Subgroups — #18007331

Order 4736.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558252	C₂	1	1	yes	maximal, characteristic
3	#14557892	C₂	1	1	yes	maximal, characteristic
4	#14557891	C₂	1	1	yes	maximal, characteristic
5	#14557882	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557821	C₂	1	1	yes	maximal, characteristic
7	#14557819	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557816	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557753	C₂	1	1	yes	maximal, characteristic
10	#14557741	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557662	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557640	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557638	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#464	C₂	1	1	yes	maximal, characteristic
16	C₂×#10596942	C₂²	1	1	yes	characteristic
17	#11924536	—	2	1	no	—
18	#11924536	C₂²	1	1	yes	characteristic
19	#11924535	—	2	1	no	—
20	#11924535	—	2	1	no	—
21	#11924535	C₂²	1	1	yes	characteristic
22	#11924535	C₂²	1	1	yes	characteristic
23	#11924534	C₂²	1	1	yes	characteristic
24	#11924529	—	2	1	no	—
25	#11924529	—	2	1	no	—
26	#11924523	C₂²	1	1	yes	characteristic
27	#11924513	—	2	1	no	—
28	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
29	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
30	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
31	C₄×Dic₇₄	—	2	1	no	—
32	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
33	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
34	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
35	#11924492	—	2	1	no	—
36	#11924468	C₂²	1	1	yes	characteristic
37	#11924468	C₂²	1	1	yes	characteristic
38	#11924467	C₂²	1	1	yes	characteristic
39	#11924465	C₂²	1	1	yes	characteristic
40	#11924465	C₂²	1	1	yes	characteristic
41	#11924461	C₂²	1	1	yes	characteristic
42	#11924461	C₂²	1	1	yes	characteristic
43	#11924460	C₂²	1	1	yes	characteristic
44	#11924460	C₂²	1	1	yes	characteristic
45	#11924460	C₂²	1	1	yes	characteristic
46	#11924459	C₂²	1	1	yes	characteristic
47	#11924457	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	#11924447	C₂²	1	1	yes	characteristic
50	#11924446	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	#124×C₃₇	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₇	—	2	1	no	—
63	C₂×Dic₇₄	—	2	1	no	—
64	C₂×Dic₇₄	—	2	1	no	—
65	C₂×Dic₇₄	—	2	1	no	—
66	C₂×Dic₇₄	D₄	1	1	yes	characteristic
67	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
68	Q₁₆×C₃₇	—	2	2	no	—
69	SD₁₆×C₃₇	—	2	2	no	—
70	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₃₇	—	2	1	no	—
73	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₃₇	—	2	1	no	—
75	C₂²⋊C₄×C₃₇	—	2	1	no	—
76	C₄²×C₃₇	—	2	1	no	—
77	C₄²×C₃₇	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₃₇	—	2	1	no	—
79	C₂²⋊Dic₃₇	—	2	1	no	—
80	C₂²⋊Dic₃₇	—	2	1	no	—
81	#10596942	—	2	2	no	—
82	C₄⋊Dic₃₇	—	2	1	no	—
83	C₄⋊Dic₃₇	—	2	1	no	—
84	C₄⋊Dic₃₇	—	2	1	no	—
85	C₄⋊Dic₃₇	—	2	1	no	—
86	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
