Subgroups — C₇×#10536 #13187752

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₇×#1982	C₂	1	2	yes	maximal, complemented
3	C₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#331	C₂	1	2	yes	maximal, complemented
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₈	—	2	1	no	—
7	C₂²×C₇×D₈	—	2	1	no	—
8	C₂×C₇×#127	C₂²	1	2	yes	—
9	C₇×#493	—	2	4	no	—
10	C₂×C₇×C₄⋊C₈	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×#102	—	2	2	no	—
14	C₂×C₇×#102	—	2	2	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₇×#336	—	2	1	no	—
18	C₇×#331	—	2	4	no	—
19	#10536	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×D₄	—	2	2	no	—
21	C₂²×C₇×D₄	—	2	2	no	—
22	C₂²×C₇×D₄	—	2	2	no	—
23	C₂×C₇×D₈	—	2	4	no	—
24	C₂×C₇×D₈	—	2	4	no	—
25	C₂×C₇×D₈	—	4	2	no	—
26	C₂×C₇×D₈	—	4	2	no	—
27	C₂×C₇×D₈	—	4	2	no	—
28	C₂×C₇×D₈	—	4	2	no	—
29	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₇×#127	—	2	2	no	—
34	C₇×#127	—	2	2	no	—
35	C₇×#127	—	2	2	no	—
36	C₇×#127	D₄	1	4	yes	—
37	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	—	2	1	no	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊C₈	—	2	2	no	—
42	C₇×C₄⋊C₈	—	2	2	no	—
43	C₇×C₄⋊C₈	—	2	2	no	—
44	C₇×C₄⋊C₈	—	4	2	no	—
45	C₇×#102	—	4	4	no	—
46	C₇×#102	—	4	4	no	—
47	C₄×C₇×C₈	—	2	2	no	—
48	C₇×#95	—	2	1	no	—
49	C₂×#493	C₂×C₇	1	1	yes	characteristic
50	#1982	C₂×C₇	1	2	yes	complemented
51	#1939	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂×#331	C₂×C₇	1	2	yes	complemented
53	#1277	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂⁴×C₇	—	4	2	no	—
55	C₂×C₇×D₄	—	2	4	no	—
56	C₂×C₇×D₄	—	2	4	no	—
57	C₂×C₇×D₄	—	2	4	no	—
58	C₂×C₇×D₄	—	4	2	no	—
59	C₂×C₇×D₄	—	4	2	no	—
60	C₂×C₇×D₄	—	4	2	no	—
61	C₂×C₇×D₄	—	4	2	no	—
62	C₂×C₇×D₄	—	4	2	no	—
63	C₂×C₇×D₄	—	4	2	no	—
64	C₂×C₇×D₄	—	4	2	no	—
65	C₂×C₇×D₄	—	4	2	no	—
66	C₂×C₇×D₄	—	4	2	no	—
67	C₂×C₇×D₄	—	4	2	no	—
68	C₂×C₇×D₄	—	4	2	no	—
69	C₂×C₇×D₄	—	8	2	no	—
70	C₂×C₇×D₄	—	8	2	no	—
71	C₂²×C₄×C₇	—	2	1	no	—
72	C₂²×C₄×C₇	—	2	1	no	—
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	#41	1	1	yes	characteristic
76	C₇×D₈	—	4	8	no	—
77	C₇×D₈	—	4	8	no	—
78	C₂×C₇×C₈	—	2	1	no	—
79	C₂×C₇×C₈	—	2	1	no	—
80	C₂×C₇×C₈	—	2	1	no	—
81	C₂×C₇×C₈	—	2	1	no	—
82	C₂×C₇×C₈	—	4	1	no	—
83	C₂×C₇×C₈	—	4	2	no	—
84	C₂×C₇×C₈	—	4	2	no	—
85	C₂×C₇×C₈	—	4	1	no	—
86	C₂×C₇×C₈	—	4	2	no	—
87	C₂×C₇×C₈	—	4	1	no	—
88	C₂×C₇×C₈	D₈	1	2	yes	—
89	C₂×C₇×C₈	SD₁₆	1	2	yes	—
90	C₇×C₄⋊C₄	—	2	1	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	D₈	1	2	yes	complemented
93	C₄²×C₇	—	4	1	no	—
94	C₄²×C₇	—	4	1	no	—
95	C₄²×C₇	C₂×D₄	1	2	yes	—
96	C₄²×C₇	#41	1	2	yes	—
97	C₂²×D₈	—	2	1	no	—
98	C₂²×D₈	—	2	1	no	—
99	C₂×#127	C₂²×C₇	1	2	yes	—
100	#493	—	2	4	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
