Subgroups — #13509322

Order 1792.

482 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11739361	C₂	1	1	yes	maximal, characteristic
2	D₇×#499	C₂	1	1	yes	maximal, characteristic
3	C₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729108	C₂	1	1	yes	maximal, characteristic, complemented
5	#11729108	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727100	C₂	1	2	yes	maximal
7	#11727095	C₂	1	2	yes	maximal, complemented
8	#11727080	C₂	1	2	yes	maximal
9	#11727076	C₂	1	2	yes	maximal, complemented
10	C₄×#6372	C₂	1	1	yes	maximal, characteristic
11	#11726799	C₂	1	1	yes	maximal, characteristic, complemented
12	#90379	—	2	1	no	—
13	D₇×#125	—	2	1	no	—
14	C₇×#499	C₂²	1	1	yes	characteristic
15	C₇×#499	C₂²	1	1	yes	characteristic
16	#89836	—	2	1	no	—
17	#89836	—	2	1	no	—
18	C₂×#6372	C₂²	1	1	yes	characteristic
19	C₂×#6372	C₂²	1	1	yes	characteristic
20	#89652	C₂²	1	2	yes	—
21	D₇×#107	C₂²	1	2	yes	—
22	#89650	C₂²	1	2	yes	—
23	#89646	C₂²	1	2	yes	—
24	#89632	—	2	1	no	—
25	D₇×#106	—	2	1	no	—
26	#89630	—	2	1	no	—
27	#89627	—	2	1	no	—
28	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
29	#89455	C₂²	1	1	yes	characteristic
30	#89455	C₂²	1	1	yes	characteristic
31	C₇×#367	C₂²	1	2	yes	—
32	C₇×#366	C₂²	1	2	yes	—
33	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
34	#89300	C₂²	1	2	yes	—
35	#89299	C₂²	1	2	yes	—
36	#89297	C₂²	1	1	yes	characteristic
37	#89297	C₂²	1	1	yes	characteristic
38	#89285	C₂²	1	2	yes	—
39	#89285	C₂²	1	2	yes	—
40	#89284	C₂²	1	2	yes	—
41	#89284	C₂²	1	2	yes	—
42	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
43	#33732	—	7	1	no	maximal, Sylow
44	C₇×#125	—	2	1	no	—
45	C₇×#125	—	2	1	no	—
46	C₂×C₈×D₇	D₄	1	1	yes	characteristic
47	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
48	#6459	—	4	2	no	—
49	#6455	—	2	2	no	—
50	D₇×C₄⋊C₄	—	2	2	no	—
51	D₇×C₄⋊C₄	—	4	2	no	—
52	#6453	—	2	2	no	—
53	#6452	—	4	2	no	—
54	#6451	—	2	2	no	—
55	#6450	—	4	2	no	—
56	C₄²×D₇	D₄	1	1	yes	characteristic
57	#6433	—	2	1	no	—
58	#6433	—	2	1	no	—
59	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
61	C₇×#107	C₂³	1	2	yes	—
62	C₇×#107	C₂³	1	2	yes	—
63	C₇×#106	—	2	1	no	—
64	C₇×#106	—	2	1	no	—
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	#6392	C₂³	1	2	yes	—
67	#6392	C₂³	1	2	yes	—
68	#6391	—	2	1	no	—
69	#6391	—	2	1	no	—
70	C₈×Dic₇	D₄	1	1	yes	characteristic
71	C₈×Dic₇	C₂³	1	1	yes	characteristic
72	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
74	#6382	—	2	2	no	—
75	#6382	—	2	2	no	—
76	#6381	—	2	2	no	—
77	#6381	—	2	2	no	—
78	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
79	#6372	Q₈	1	4	yes	complemented
80	#6372	Q₈	1	4	yes	complemented
81	#3160	—	7	1	no	—
82	C₂×#499	—	7	1	no	—
83	#2255	—	7	2	no	—
84	#2255	—	7	2	no	—
85	#2253	—	7	1	no	—
86	#2253	—	7	1	no	—
87	#2253	—	7	1	no	—
88	#2253	D₇	1	1	yes	characteristic, complemented
89	#2159	—	7	2	no	—
90	#2159	—	7	2	no	—
91	C₄×M₃₂	—	7	1	no	—
92	C₂×C₄×D₇	—	4	2	no	—