87	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
90	#10596936	—	2	1	no	—
91	#10596936	—	2	1	no	—
92	#10596936	—	2	1	no	—
93	#10596936	—	2	1	no	—
94	#10596936	—	2	1	no	—
95	#10596936	D₄	1	1	yes	characteristic
96	#10596936	D₄	1	1	yes	characteristic
97	C₄×Dic₃₇	—	2	1	no	—
98	C₄×Dic₃₇	—	2	1	no	—
99	C₄×Dic₃₇	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
103	C₂×#64436	C₂³	1	1	yes	characteristic
104	C₂×#64436	C₂³	1	1	yes	characteristic
105	C₂³×C₃₇	—	2	1	no	—
106	Q₈×C₃₇	—	2	1	no	—
107	Q₈×C₃₇	—	2	1	no	—
108	Q₈×C₃₇	—	2	1	no	—
109	Q₈×C₃₇	#41	1	2	yes	—
110	D₄×C₃₇	—	2	1	no	—
111	D₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₃₇	—	2	1	no	—
119	C₂×Dic₃₇	—	2	1	no	—
120	C₂×Dic₃₇	—	2	1	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	—	4	2	no	—
123	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
127	Dic₇₄	—	2	2	no	—
128	Dic₇₄	—	2	1	no	—
129	Dic₇₄	—	2	1	no	—
130	Dic₇₄	—	8	1	no	—
131	Dic₇₄	—	8	1	no	—
132	Dic₇₄	—	8	1	no	—
133	C₈×C₃₇	—	2	1	no	—
134	C₈×C₃₇	—	2	1	no	—
135	#64436	—	2	1	no	—
136	#64436	—	2	1	no	—
137	C₂²×C₃₇	—	4	1	no	—
138	C₂²×C₃₇	—	4	1	no	—
139	C₂²×C₃₇	—	4	1	no	—
140	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	1	no	—
143	C₄×C₃₇	—	2	1	no	—
144	C₄×C₃₇	—	4	1	no	—
145	C₄×C₃₇	—	4	1	no	—
146	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
147	C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₃₇	—	2	1	no	—
149	Dic₃₇	—	2	1	no	—
150	Dic₃₇	—	2	1	no	—
151	Dic₃₇	—	4	1	no	—
152	Dic₃₇	—	4	1	no	—
153	Dic₃₇	—	4	1	no	—
154	Dic₃₇	—	4	1	no	—
155	Dic₃₇	#137	1	2	yes	—
156	#3369	—	37	1	no	maximal, Sylow
157	C₂×C₃₇	—	8	1	no	—
158	C₂×C₃₇	#548	1	1	yes	characteristic
159	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
160	C₂×C₃₇	#577	1	1	yes	characteristic
161	Q₈²	—	37	1	no	—
162	#556	—	37	1	no	—
163	#489	—	37	1	no	—
164	#489	—	37	1	no	—
165	#487	—	37	1	no	—
166	#476	—	37	1	no	—
167	#475	—	37	1	no	—
168	#475	—	37	1	no	—
169	#464	—	37	1	no	—
170	#464	D₃₇	1	1	yes	characteristic
171	#462	—	37	1	no	—
172	#446	—	37	1	no	—
173	#441	—	37	1	no	—
174	#441	—	37	1	no	—
175	C₄×SD₁₆	—	37	1	no	—
176	C₃₇	#3369	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	—	37	1	no	—
178	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
179	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	37	1	no	—
181	C₄⋊Q₈	—	37	1	no	—
182	C₄⋊Q₈	—	37	1	no	—
183	C₄⋊Q₈	—	37	1	no	—
184	C₄⋊Q₈	—	74	1	no	—
185	C₄⋊Q₈	—	74	1	no	—
186	C₄⋊Q₈	—	74	1	no	—
187	#126	—	74	1	no	—
188	#126	—	74	1	no	—
189	#124	—	37	1	no	—
190	#124	C₂×D₃₇	1	1	yes	characteristic
191	C₂²⋊Q₈	—	74	1	no	—
192	C₄×Q₈	—	37	1	no	—
193	C₄×Q₈	—	37	1	no	—
194	C₄×Q₈	—	37	1	no	—
195	C₄×Q₈	—	37	1	no	—
196	C₄×Q₈	—	37	1	no	—
197	C₄×Q₈	—	74	1	no	—
198	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
199	C₄×D₄	—	37	1	no	—
200	#117	—	74	1	no	—
201	#106	—	37	1	no	—
202	#106	—	37	1	no	—
203	#106	—	37	1	no	—
204	C₄⋊C₈	—	37	1	no	—
205	C₄⋊C₈	—	37	1	no	—
206	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
207	#103	—	37	1	no	—
208	#103	—	37	1	no	—