104	C₂×#102	—	2	2	no	—
105	C₂×#102	—	2	2	no	—
106	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
107	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#331	—	2	4	no	—
110	C₂³×C₇	—	4	4	no	—
111	C₂³×C₇	—	8	2	no	—
112	C₂³×C₇	—	8	2	no	—
113	C₂³×C₇	—	8	2	no	—
114	C₂³×C₇	—	8	2	no	—
115	C₂³×C₇	—	8	2	no	—
116	C₂³×C₇	—	8	2	no	—
117	C₂³×C₇	#121	1	1	yes	characteristic
118	C₇×D₄	—	4	4	no	—
119	C₇×D₄	—	4	4	no	—
120	C₇×D₄	—	4	4	no	—
121	C₇×D₄	—	4	4	no	—
122	C₇×D₄	—	8	4	no	—
123	C₇×D₄	—	8	4	no	—
124	C₇×D₄	—	8	2	no	—
125	C₇×D₄	—	8	2	no	—
126	C₇×D₄	—	8	2	no	—
127	C₇×D₄	—	8	2	no	—
128	C₂×C₄×C₇	—	2	1	no	—
129	C₂×C₄×C₇	—	2	1	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	4	1	no	—
132	C₂×C₄×C₇	—	4	1	no	—
133	C₂×C₄×C₇	—	4	1	no	—
134	C₂×C₄×C₇	—	4	1	no	—
135	C₂×C₄×C₇	—	4	1	no	—
136	C₂×C₄×C₇	#120	1	1	yes	characteristic
137	C₂×C₄×C₇	#121	1	2	yes	—
138	C₂×C₄×C₇	#123	1	2	yes	—
139	C₂×C₄×C₇	#124	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₇	#135	1	1	yes	characteristic
144	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
147	C₇×C₈	—	2	2	no	—
148	C₇×C₈	—	2	2	no	—
149	C₇×C₈	—	8	1	no	—
150	C₇×C₈	—	8	2	no	—
151	C₂²×D₄	—	2	2	no	—
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	2	2	no	—
154	C₂×D₈	—	2	4	no	—
155	C₂×D₈	—	2	4	no	—
156	C₂×D₈	—	4	2	no	—
157	C₂×D₈	—	4	2	no	—
158	C₂×D₈	—	4	2	no	—
159	C₂×D₈	—	4	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
163	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
164	#127	—	2	2	no	—
165	#127	—	2	2	no	—
166	#127	—	2	2	no	—
167	#127	C₇×D₄	1	4	yes	—
168	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	4	2	no	—
176	#102	—	4	4	no	—
177	#102	—	4	4	no	—
178	C₄×C₈	—	2	2	no	—
179	#95	—	2	1	no	—
180	C₂²×C₇	—	8	4	no	—
181	C₂²×C₇	—	8	4	no	—
182	C₂²×C₇	—	8	4	no	—
183	C₂²×C₇	—	16	2	no	—
184	C₂²×C₇	—	16	2	no	—
185	C₂²×C₇	—	16	2	no	—
186	C₂²×C₇	—	16	2	no	—
187	C₂²×C₇	#394	1	1	yes	characteristic
188	C₂²×C₇	#459	1	1	yes	characteristic
189	C₂²×C₇	#460	1	1	yes	characteristic
190	C₂²×C₇	#466	1	1	yes	characteristic
191	C₂²×C₇	#469	1	1	yes	characteristic
192	C₂²×C₇	#472	1	1	yes	characteristic
193	C₂²×C₇	#472	1	1	yes	characteristic
194	C₄×C₇	—	2	2	no	—
195	C₄×C₇	—	4	2	no	—
196	C₄×C₇	—	8	1	no	—
197	C₄×C₇	#447	1	1	yes	characteristic
198	C₄×C₇	#450	1	1	yes	characteristic
199	C₄×C₇	#459	1	2	yes	—
200	C₄×C₇	#463	1	2	yes	—
201	C₄×C₇	#486	1	1	yes	characteristic
202	C₄×C₇	#490	1	1	yes	characteristic
203	C₂⁴	—	4	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	8	2	no	—
219	C₂×D₄	—	8	2	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₇×#41	1	1	yes	characteristic
225	D₈	—	4	8	no	—
226	D₈	—	4	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	C₇×D₈	1	2	yes	—
