Subgroups — #13509337

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11737170	C₂	1	2	yes	maximal
2	C₄×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729258	C₂	1	1	yes	maximal, characteristic
4	#11729253	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6403	C₂	1	2	yes	maximal, complemented
6	#11727082	C₂	1	1	yes	maximal, characteristic
7	#11727074	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726981	C₂	1	2	yes	maximal
9	#11726969	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic
12	#90269	—	2	2	no	—
13	C₂×C₇×Q₃₂	C₂²	1	1	yes	characteristic
14	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
15	#89960	C₂²	1	2	yes	—
16	#89953	C₂²	1	2	yes	—
17	C₂×#6403	C₂²	1	2	yes	—
18	C₄×#1144	—	2	2	no	—
19	C₂×Dic₅₆	C₂²	1	1	yes	characteristic
20	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
21	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
22	#89659	C₂²	1	2	yes	—
23	#89646	C₂²	1	2	yes	—
24	#89585	—	2	2	no	—
25	#89564	—	2	2	no	—
26	C₄×Dic₂₈	C₂²	1	2	yes	—
27	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
28	C₇×#366	C₂²	1	1	yes	characteristic
29	C₇×#358	C₂²	1	2	yes	—
30	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
31	#89359	C₂²	1	2	yes	—
32	#89299	C₂²	1	1	yes	characteristic
33	#89298	C₂²	1	2	yes	—
34	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
36	#89287	C₂²	1	2	yes	—
37	#89284	C₂²	1	2	yes	—
38	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
39	#33734	—	7	1	no	maximal, Sylow
40	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
41	C₄×C₇×Q₈	—	2	2	no	—
42	#6509	—	2	2	no	—
43	#6507	—	2	2	no	—
44	C₂×#1144	—	2	2	no	—
45	C₂×Dic₂₈	C₂³	1	2	yes	—
46	C₂×C₈×D₇	D₄	1	1	yes	characteristic
47	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
48	#6460	—	2	2	no	—
49	#6459	—	4	2	no	—
50	#6452	—	4	2	no	—
51	#6451	—	2	2	no	—
52	#6435	—	4	2	no	—
53	C₄²×D₇	D₄	1	1	yes	characteristic
54	C₄×Dic₁₄	—	2	2	no	—
55	C₄×Dic₁₄	—	4	2	no	—
56	C₇×Q₃₂	—	2	2	no	—
57	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
59	C₇×#107	C₂³	1	2	yes	—
60	C₇×#103	—	2	2	no	—
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	#6409	—	2	2	no	—
63	#6403	—	2	4	no	—
64	#6392	C₂³	1	2	yes	—
65	#6390	—	2	2	no	—
66	C₈×Dic₇	D₄	1	1	yes	characteristic
67	C₈×Dic₇	C₂³	1	1	yes	characteristic
68	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
70	#6384	—	2	2	no	—
71	#6381	—	2	2	no	—
72	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
73	Dic₅₆	—	2	2	no	—
74	D₇×C₁₆	—	2	2	no	—
75	D₇×C₁₆	—	2	2	no	—
76	#3170	—	7	2	no	—
77	#2252	—	7	1	no	—
78	#2247	—	7	1	no	—
79	#2241	—	7	2	no	—
80	#2216	—	7	1	no	—
81	#2216	—	7	1	no	—
82	#2216	—	7	2	no	—
83	C₄×Q₃₂	—	7	2	no	—
84	C₄×Q₃₂	—	7	1	no	—
85	C₄×Q₃₂	D₇	1	1	yes	characteristic, complemented
86	C₂×C₄×C₁₆	—	7	1	no	—
87	C₂×C₇×Q₈	—	2	2	no	—
88	C₂×C₄×D₇	D₈	1	1	yes	characteristic
89	C₂×C₄×D₇	D₈	1	1	yes	characteristic
90	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₁₄	—	2	2	no	—
92	C₂×Dic₁₄	—	4	2	no	—
93	C₇×Q₁₆	—	2	2	no	—
94	C₇×Q₁₆	—	2	2	no	—
95	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	4	2	no	—
