Subgroups — #13509307

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11739384	C₂	1	2	yes	maximal
2	C₇×#2254	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729140	C₂	1	2	yes	maximal, complemented
4	#11727100	C₂	1	2	yes	maximal
5	#11727099	C₂	1	1	yes	maximal, characteristic
6	D₇×#367	C₂	1	1	yes	maximal, characteristic
7	#11727097	C₂	1	1	yes	maximal, characteristic, complemented
8	#11727095	C₂	1	2	yes	maximal, complemented
9	#11727094	C₂	1	1	yes	maximal, characteristic, complemented
10	#11726805	C₂	1	1	yes	maximal, characteristic
11	#11726799	C₂	1	1	yes	maximal, characteristic, complemented
12	#90420	—	2	2	no	—
13	C₇×#500	C₂²	1	2	yes	—
14	#89852	—	2	2	no	—
15	C₂×#6372	C₂²	1	1	yes	characteristic
16	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
17	#89653	—	2	2	no	—
18	#89652	C₂²	1	2	yes	—
19	D₇×#107	C₂²	1	2	yes	—
20	#89650	C₂²	1	2	yes	—
21	#89648	—	2	2	no	—
22	#89646	C₂²	1	2	yes	—
23	#89461	C₂²	1	1	yes	characteristic
24	#89455	C₂²	1	2	yes	—
25	C₇×#367	C₂²	1	1	yes	characteristic
26	C₇×#367	C₂²	1	2	yes	—
27	C₇×#367	C₂²	1	1	yes	characteristic
28	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
29	#89300	C₂²	1	1	yes	characteristic
30	#89300	C₂²	1	1	yes	characteristic
31	#89299	C₂²	1	2	yes	—
32	#89297	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
34	#89285	C₂²	1	2	yes	—
35	#89285	C₂²	1	2	yes	—
36	#89284	C₂²	1	2	yes	—
37	#89284	C₂²	1	2	yes	—
38	#89256	C₂²	1	1	yes	characteristic
39	#33730	—	7	1	no	maximal, Sylow
40	C₇×C₄⋊Q₈	—	2	2	no	—
41	#6509	—	4	2	no	—
42	#6507	—	4	2	no	—
43	C₂×#1131	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
45	#6459	—	4	2	no	—
46	#6455	—	2	2	no	—
47	D₇×C₄⋊C₄	—	2	2	no	—
48	#6453	—	2	2	no	—
49	#6452	—	4	2	no	—
50	#6451	—	2	2	no	—
51	#6435	D₄	1	1	yes	characteristic
52	#6433	—	2	2	no	—
53	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
55	C₇×#107	—	2	2	no	—
56	C₇×#107	C₂³	1	2	yes	—
57	C₇×#107	C₂³	1	2	yes	—
58	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
59	#6392	C₂³	1	2	yes	—
60	#6392	C₂³	1	2	yes	—
61	#6391	—	2	2	no	—
62	#6389	D₄	1	1	yes	characteristic
63	C₈×Dic₇	C₂³	1	1	yes	characteristic
64	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
66	#6382	—	2	2	no	—
67	#6382	—	2	2	no	—
68	#6381	—	2	2	no	—
69	#6381	—	2	2	no	—
70	#6377	D₄	1	1	yes	characteristic
71	#6372	Q₈	1	4	yes	complemented
72	D₇×C₁₆	Q₈	1	4	yes	complemented
73	#3164	—	7	2	no	—
74	#2255	—	7	2	no	—
75	#2255	—	7	2	no	—
76	#2254	—	7	1	no	—
77	#2254	D₇	1	1	yes	characteristic, complemented
78	#2253	—	7	1	no	—
79	#2253	—	7	1	no	—
80	#2159	—	7	2	no	—
81	#2158	—	7	1	no	—
82	C₂×#367	—	7	1	no	—
83	#2108	—	7	1	no	—
84	C₂×C₇×Q₈	—	4	2	no	—
85	C₂×C₄×D₇	—	4	2	no	—
86	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₄	—	4	2	no	—
88	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	(C₂×D₇)⋊C₄	—	4	2	no	—