238	C₂×C₈	C₇×SD₁₆	1	2	yes	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	C₇×D₈	1	2	yes	complemented
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	C₂×C₇×D₄	1	2	yes	—
245	C₄²	C₇×#41	1	2	yes	—
246	C₂×C₇	—	16	4	no	—
247	C₂×C₇	#1667	1	1	yes	characteristic
248	C₂×C₇	#1668	1	1	yes	characteristic
249	C₂×C₇	#1687	1	1	yes	characteristic
250	C₂×C₇	#1688	1	1	yes	characteristic
251	C₂×C₇	#2041	1	1	yes	characteristic
252	C₂×C₇	#2045	1	1	yes	characteristic
253	C₂×C₇	#2054	1	1	yes	characteristic
254	C₂³	—	4	4	no	—
255	C₂³	—	8	2	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	C₇×#121	1	1	yes	characteristic
262	D₄	—	4	4	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	8	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	2	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	C₇×#120	1	1	yes	characteristic
281	C₂×C₄	C₇×#121	1	2	yes	—
282	C₂×C₄	C₇×#123	1	2	yes	—
283	C₂×C₄	C₇×#124	1	1	yes	characteristic
284	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
285	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
286	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
287	C₂×C₄	C₇×#135	1	1	yes	characteristic
288	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	2	no	—
295	C₇	#10536	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	16	2	no	—
300	C₂²	—	16	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	C₇×#394	1	1	yes	characteristic
304	C₂²	C₇×#459	1	1	yes	characteristic
305	C₂²	C₇×#460	1	1	yes	characteristic
306	C₂²	C₇×#466	1	1	yes	characteristic
307	C₂²	C₇×#469	1	1	yes	characteristic
308	C₂²	C₇×#472	1	1	yes	characteristic
309	C₂²	C₇×#472	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	C₇×#447	1	1	yes	characteristic
314	C₄	C₇×#450	1	1	yes	characteristic
315	C₄	C₇×#459	1	2	yes	—
316	C₄	C₇×#463	1	2	yes	—
317	C₄	C₇×#486	1	1	yes	characteristic
318	C₄	C₇×#490	1	1	yes	characteristic
319	C₂	—	16	4	no	—
320	C₂	C₇×#1667	1	1	yes	characteristic
321	C₂	C₇×#1668	1	1	yes	characteristic
322	C₂	C₇×#1687	1	1	yes	characteristic
323	C₂	C₇×#1688	1	1	yes	characteristic
324	C₂	C₇×#2041	1	1	yes	characteristic
325	C₂	C₇×#2045	1	1	yes	characteristic
326	C₂	C₇×#2054	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#1982	C₂	semidirect
3. C₇×#1939	C₂	semidirect
4. C₂×C₇×#331	C₂	semidirect
5. C₇×#1277	C₂	semidirect
19. #10536	C₇	direct (N × H)
50. #1982	C₂×C₇	semidirect
51. #1939	C₂×C₇	semidirect
52. C₂×#331	C₂×C₇	semidirect
53. #1277	C₂×C₇	semidirect
92. C₇×C₄⋊C₄	D₈	semidirect
241. C₄⋊C₄	C₇×D₈	semidirect
295. C₇	#10536	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#493	2
C₇×#1982	2
C₇×#1939	2
C₂×C₇×#331	2
C₇×#1277	2
#10536	7

Subgroups — C7×#10536 #13187752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10536 #13187752