99	C₄²×C₇	—	4	2	no	—
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	#1144	—	4	4	no	—
102	(C₂×D₇)⋊C₄	—	4	2	no	—
103	(C₂×D₇)⋊C₄	—	4	2	no	—
104	C₄⋊Dic₇	—	2	2	no	—
105	C₄⋊Dic₇	—	4	2	no	—
106	Dic₇⋊C₄	—	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	C₄×Dic₇	—	4	2	no	—
112	C₄×Dic₇	D₈	1	1	yes	characteristic
113	C₄×Dic₇	D₈	1	1	yes	characteristic
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	Dic₂₈	—	2	2	no	—
118	Dic₂₈	—	2	2	no	—
119	C₈×D₇	—	2	1	no	—
120	C₈×D₇	—	2	1	no	—
121	C₈×D₇	—	2	2	no	—
122	C₇×C₁₆	—	2	1	no	—
123	C₇×C₁₆	#41	1	2	yes	—
124	C₇⋊C₁₆	—	2	1	no	—
125	C₇⋊C₁₆	#41	1	2	yes	—
126	#533	—	14	2	no	—
127	C₂×Q₃₂	—	7	2	no	—
128	C₂×Q₃₂	—	7	1	no	—
129	C₂×Q₃₂	C₂×D₇	1	1	yes	characteristic
130	C₂²×C₁₆	—	7	1	no	—
131	C₂²×C₁₆	—	7	1	no	—
132	#500	—	7	2	no	—
133	#494	—	7	2	no	—
134	#487	—	14	2	no	—
135	#467	—	7	2	no	—
136	#467	—	7	2	no	—
137	#467	—	14	2	no	—
138	C₄×Q₁₆	—	7	2	no	—
139	C₄×Q₁₆	—	14	2	no	—
140	C₄×Q₁₆	C₂×D₇	1	2	yes	—
141	C₂×C₄×C₈	—	7	1	no	—
142	#366	—	7	1	no	—
143	#366	—	7	2	no	—
144	#366	C₂×D₇	1	1	yes	characteristic
145	#358	—	7	2	no	—
146	#358	—	7	2	no	—
147	#358	—	7	2	no	—
148	#358	C₂×D₇	1	2	yes	—
149	C₄×C₁₆	—	7	1	no	—
150	C₄×C₁₆	—	7	1	no	—
151	C₄×C₁₆	—	7	1	no	—
152	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
153	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
154	C₇×Q₈	—	4	2	no	—
155	C₇×Q₈	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×Dic₇	—	4	2	no	—
162	C₂×Dic₇	—	4	2	no	—
163	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
164	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
165	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
166	C₄×D₇	—	2	1	no	—
167	C₄×D₇	—	2	1	no	—
168	C₄×D₇	—	2	2	no	—
169	C₄×D₇	Q₃₂	1	4	yes	complemented
170	Dic₁₄	—	4	2	no	—
171	Dic₁₄	—	4	2	no	—
172	Dic₁₄	—	8	4	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	C₂×#41	1	1	yes	characteristic
175	C₇×C₈	C₂×#41	1	1	yes	characteristic
176	C₇⋊C₈	—	2	1	no	—
177	C₇⋊C₈	#135	1	2	yes	—
178	C₂×Q₁₆	—	7	2	no	—
179	C₂×Q₁₆	—	14	2	no	—
180	C₂×Q₁₆	C₂²×D₇	1	2	yes	—
181	C₂²×C₈	—	7	1	no	—
182	C₂²×C₈	—	7	1	no	—
183	C₄⋊Q₈	—	14	2	no	—
184	C₄⋊Q₈	—	14	2	no	—
185	#125	—	28	2	no	—
186	C₂²⋊Q₈	—	14	2	no	—
187	C₂²⋊Q₈	—	14	2	no	—
188	C₂²⋊Q₈	—	28	2	no	—
189	C₄×Q₈	—	2	2	no	—
190	C₄×Q₈	—	14	2	no	—
191	C₄×Q₈	—	28	2	no	—
192	#117	—	28	2	no	—
193	C₂×C₄²	—	7	1	no	—
194	Q₃₂	—	2	2	no	—
195	Q₃₂	—	14	4	no	—
196	Q₃₂	—	14	2	no	—
197	C₂×C₁₆	—	7	1	no	—
198	C₂×C₁₆	—	7	1	no	—
199	C₂×C₁₆	—	14	2	no	—
200	C₂×C₁₆	—	14	2	no	—
201	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
202	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
203	#107	—	7	2	no	—
204	#107	—	14	2	no	—
205	#107	C₂²×D₇	1	2	yes	—
206	#103	—	2	2	no	—
207	#103	—	14	2	no	—
208	#103	—	14	2	no	—
209	#103	—	14	2	no	—
210	C₄×C₈	—	7	1	no	—
211	C₄×C₈	—	7	1	no	—
212	C₄×C₈	—	7	1	no	—
213	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
214	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
215	C₂×D₇	—	2	1	no	—
216	C₂×D₇	—	2	1	no	—
217	C₂×D₇	C₂×Q₃₂	1	2	yes	—
218	C₄×C₇	—	2	1	no	—
