Subgroups — C₇×#12811 #13190027

Order 1792.

510 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#357	C₂	1	2	yes	maximal
4	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×D₄	—	2	1	no	—
8	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
9	C₄×C₇×D₈	—	2	2	no	—
10	C₄×C₇×D₈	C₄	1	4	yes	complemented
11	C₂×C₇×#107	—	2	1	no	—
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#102	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×#384	—	2	1	no	—
17	C₇×C₄⋊C₁₆	—	2	2	no	—
18	C₇×#357	—	2	4	no	—
19	#12811	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×D₄	—	2	1	no	—
21	C₂³×C₄×C₇	—	4	1	no	—
22	C₂×C₇×D₈	—	2	1	no	—
23	C₂×C₇×D₈	—	2	1	no	—
24	C₂×C₇×D₈	—	2	1	no	—
25	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
26	C₂×C₇×D₈	D₄	1	2	yes	—
27	C₂×C₇×D₈	Q₈	1	2	yes	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₄×C₇×D₄	—	2	4	no	—
31	C₄×C₇×D₄	—	4	2	no	—
32	C₂×C₇×C₄⋊C₄	—	2	1	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	4	1	no	—
35	C₂×C₇×C₄⋊C₄	—	4	1	no	—
36	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₇×C₁₆	—	2	2	no	—
39	C₂×C₇×C₁₆	—	2	2	no	—
40	C₂×C₇×C₁₆	—	2	2	no	—
41	C₇×#107	—	2	1	no	—
42	C₇×#107	—	2	2	no	—
43	C₇×#107	—	2	2	no	—
44	C₇×#107	—	4	2	no	—
45	C₇×#107	C₂×C₄	1	2	yes	—
46	C₇×#102	—	2	4	no	—
47	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₇×C₈	D₄	1	2	yes	—
49	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
50	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
51	C₂×#357	C₂×C₇	1	2	yes	—
52	#1845	C₂×C₇	1	1	yes	characteristic, complemented
53	#1380	C₂×C₇	1	2	yes	complemented
54	C₂⁴×C₇	—	4	1	no	—
55	C₂×C₇×D₄	—	2	2	no	—
56	C₂×C₇×D₄	—	2	2	no	—
57	C₂×C₇×D₄	—	2	2	no	—
58	C₂×C₇×D₄	—	4	1	no	—
59	C₂×C₇×D₄	—	4	1	no	—
60	C₂×C₇×D₄	—	4	1	no	—
61	C₂²×C₄×C₇	—	4	1	no	—
62	C₂²×C₄×C₇	—	4	4	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	—	8	2	no	—
65	C₂²×C₄×C₇	—	8	2	no	—
66	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₇×D₈	—	2	2	no	—
70	C₇×D₈	—	2	2	no	—
71	C₇×D₈	—	2	2	no	—
72	C₇×D₈	C₄⋊C₄	1	4	yes	complemented
73	C₂×C₇×C₈	—	2	1	no	—
74	C₂×C₇×C₈	—	2	1	no	—
75	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₇×C₈	#41	1	1	yes	characteristic
81	C₂×C₇×C₈	#41	1	1	yes	characteristic
82	C₇×C₄⋊C₄	—	2	2	no	—
83	C₇×C₄⋊C₄	—	4	1	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₄⋊C₄	—	8	2	no	—
87	C₇×C₄⋊C₄	—	8	2	no	—
88	C₇×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₇	SD₁₆	1	2	yes	—
91	C₇×C₁₆	—	4	2	no	—
92	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₄×D₄	—	2	1	no	—
94	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
95	C₄×D₈	—	2	2	no	—
96	C₄×D₈	C₄×C₇	1	4	yes	complemented
97	C₂×#107	—	2	1	no	—
98	C₂×#107	C₂²×C₇	1	2	yes	—
99	C₂×#107	C₂²×C₇	1	1	yes	characteristic
100	C₂×#102	—	2	1	no	—
101	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
102	#384	—	2	1	no	—
103	C₄⋊C₁₆	—	2	2	no	—
104	#357	—	2	4	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	—	4	2	no	—
107	C₂³×C₇	—	4	2	no	—
108	C₂³×C₇	—	8	1	no	—
109	C₂³×C₇	—	8	1	no	—
110	C₂³×C₇	—	8	1	no	—
111	C₂³×C₇	—	8	1	no	—
112	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
113	C₇×D₄	—	2	4	no	—
114	C₇×D₄	—	4	2	no	—
115	C₇×D₄	—	4	2	no	—
116	C₇×D₄	—	4	2	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	8	1	no	—
121	C₂×C₄×C₇	—	8	1	no	—
122	C₂×C₄×C₇	—	8	8	no	—
123	C₂×C₄×C₇	—	8	2	no	—
124	C₂×C₄×C₇	—	8	2	no	—
125	C₂×C₄×C₇	—	8	2	no	—
126	C₂×C₄×C₇	#102	1	2	yes	—
127	C₂×C₄×C₇	#102	1	2	yes	—
128	C₂×C₄×C₇	D₁₆	1	2	yes	—
129	C₂×C₄×C₇	SD₃₂	1	2	yes	—
130	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₇	#137	1	1	yes	characteristic
136	C₇×C₈	—	2	2	no	—
137	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₇×C₈	#117	1	1	yes	characteristic
139	C₇×C₈	#121	1	1	yes	characteristic
140	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×D₄	—	2	1	no	—
142	C₂³×C₄	—	4	1	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
147	C₂×D₈	C₇×D₄	1	2	yes	—
148	C₂×D₈	C₇×Q₈	1	2	yes	—
149	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
151	C₄×D₄	—	2	4	no	—
152	C₄×D₄	—	4	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	4	1	no	—
156	C₂×C₄⋊C₄	—	4	1	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	#107	—	2	1	no	—
163	#107	—	2	2	no	—
164	#107	—	2	2	no	—
165	#107	—	4	2	no	—