95	(C₂×D₇)⋊C₄	D₈	1	2	yes	—
96	C₄⋊Dic₇	—	2	2	no	—
97	C₄⋊Dic₇	—	2	2	no	—
98	C₄⋊Dic₇	—	4	2	no	—
99	Dic₇⋊C₄	—	4	2	no	—
100	Dic₇⋊C₄	—	4	2	no	—
101	Dic₇⋊C₄	—	4	2	no	—
102	Dic₇⋊C₄	—	4	2	no	—
103	Dic₇⋊C₄	—	4	2	no	—
104	Dic₇⋊C₄	D₈	1	2	yes	—
105	C₄×Dic₇	—	4	2	no	—
106	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	#1131	Q₁₆	1	4	yes	—
110	C₈×D₇	C₂×Q₈	1	2	yes	—
111	C₈×D₇	C₂×Q₈	1	2	yes	—
112	C₇×C₁₆	C₂×Q₈	1	2	yes	—
113	C₇×C₁₆	C₂×Q₈	1	2	yes	—
114	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
115	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
116	#544	—	14	2	no	—
117	C₂×M₃₂	—	7	1	no	—
118	C₂²×C₁₆	—	7	1	no	—
119	#501	—	14	2	no	—
120	#501	—	14	2	no	—
121	#500	—	7	2	no	—
122	#500	C₂×D₇	1	2	yes	—
123	#499	—	7	2	no	—
124	#499	—	7	2	no	—
125	#428	—	14	2	no	—
126	#427	—	7	2	no	—
127	C₂×#107	—	7	2	no	—
128	#405	—	7	1	no	—
129	#367	—	7	1	no	—
130	#367	—	7	1	no	—
131	#367	—	7	2	no	—
132	#367	—	7	2	no	—
133	#367	C₂×D₇	1	1	yes	characteristic
134	#367	C₂×D₇	1	1	yes	characteristic
135	#367	C₂×D₇	1	2	yes	—
136	#366	—	7	2	no	—
137	#366	—	7	2	no	—
138	#366	—	7	2	no	—
139	#346	—	7	1	no	—
140	#346	—	7	1	no	—
141	C₄×C₁₆	—	7	1	no	—
142	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
143	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
144	C₇×Q₈	—	16	2	no	—
145	C₂×C₄×C₇	—	4	2	no	—
146	C₂×C₄×C₇	—	4	2	no	—
147	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₇	—	4	2	no	—
150	C₂×Dic₇	—	4	2	no	—
151	C₂×Dic₇	C₂×D₈	1	2	yes	—
152	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
153	C₄×D₇	—	16	2	no	—
154	C₄×D₇	C₄⋊Q₈	1	2	yes	—
155	C₄×D₇	C₂×Q₁₆	1	2	yes	—
156	Dic₁₄	—	16	2	no	—
157	C₇×C₈	C₂×Q₁₆	1	2	yes	—
158	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
159	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
160	C₇⋊C₈	C₄⋊Q₈	1	2	yes	—
161	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
162	C₂×M₁₆	—	7	1	no	—
163	C₂²×C₈	—	7	1	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	14	2	no	—
166	C₄⋊Q₈	—	28	2	no	—
167	#125	—	14	2	no	—
168	#125	—	14	2	no	—
169	#125	—	28	2	no	—
170	C₂²⋊Q₈	—	28	2	no	—
171	C₂²⋊Q₈	—	28	2	no	—
172	#117	—	7	1	no	—
173	#117	—	14	2	no	—
174	C₂×C₄⋊C₄	—	14	2	no	—
175	M₃₂	—	7	4	no	—
176	C₂×C₁₆	—	7	1	no	—
177	C₂×C₁₆	—	7	1	no	—
178	C₂×C₁₆	—	7	4	no	—
179	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
180	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
181	#107	—	2	2	no	—
182	#107	—	7	2	no	—
183	#107	—	7	2	no	—
184	#107	—	14	2	no	—
185	#107	—	14	2	no	—
186	#107	C₂²×D₇	1	2	yes	—
187	#107	C₂²×D₇	1	2	yes	—
188	#106	—	14	2	no	—
189	#106	—	14	2	no	—
190	#106	—	14	2	no	—
191	#97	—	7	1	no	—
192	#97	—	7	1	no	—
193	C₄×C₈	—	7	1	no	—
194	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
195	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₇	—	2	2	no	—