219	C₄×C₇	—	8	2	no	—
220	C₄×C₇	—	8	2	no	—
221	C₄×C₇	C₂×Q₃₂	1	2	yes	—
222	C₄×C₇	#532	1	1	yes	characteristic
223	C₄×C₇	C₂×#135	1	1	yes	characteristic
224	Dic₇	—	2	1	no	—
225	Dic₇	—	8	2	no	—
226	Dic₇	—	8	2	no	—
227	Dic₇	C₂×Q₃₂	1	2	yes	—
228	Dic₇	#508	1	2	yes	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	14	2	no	—
231	C₂×Q₈	—	28	2	no	—
232	C₂²×C₄	—	7	1	no	—
233	C₂²×C₄	—	7	1	no	—
234	C₂²×C₄	—	7	1	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	14	2	no	—
238	Q₁₆	—	14	2	no	—
239	Q₁₆	—	28	4	no	—
240	C₂×C₈	—	7	1	no	—
241	C₂×C₈	—	7	1	no	—
242	C₂×C₈	—	14	2	no	—
243	C₂×C₈	—	14	1	no	—
244	C₂×C₈	—	14	1	no	—
245	C₂×C₈	D₄×D₇	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	14	2	no	—
250	C₄⋊C₄	—	28	2	no	—
251	C₄⋊C₄	—	28	2	no	—
252	C₄⋊C₄	—	28	2	no	—
253	C₄⋊C₄	—	28	2	no	—
254	C₄⋊C₄	—	28	2	no	—
255	C₄⋊C₄	—	28	2	no	—
256	C₂²⋊C₄	—	28	2	no	—
257	C₂²⋊C₄	—	28	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	7	1	no	—
260	C₄²	—	7	1	no	—
261	C₄²	—	7	1	no	—
262	C₄²	—	28	2	no	—
263	C₄²	D₄×D₇	1	1	yes	characteristic
264	C₁₆	—	2	1	no	—
265	C₁₆	—	7	2	no	—
266	C₁₆	—	14	1	no	—
267	C₁₆	#1157	1	2	yes	—
268	C₂×C₇	#3169	1	1	yes	characteristic
269	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
270	C₂×C₇	C₂×#508	1	1	yes	characteristic
271	D₇	—	2	2	no	—
272	C₂³	—	7	1	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	28	2	no	—
276	Q₈	—	28	2	no	—
277	Q₈	—	56	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	7	1	no	—
281	C₂×C₄	—	7	1	no	—
282	C₂×C₄	—	7	4	no	—
283	C₂×C₄	—	7	1	no	—
284	C₂×C₄	—	14	2	no	—
285	C₂×C₄	—	14	1	no	—
286	C₂×C₄	—	14	1	no	—
287	C₂×C₄	—	28	2	no	—
288	C₂×C₄	—	28	2	no	—
289	C₂×C₄	D₇×D₈	1	1	yes	characteristic
290	C₂×C₄	D₇×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	7	2	no	—
294	C₈	—	14	1	no	—
295	C₈	C₂×#1157	1	1	yes	characteristic
296	C₈	D₇×#41	1	1	yes	characteristic
297	C₇	#33734	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	7	2	no	—
299	C₂²	—	14	1	no	—
300	C₂²	—	14	1	no	—
301	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	7	2	no	—
304	C₄	—	7	2	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	14	1	no	—
308	C₄	—	56	2	no	—
309	C₄	—	56	2	no	—
310	C₄	D₇×Q₃₂	1	2	yes	—
311	C₄	#90239	1	1	yes	characteristic
312	C₄	D₇×#135	1	1	yes	characteristic
313	C₂	—	14	2	no	—
314	C₂	#11737043	1	1	yes	characteristic
315	C₂	C₂×D₇×Q₃₂	1	1	yes	characteristic
316	C₂	D₇×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₄×C₇×Q₃₂	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₇	C₂×D₈
hypercenter	C₄×Q₃₂	D₇
hypocenter	C₇	#33734
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×Q₃₂	C₂	semidirect
4. #11729253	C₂	semidirect
5. C₄×#6403	C₂	semidirect
7. #11727074	C₂	semidirect
9. #11726969	C₂	semidirect
10. C₄×Dic₅₆	C₂	semidirect
85. C₄×Q₃₂	D₇	semidirect
169. C₄×D₇	Q₃₂	semidirect
297. C₇	#33734	semidirect

Maximal subgroups

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

subgroup	index
#11737170	2
C₄×C₇×Q₃₂	2
#11729258	2
#11729253	2
C₄×#6403	2
#11727082	2
#11727074	2
#11726981	2
#11726969	2
C₄×Dic₅₆	2
C₄×D₇×C₁₆	2
#33734	7