Subgroups — #13252958

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3169	C₂	1	1	yes	maximal, characteristic
2	#11729572	C₂	1	2	yes	maximal, complemented
3	#11729565	C₂	1	2	yes	maximal, complemented
4	C₄×#6392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#1136	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#532	C₂²	1	2	yes	—
7	C₇×#500	C₄	1	1	yes	characteristic, complemented
8	C₇×#493	C₄	1	1	yes	characteristic, complemented
9	C₇×#486	C₄	1	2	yes	complemented
10	C₇×#466	—	2	1	no	—
11	C₇×#466	—	2	1	no	—
12	C₄×C₇×D₈	—	2	1	no	—
13	C₄×C₇×D₈	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6411	—	2	4	no	—
16	C₂×#6392	—	2	1	no	—
17	C₄×C₄⋊Dic₇	C₂²	1	2	yes	—
18	#89697	C₂²	1	1	yes	characteristic
19	C₄×#1136	C₂²	1	1	yes	characteristic
20	#89346	—	2	1	no	—
21	#89332	—	2	4	no	—
22	#89249	—	2	1	no	—
23	C₈×C₇⋊C₈	—	2	1	no	—
24	#9719	—	7	1	no	maximal, Sylow
25	C₂×C₇×#41	—	2	2	no	—
26	C₂×C₇×D₈	—	2	1	no	—
27	C₂×C₇×D₈	—	2	1	no	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₇×#127	C₂×C₄	1	2	yes	—
32	C₇×#124	—	2	2	no	—
33	C₇×#121	—	2	2	no	—
34	C₇×#121	—	2	2	no	—
35	C₄×C₇×D₄	—	2	2	no	—
36	C₄×C₇×D₄	—	2	2	no	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₇	—	2	2	no	—
39	C₂×C₄×Dic₇	—	2	2	no	—
40	C₂×#1136	D₄	1	2	yes	—
41	C₇×#107	—	2	1	no	—
42	C₇×#107	—	2	1	no	—
43	C₇×#102	—	2	2	no	—
44	C₇×#102	—	2	2	no	—
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	D₄	1	1	yes	characteristic
49	#6411	—	4	8	no	—
50	C₇⋊#95	—	2	2	no	—
51	C₂²⋊(C₇⋊C₈)	D₄	1	2	yes	—
52	#6392	—	2	4	no	—
53	C₄⋊(C₇⋊C₈)	—	2	1	no	—
54	#6377	—	2	1	no	—
55	C₄×C₇⋊C₈	—	2	1	no	—
56	C₄×C₇⋊C₈	—	2	1	no	—
57	#3169	D₇	1	1	yes	characteristic
58	#1980	—	7	2	no	—
59	C₄×#107	—	7	1	no	—
60	#1502	—	7	2	no	—
61	C₈×M₁₆	—	7	1	no	—
62	C₇×#41	—	4	4	no	—
63	C₇×#41	—	4	4	no	—
64	C₂×C₇×Q₈	—	2	2	no	—
65	C₂×C₇×D₄	—	2	2	no	—
66	C₂×C₇×D₄	—	2	2	no	—
67	C₂×C₇×D₄	—	2	2	no	—
68	C₂×C₇×D₄	—	4	2	no	—
69	C₂²×C₄×C₇	—	4	2	no	—
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₇	—	4	2	no	—
74	C₇×D₈	—	4	2	no	—
75	C₇×D₈	—	4	2	no	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	—	4	1	no	—
78	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₇×C₄⋊C₄	—	2	2	no	—
83	C₇×C₄⋊C₄	—	2	2	no	—
84	C₇×C₂²⋊C₄	—	4	2	no	—
85	C₇×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇	#41	1	1	yes	characteristic
89	C₄²×C₇	#41	1	1	yes	characteristic
90	C₄⋊Dic₇	—	2	4	no	—
91	C₄⋊Dic₇	—	4	2	no	—
92	C₄×Dic₇	—	4	4	no	—
93	#1136	—	2	2	no	—
94	#1136	D₈	1	4	yes	complemented
95	C₂×C₇⋊C₈	—	2	2	no	—
96	C₂×C₇⋊C₈	—	2	2	no	—
97	#532	C₂×D₇	1	2	yes	—
98	#500	Dic₇	1	1	yes	characteristic, complemented
99	#493	Dic₇	1	1	yes	characteristic, complemented
100	#486	Dic₇	1	2	yes	complemented
101	#466	—	2	1	no	—
102	#466	—	2	1	no	—
103	C₄×D₈	—	2	1	no	—
104	C₄×D₈	—	2	1	no	—
105	#431	—	7	1	no	—
106	C₂×#107	—	14	1	no	—
107	C₂×#104	—	14	4	no	—
108	C₄×M₁₆	—	7	1	no	—
109	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	7	2	no	—
111	#340	—	14	1	no	—
112	#327	—	14	4	no	—
113	#322	—	14	1	no	—
114	C₈²	—	14	1	no	—
115	C₂³×C₇	—	4	2	no	—
116	C₂³×C₇	#127	1	1	yes	characteristic
117	C₇×Q₈	—	4	4	no	—
118	C₇×D₄	—	4	2	no	—
119	C₇×D₄	—	4	2	no	—
120	C₇×D₄	—	4	4	no	—
121	C₇×D₄	—	8	2	no	—
122	C₇×D₄	—	8	2	no	—
123	C₇×D₄	—	8	2	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	4	1	no	—
127	C₂×C₄×C₇	—	4	2	no	—
128	C₂×C₄×C₇	—	8	4	no	—
129	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₇	#124	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₇	#135	1	2	yes	—
137	C₂×Dic₇	—	4	4	no	—
138	C₂×Dic₇	—	8	2	no	—
139	C₂×Dic₇	—	8	2	no	—
140	C₇×C₈	—	2	1	no	—
141	C₇×C₈	—	2	1	no	—
142	C₇×C₈	—	2	1	no	—
143	C₇×C₈	#104	1	2	yes	complemented
144	C₇⋊C₈	—	2	4	no	—
145	C₇⋊C₈	—	4	2	no	—
146	C₂×#41	—	2	2	no	—
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×M₁₆	—	7	2	no	—