166	#107	C₂×C₄×C₇	1	2	yes	—
167	#102	—	2	4	no	—
168	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
169	C₄×C₈	C₇×D₄	1	2	yes	—
170	C₂²×C₇	—	4	4	no	—
171	C₂²×C₇	—	8	2	no	—
172	C₂²×C₇	—	8	2	no	—
173	C₂²×C₇	—	8	2	no	—
174	C₂²×C₇	—	8	2	no	—
175	C₂²×C₇	—	8	2	no	—
176	C₂²×C₇	—	8	2	no	—
177	C₂²×C₇	C₂×#102	1	1	yes	characteristic
178	C₂²×C₇	#459	1	1	yes	characteristic
179	C₂²×C₇	#474	1	1	yes	characteristic
180	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
181	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
182	C₂²×C₇	#509	1	1	yes	characteristic
183	C₂²×C₇	#510	1	1	yes	characteristic
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	8	2	no	—
186	C₄×C₇	—	16	2	no	—
187	C₄×C₇	#357	1	4	yes	complemented
188	C₄×C₇	#380	1	1	yes	characteristic
189	C₄×C₇	#417	1	1	yes	characteristic
190	C₄×C₇	#461	1	1	yes	characteristic
191	C₄×C₇	#476	1	1	yes	characteristic
192	C₂⁴	—	4	1	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	2	no	—
203	C₂²×C₄	—	8	2	no	—
204	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
205	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
207	D₈	—	2	2	no	—
208	D₈	—	2	2	no	—
209	D₈	—	2	2	no	—
210	D₈	C₇×C₄⋊C₄	1	4	yes	complemented
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
215	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
216	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
218	C₂×C₈	C₇×#41	1	1	yes	characteristic
219	C₂×C₈	C₇×#41	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	8	2	no	—
225	C₄⋊C₄	—	8	2	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
228	C₄²	C₇×SD₁₆	1	2	yes	—
229	C₁₆	—	4	2	no	—
230	C₂×C₇	—	8	4	no	—
231	C₂×C₇	#1802	1	1	yes	characteristic
232	C₂×C₇	C₂×#357	1	1	yes	characteristic
233	C₂×C₇	#2139	1	1	yes	characteristic
234	C₂×C₇	#2206	1	1	yes	characteristic
235	C₂×C₇	#2208	1	1	yes	characteristic
236	C₂×C₇	#2224	1	1	yes	characteristic
237	C₂×C₇	#2226	1	1	yes	characteristic
238	C₂³	—	4	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	8	1	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	8	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	C₇×#102	1	2	yes	—
260	C₂×C₄	C₇×#102	1	2	yes	—
261	C₂×C₄	C₇×D₁₆	1	2	yes	—
262	C₂×C₄	C₇×SD₃₂	1	2	yes	—
263	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₇×#121	1	1	yes	characteristic
265	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
267	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂×C₄	C₇×#137	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
271	C₈	C₇×#117	1	1	yes	characteristic
272	C₈	C₇×#121	1	1	yes	characteristic
273	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
274	C₇	#12811	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	4	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₂×C₇×#102	1	1	yes	characteristic
283	C₂²	C₇×#459	1	1	yes	characteristic
284	C₂²	C₇×#474	1	1	yes	characteristic
285	C₂²	C₂×C₇×D₁₆	1	1	yes	characteristic
286	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
287	C₂²	C₇×#509	1	1	yes	characteristic
288	C₂²	C₇×#510	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	16	2	no	—
292	C₄	C₇×#357	1	4	yes	complemented
293	C₄	C₇×#380	1	1	yes	characteristic
294	C₄	C₇×#417	1	1	yes	characteristic
295	C₄	C₇×#461	1	1	yes	characteristic
296	C₄	C₇×#476	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	C₇×#1802	1	1	yes	characteristic
299	C₂	C₂×C₇×#357	1	1	yes	characteristic
300	C₂	C₇×#2139	1	1	yes	characteristic
301	C₂	C₇×#2206	1	1	yes	characteristic
302	C₂	C₇×#2208	1	1	yes	characteristic
303	C₂	C₇×#2224	1	1	yes	characteristic
304	C₂	C₇×#2226	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
4. C₇×#1845	C₂	semidirect
5. C₇×#1380	C₂	semidirect
10. C₄×C₇×D₈	C₄	semidirect
19. #12811	C₇	direct (N × H)
50. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
52. #1845	C₂×C₇	semidirect
53. #1380	C₂×C₇	semidirect
72. C₇×D₈	C₄⋊C₄	semidirect
96. C₄×D₈	C₄×C₇	semidirect
187. C₄×C₇	#357	semidirect
210. D₈	C₇×C₄⋊C₄	semidirect
274. C₇	#12811	direct (N × H)
292. C₄	C₇×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₈	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×#357	2
C₇×#1845	2
C₇×#1380	2
#12811	7

Subgroups — C7×#12811 #13190027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12811 #13190027