209	#103	—	37	1	no	—
210	#103	—	37	1	no	—
211	#103	—	37	1	no	—
212	#103	—	37	1	no	—
213	#103	C₂×D₃₇	1	1	yes	characteristic
214	#102	—	37	1	no	—
215	#102	—	37	1	no	—
216	#102	C₂×D₃₇	1	1	yes	characteristic
217	#97	—	37	1	no	—
218	#97	—	37	1	no	—
219	C₄×C₈	—	37	1	no	—
220	C₂×Q₈	—	37	1	no	—
221	C₂×Q₈	—	37	1	no	—
222	C₂×Q₈	—	74	1	no	—
223	C₂×Q₈	—	74	1	no	—
224	C₂×Q₈	—	74	1	no	—
225	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
227	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
228	C₂²×C₄	—	74	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	74	2	no	—
231	SD₁₆	—	2	2	no	—
232	C₂×C₈	—	37	1	no	—
233	C₂×C₈	—	37	1	no	—
234	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	37	1	no	—
238	C₄⋊C₄	—	37	1	no	—
239	C₄⋊C₄	—	37	1	no	—
240	C₄⋊C₄	—	37	1	no	—
241	C₄⋊C₄	—	37	1	no	—
242	C₄⋊C₄	—	37	1	no	—
243	C₄⋊C₄	—	74	1	no	—
244	C₄⋊C₄	—	74	1	no	—
245	C₄⋊C₄	—	74	1	no	—
246	C₄⋊C₄	—	74	1	no	—
247	C₄⋊C₄	—	74	1	no	—
248	C₄⋊C₄	—	74	1	no	—
249	C₄⋊C₄	—	74	1	no	—
250	C₄⋊C₄	—	74	1	no	—
251	C₄⋊C₄	—	74	1	no	—
252	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	74	1	no	—
256	C₂²⋊C₄	—	74	1	no	—
257	C₂²⋊C₄	—	74	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	37	1	no	—
260	C₄²	—	37	1	no	—
261	C₄²	—	74	1	no	—
262	C₄²	—	74	1	no	—
263	C₄²	—	74	1	no	—
264	C₄²	—	74	1	no	—
265	C₄²	C₂²×D₃₇	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	296	1	no	—
271	Q₈	—	296	1	no	—
272	Q₈	—	296	1	no	—
273	Q₈	—	74	2	no	—
274	Q₈	—	74	1	no	—
275	Q₈	—	74	1	no	—
276	Q₈	#10596964	1	2	yes	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	37	1	no	—
283	C₂×C₄	—	37	1	no	—
284	C₂×C₄	—	37	1	no	—
285	C₂×C₄	—	37	1	no	—
286	C₂×C₄	—	74	1	no	—
287	C₂×C₄	—	74	1	no	—
288	C₂×C₄	—	74	1	no	—
289	C₂×C₄	—	74	1	no	—
290	C₂×C₄	—	148	2	no	—
291	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
292	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	74	1	no	—
297	C₈	—	74	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	37	2	no	—
308	C₄	—	74	1	no	—
309	C₄	—	74	1	no	—
310	C₄	—	74	1	no	—
311	C₄	—	148	1	no	—
312	C₄	—	148	1	no	—
313	C₄	—	148	1	no	—
314	C₄	—	148	1	no	—
315	C₄	C₂×#10596964	1	1	yes	characteristic
316	C₄	#11924583	1	1	yes	characteristic
317	C₂	—	8	1	no	—
318	C₂	#14558189	1	1	yes	characteristic
319	C₂	#137×D₃₇	1	1	yes	characteristic
320	C₂	#14558340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#464	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×D₄×C₃₇	C₂³
hypercenter	#464	D₃₇
hypocenter	C₃₇	#3369
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₇₄	C₂	semidirect
5. #14557882	C₂	semidirect
7. #14557819	C₂	semidirect
8. #14557816	C₂	semidirect
10. #14557741	C₂	semidirect
11. #14557662	C₂	semidirect
12. #14557640	C₂	semidirect
13. #14557638	C₂	semidirect
176. C₃₇	#3369	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₇₄	2
#14558252	2
#14557892	2
#14557891	2
#14557882	2
#14557821	2
#14557819	2
#14557816	2
#14557753	2
#14557741	2
#14557662	2
#14557640	2
#14557638	2
SD₁₆×Dic₃₇	2
C₃₇×#464	2
#3369	37