150	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
152	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
153	#127	C₂×Dic₇	1	2	yes	—
154	#124	—	2	2	no	—
155	#121	—	2	2	no	—
156	#121	—	2	2	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	14	2	no	—
160	C₂×C₄²	—	14	2	no	—
161	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
162	#107	—	2	1	no	—
163	#107	—	2	1	no	—
164	#107	—	14	4	no	—
165	C₄⋊C₈	—	14	1	no	—
166	#104	—	28	8	no	—
167	#102	—	2	2	no	—
168	#102	—	2	2	no	—
169	C₂²⋊C₈	—	7	2	no	—
170	#97	—	14	1	no	—
171	C₄×C₈	—	14	1	no	—
172	C₄×C₈	—	14	1	no	—
173	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
174	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
177	#95	—	14	2	no	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	—	8	2	no	—
181	C₂²×C₇	—	8	2	no	—
182	C₂²×C₇	—	8	2	no	—
183	C₂²×C₇	#390	1	1	yes	characteristic
184	C₂²×C₇	#493	1	1	yes	characteristic
185	C₂²×C₇	#495	1	1	yes	characteristic
186	C₄×C₇	—	2	1	no	—
187	C₄×C₇	—	2	1	no	—
188	C₄×C₇	—	2	1	no	—
189	C₄×C₇	—	8	2	no	—
190	C₄×C₇	C₂×#104	1	1	yes	characteristic
191	C₄×C₇	#421	1	1	yes	characteristic
192	C₄×C₇	C₄×D₈	1	2	yes	—
193	C₄×C₇	#466	1	2	yes	—
194	Dic₇	—	8	4	no	—
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	28	2	no	—
204	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
207	D₈	—	4	2	no	—
208	D₈	—	4	2	no	—
209	M₁₆	—	7	4	no	—
210	M₁₆	—	14	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	14	2	no	—
214	C₂×C₈	—	14	2	no	—
215	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	14	4	no	—
222	C₄⋊C₄	—	28	2	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	28	4	no	—
226	C₄²	#1159	1	1	yes	characteristic
227	C₄²	#1159	1	1	yes	characteristic
228	C₄²	C₂²×Dic₇	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
230	C₂×C₇	—	2	1	no	—
231	C₂×C₇	—	2	1	no	—
232	C₂×C₇	—	16	2	no	—
233	C₂×C₇	#1970	1	1	yes	characteristic
234	C₂×C₇	#1976	1	1	yes	characteristic
235	C₂×C₇	#1980	1	1	yes	characteristic
236	C₂³	—	4	2	no	—
237	C₂³	#6503	1	1	yes	characteristic
238	Q₈	—	4	4	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	28	4	no	—
251	C₂×C₄	—	56	2	no	—
252	C₂×C₄	—	56	2	no	—
253	C₂×C₄	D₇×D₈	1	2	yes	—
254	C₂×C₄	#6475	1	2	yes	—
255	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
257	C₂×C₄	#6499	1	1	yes	characteristic
258	C₂×C₄	#6501	1	1	yes	characteristic
259	C₂×C₄	#6501	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	14	4	no	—
265	C₈	—	28	2	no	—
266	C₈	#6411	1	2	yes	complemented
267	C₇	#9719	1	1	yes	characteristic, complemented, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	#89923	1	1	yes	characteristic
274	C₂²	#89927	1	1	yes	characteristic
275	C₂²	#89993	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	56	4	no	—
281	C₄	D₈×Dic₇	1	2	yes	—
282	C₄	#89929	1	2	yes	—
283	C₄	C₂×#6411	1	1	yes	characteristic
284	C₄	#90008	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	16	2	no	—
288	C₂	#11729572	1	1	yes	characteristic
289	C₂	#11729792	1	1	yes	characteristic
290	C₂	#11729840	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6501
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting, prime radical	C₇×#3169	C₂
socle	C₂²×C₇	#390
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#3169	D₇
hypocenter	C₇	#9719
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
2. #11729572	C₂	semidirect
3. #11729565	C₂	semidirect
4. C₄×#6392	C₂	semidirect
5. C₈×#1136	C₂	semidirect
7. C₇×#500	C₄	semidirect
8. C₇×#493	C₄	semidirect
9. C₇×#486	C₄	semidirect
94. #1136	D₈	semidirect
98. #500	Dic₇	semidirect
99. #493	Dic₇	semidirect
100. #486	Dic₇	semidirect
143. C₇×C₈	#104	semidirect
266. C₈	#6411	semidirect
267. C₇	#9719	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3169	2
#11729572	2
#11729565	2
C₄×#6392	2
C₈×#1136	2
#9719	7