93	C₂×C₄×D₇	D₈	1	2	yes	—
94	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
95	C₂×Dic₁₄	—	4	2	no	—
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	4	2	no	—
101	C₇×C₄⋊C₄	—	4	2	no	—
102	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
103	(C₂×D₇)⋊C₄	—	4	2	no	—
104	C₄⋊Dic₇	—	2	2	no	—
105	C₄⋊Dic₇	—	2	2	no	—
106	C₄⋊Dic₇	—	4	2	no	—
107	Dic₇⋊C₄	—	4	2	no	—
108	Dic₇⋊C₄	—	4	2	no	—
109	Dic₇⋊C₄	—	4	2	no	—
110	Dic₇⋊C₄	—	4	2	no	—
111	Dic₇⋊C₄	—	4	2	no	—
112	C₄×Dic₇	—	4	2	no	—
113	C₄×Dic₇	D₈	1	2	yes	—
114	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
117	C₈×D₇	Q₁₆	1	4	yes	—
118	C₈×D₇	C₂×Q₈	1	2	yes	—
119	C₈×D₇	C₂×Q₈	1	2	yes	—
120	C₇×C₁₆	C₂×Q₈	1	2	yes	—
121	C₇×C₁₆	C₂×Q₈	1	2	yes	—
122	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
123	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
124	#533	—	14	1	no	—
125	C₂×#125	—	14	1	no	—
126	C₂×M₃₂	—	7	1	no	—
127	C₂×M₃₂	—	7	1	no	—
128	#500	—	7	2	no	—
129	#499	—	7	2	no	—
130	#499	—	7	1	no	—
131	#499	—	7	1	no	—
132	#499	—	14	1	no	—
133	#499	—	14	1	no	—
134	#499	—	14	1	no	—
135	#499	C₂×D₇	1	1	yes	characteristic
136	#499	C₂×D₇	1	1	yes	characteristic
137	#498	—	14	1	no	—
138	#427	—	7	2	no	—
139	#427	—	14	1	no	—
140	C₂×#107	—	7	2	no	—
141	C₂×#106	—	14	1	no	—
142	C₂×C₄×C₈	—	7	1	no	—
143	#367	—	7	2	no	—
144	#367	—	7	2	no	—
145	#367	—	7	2	no	—
146	#367	C₂×D₇	1	2	yes	—
147	#366	—	7	2	no	—
148	#366	—	7	2	no	—
149	#366	—	7	2	no	—
150	#366	C₂×D₇	1	2	yes	—
151	#346	—	7	1	no	—
152	#346	—	7	1	no	—
153	C₄×C₁₆	—	7	1	no	—
154	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
155	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×Dic₇	—	4	2	no	—
161	C₂×Dic₇	—	4	2	no	—
162	C₂×Dic₇	C₂×D₈	1	2	yes	—
163	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
164	C₄×D₇	—	4	2	no	—
165	C₄×D₇	—	8	4	no	—
166	C₄×D₇	C₄⋊Q₈	1	2	yes	—
167	C₄×D₇	C₂×Q₁₆	1	2	yes	—
168	Dic₁₄	—	8	4	no	—
169	C₇×C₈	C₂×Q₁₆	1	2	yes	—
170	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
171	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
172	C₇⋊C₈	C₄⋊Q₈	1	2	yes	—
173	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
174	C₂²×C₈	—	7	1	no	—
175	C₂²×C₈	—	7	1	no	—
176	C₄⋊Q₈	—	14	2	no	—
177	#125	—	2	1	no	—
178	#125	—	2	1	no	—
179	#125	—	14	2	no	—
180	#125	—	14	1	no	—
181	#125	—	14	1	no	—
182	#125	—	28	2	no	—
183	C₂²⋊Q₈	—	28	2	no	—
184	C₄×Q₈	—	28	2	no	—
185	#117	—	14	2	no	—
186	C₂×C₄⋊C₄	—	14	2	no	—
187	C₂×C₄⋊C₄	—	28	2	no	—
188	C₂×C₄²	—	7	1	no	—
189	M₃₂	—	7	4	no	—
190	M₃₂	—	7	4	no	—
191	C₂×C₁₆	—	7	1	no	—
192	C₂×C₁₆	—	7	1	no	—
193	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
194	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
195	#107	—	7	2	no	—
196	#107	—	7	2	no	—
197	#107	—	14	2	no	—
198	#107	—	14	2	no	—
199	#107	C₂²×D₇	1	2	yes	—
200	#107	C₂²×D₇	1	2	yes	—
201	#106	—	2	1	no	—
202	#106	—	2	1	no	—
203	#106	—	14	2	no	—
204	#106	—	14	2	no	—
205	#106	—	14	1	no	—
206	#106	—	14	1	no	—
