	C₂³	—	4	1	no	—
482	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
483	Q₈	—	4	8	no	—
484	D₄	—	4	1	no	—
485	D₄	—	4	1	no	—
486	D₄	—	4	1	no	—
487	D₄	—	4	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	C₇×#135	1	2	yes	—
526	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
527	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
528	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
529	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
530	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
531	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
532	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
533	C₈	—	2	2	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₇	#27129	1	1	yes	characteristic, direct product factor, Sylow
538	C₂²	—	8	1	no	—
539	C₂²	—	8	1	no	—
540	C₂²	—	8	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	C₇×#525	1	1	yes	characteristic
546	C₂²	C₇×#547	1	1	yes	characteristic
547	C₂²	C₇×#548	1	1	yes	characteristic
548	C₂²	C₂×C₇×#135	1	1	yes	characteristic
549	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
550	C₂²	C₇×#577	1	1	yes	characteristic
551	C₂²	C₇×#577	1	1	yes	characteristic
552	C₄	—	2	2	no	—
553	C₄	—	2	2	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	4	2	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	1	no	—
561	C₄	—	8	1	no	—
562	C₄	—	8	1	no	—
563	C₄	—	8	1	no	—
564	C₄	—	8	1	no	—
565	C₄	C₇×#529	1	1	yes	characteristic
566	C₄	C₇×#541	1	1	yes	characteristic
567	C₄	C₇×#556	1	1	yes	characteristic
568	C₄	C₇×#556	1	1	yes	characteristic
569	C₂	—	16	1	no	—
570	C₂	C₇×#2469	1	1	yes	characteristic
571	C₂	C₇×#3105	1	1	yes	characteristic
572	C₂	C₇×#3126	1	1	yes	characteristic
573	C₂	C₇×#3340	1	1	yes	characteristic
574	C₂	C₇×#3345	1	1	yes	characteristic
575	C₂	C₇×#3365	1	1	yes	characteristic
576	C₂	C₇×#3373	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
2. C₇×#2365	C₂	semidirect
3. C₇×#2101	C₂	semidirect
5. C₇×#2058	C₂	semidirect
7. C₇×#1949	C₂	semidirect
12. C₇×#1921	C₂	semidirect
13. C₇×#1919	C₂	semidirect
14. C₇×#1916	C₂	semidirect
15. C₄×C₇×#106	C₂	semidirect
59. #27129	C₇	direct (N × H)
117. #2365	C₂×C₇	semidirect
118. #2101	C₂×C₇	semidirect
120. #2058	C₂×C₇	semidirect
122. #1949	C₂×C₇	semidirect
127. #1921	C₂×C₇	semidirect
128. #1919	C₂×C₇	semidirect
129. #1916	C₂×C₇	semidirect
130. C₄×#106	C₂×C₇	semidirect
537. C₇	#27129	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×#2513	2
C₇×#2365	2
C₇×#2101	2
C₇×#2065	2
C₇×#2058	2
C₇×#2023	2
C₇×#1949	2
C₇×#1934	2
C₇×#1926	2
C₇×#1925	2
C₇×#1921	2
C₇×#1919	2
C₇×#1916	2
C₄×C₇×#106	2
#27129	7

Subgroups — C7×#27129 #13204345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#27129 #13204345