197	C₂×D₇	#500	1	2	yes	—
198	C₄×C₇	—	2	2	no	—
199	C₄×C₇	—	8	2	no	—
200	C₄×C₇	—	8	2	no	—
201	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
202	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
203	Dic₇	—	2	2	no	—
204	Dic₇	—	8	2	no	—
205	Dic₇	—	8	2	no	—
206	Dic₇	#500	1	2	yes	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	28	2	no	—
209	C₂²×C₄	—	7	1	no	—
210	C₂²×C₄	—	28	2	no	—
211	M₁₆	—	7	4	no	—
212	C₂×C₈	—	7	1	no	—
213	C₂×C₈	—	7	1	no	—
214	C₂×C₈	—	7	2	no	—
215	C₂×C₈	—	7	2	no	—
216	C₂×C₈	D₄×D₇	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	7	2	no	—
222	C₄⋊C₄	—	14	2	no	—
223	C₄⋊C₄	—	14	2	no	—
224	C₄⋊C₄	—	28	2	no	—
225	C₄⋊C₄	—	28	2	no	—
226	C₄⋊C₄	—	28	2	no	—
227	C₄⋊C₄	—	28	2	no	—
228	C₄⋊C₄	—	28	2	no	—
229	C₄⋊C₄	—	28	2	no	—
230	C₂²⋊C₄	—	7	2	no	—
231	C₂²⋊C₄	—	28	2	no	—
232	C₄²	—	7	1	no	—
233	C₄²	—	28	2	no	—
234	C₄²	D₄×D₇	1	1	yes	characteristic
235	C₁₆	—	7	2	no	—
236	C₁₆	—	7	2	no	—
237	C₁₆	Q₈×D₇	1	2	yes	—
238	C₁₆	Q₈×D₇	1	2	yes	—
239	C₂×C₇	C₂×#500	1	1	yes	characteristic
240	C₂×C₇	#3416	1	2	yes	—
241	D₇	—	2	2	no	—
242	C₂³	—	7	1	no	—
243	Q₈	—	16	2	no	—
244	Q₈	—	112	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	7	1	no	—
248	C₂×C₄	—	7	2	no	—
249	C₂×C₄	—	7	2	no	—
250	C₂×C₄	—	7	2	no	—
251	C₂×C₄	—	28	2	no	—
252	C₂×C₄	—	28	2	no	—
253	C₂×C₄	—	112	2	no	—
254	C₂×C₄	D₇×D₈	1	2	yes	—
255	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
256	C₈	—	7	2	no	—
257	C₈	—	7	2	no	—
258	C₈	D₇×Q₁₆	1	2	yes	—
259	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
260	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
261	C₇	#33730	1	1	yes	characteristic, complemented, Sylow
262	C₂²	—	7	2	no	—
263	C₂²	—	14	2	no	—
264	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	7	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	14	2	no	—
270	C₄	—	56	2	no	—
271	C₄	—	56	2	no	—
272	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
273	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
274	C₂	—	14	2	no	—
275	C₂	D₇×#500	1	1	yes	characteristic
276	C₂	#11739469	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2254	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₇	C₂×D₈
hypercenter	#2254	D₇
hypocenter	C₇	#33730
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
2. C₇×#2254	C₂	semidirect
3. #11729140	C₂	semidirect
7. #11727097	C₂	semidirect
8. #11727095	C₂	semidirect
9. #11727094	C₂	semidirect
11. #11726799	C₂	semidirect
71. #6372	Q₈	semidirect
72. D₇×C₁₆	Q₈	semidirect
77. #2254	D₇	semidirect
261. C₇	#33730	semidirect

Maximal subgroups

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

subgroup	index
#11739384	2
C₇×#2254	2
#11729140	2
#11727100	2
#11727099	2
D₇×#367	2
#11727097	2
#11727095	2
#11727094	2
#11726805	2
#11726799	2
#33730	7