207	C₄×C₈	—	7	1	no	—
208	C₄×C₈	—	7	1	no	—
209	C₄×C₈	—	7	1	no	—
210	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
211	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
212	C₂×D₇	—	2	2	no	—
213	C₂×D₇	#500	1	2	yes	—
214	C₄×C₇	—	2	2	no	—
215	C₄×C₇	—	8	2	no	—
216	C₄×C₇	—	8	2	no	—
217	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
218	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
219	Dic₇	—	2	2	no	—
220	Dic₇	—	8	2	no	—
221	Dic₇	—	8	2	no	—
222	Dic₇	#500	1	2	yes	—
223	C₂×Q₈	—	28	2	no	—
224	C₂²×C₄	—	7	1	no	—
225	C₂²×C₄	—	7	2	no	—
226	C₂²×C₄	—	28	2	no	—
227	C₂×C₈	—	7	1	no	—
228	C₂×C₈	—	7	1	no	—
229	C₂×C₈	—	7	4	no	—
230	C₂×C₈	—	7	2	no	—
231	C₂×C₈	—	7	2	no	—
232	C₂×C₈	D₄×D₇	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	14	2	no	—
239	C₄⋊C₄	—	14	2	no	—
240	C₄⋊C₄	—	28	2	no	—
241	C₄⋊C₄	—	28	2	no	—
242	C₄⋊C₄	—	28	2	no	—
243	C₄⋊C₄	—	28	2	no	—
244	C₄⋊C₄	—	28	2	no	—
245	C₄⋊C₄	—	28	2	no	—
246	C₂²⋊C₄	—	28	2	no	—
247	C₄²	—	7	2	no	—
248	C₄²	—	7	1	no	—
249	C₄²	—	28	2	no	—
250	C₄²	D₄×D₇	1	1	yes	characteristic
251	C₁₆	—	7	2	no	—
252	C₁₆	—	7	2	no	—
253	C₁₆	Q₈×D₇	1	2	yes	—
254	C₁₆	Q₈×D₇	1	2	yes	—
255	C₂×C₇	C₂×#500	1	1	yes	characteristic
256	C₂×C₇	#3414	1	2	yes	—
257	D₇	—	2	2	no	—
258	C₂³	—	7	1	no	—
259	Q₈	—	56	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	7	1	no	—
263	C₂×C₄	—	7	2	no	—
264	C₂×C₄	—	7	2	no	—
265	C₂×C₄	—	7	2	no	—
266	C₂×C₄	—	28	2	no	—
267	C₂×C₄	—	28	2	no	—
268	C₂×C₄	—	28	2	no	—
269	C₂×C₄	—	56	4	no	—
270	C₂×C₄	D₇×D₈	1	2	yes	—
271	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
272	C₈	—	7	2	no	—
273	C₈	—	7	2	no	—
274	C₈	D₇×Q₁₆	1	2	yes	—
275	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
276	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
277	C₇	#33732	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	7	2	no	—
279	C₂²	—	14	2	no	—
280	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	7	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	14	2	no	—
286	C₄	—	56	2	no	—
287	C₄	—	56	2	no	—
288	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
289	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
290	C₂	—	14	2	no	—
291	C₂	D₇×#500	1	1	yes	characteristic
292	C₂	#11739475	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2253	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₇	C₂×D₈
hypercenter	#2253	D₇
hypocenter	C₇	#33732
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
3. C₇×#2253	C₂	semidirect
4. #11729108	C₂	semidirect
5. #11729108	C₂	semidirect
7. #11727095	C₂	semidirect
9. #11727076	C₂	semidirect
11. #11726799	C₂	semidirect
79. #6372	Q₈	semidirect
80. #6372	Q₈	semidirect
88. #2253	D₇	semidirect
277. C₇	#33732	semidirect

Maximal subgroups

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

subgroup	index
#11739361	2
D₇×#499	2
C₇×#2253	2
#11729108	2
#11727100	2
#11727095	2
#11727080	2
#11727076	2
C₄×#6372	2
#11726799	2
#33732	7