Subgroups — #18011356

Order 4736.

492 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	#14558601	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558600	C₂	1	1	yes	maximal, characteristic
3	#14558572	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#3866	C₂	1	4	yes	maximal, complemented
5	#14558543	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558526	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558525	C₂	1	1	yes	maximal, characteristic
8	#14558496	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₃₇	C₂	1	1	yes	maximal, characteristic
10	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#3866	—	2	2	no	—
12	Q₈×#3866	C₂²	1	4	yes	complemented
13	D₄×#3866	—	2	2	no	—
14	D₄×#3866	C₂²	1	4	yes	—
15	#11924615	—	2	2	no	—
16	#11924615	C₂²	1	1	yes	characteristic
17	#11924615	C₂²	1	1	yes	characteristic
18	#11924615	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#3866	—	2	2	no	—
20	#11924605	C₂²	1	2	yes	complemented
21	#11924604	C₂²	1	2	yes	complemented
22	#11924599	C₂²	1	4	yes	complemented
23	#11924598	C₂²	1	4	yes	—
24	#11924590	C₂²	1	2	yes	complemented
25	C₈×#3866	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₃₇	C₂²	1	1	yes	characteristic
28	C₂×#10596930	C₄	1	1	yes	characteristic, complemented
29	C₂×#10596940	C₄	1	1	yes	characteristic, complemented
30	C₂×#10596941	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₃₇	C₄	1	2	yes	complemented
32	SD₁₆×D₃₇	C₄	1	2	yes	complemented
33	SD₁₆×D₃₇	C₂²	1	2	yes	—
34	SD₁₆×D₃₇	C₂²	1	2	yes	—
35	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₃₇	C₄	1	1	yes	characteristic, complemented
37	C₂³×D₃₇	—	2	1	no	—
38	C₂²×#3866	—	2	4	no	—
39	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₃₇	—	2	1	no	—
42	Q₈×D₃₇	—	2	1	no	—
43	Q₈×D₃₇	—	2	1	no	—
44	Q₈×D₃₇	C₂×C₄	1	2	yes	complemented
45	Q₈×D₃₇	C₂³	1	2	yes	—
46	D₄×D₃₇	—	2	1	no	—
47	D₄×D₃₇	—	2	1	no	—
48	D₄×D₃₇	C₂×C₄	1	2	yes	—
49	D₄×D₃₇	C₂³	1	2	yes	—
50	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₇	—	2	1	no	—
52	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₇₄	—	2	1	no	—
54	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂²⋊#3866	—	2	1	no	—
56	C₂²⋊#3866	—	2	1	no	—
57	C₂²⋊#3866	—	2	4	no	—
58	C₂²⋊#3866	D₄	1	2	yes	—
59	#10596957	—	2	1	no	—
60	C₄⋊#3866	—	2	4	no	—
61	C₄⋊#3866	D₄	1	2	yes	—
62	C₄⋊#3866	C₂³	1	2	yes	—
63	C₄⋊#3866	C₂³	1	2	yes	—
64	C₄×#3866	—	2	4	no	—
65	C₄×#3866	—	2	1	no	—
66	C₄×#3866	—	2	1	no	—
67	C₄×#3866	C₂³	1	2	yes	—
68	#10596953	—	2	1	no	—
69	D₃₇⋊C₈	C₂³	1	2	yes	—
70	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
71	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
72	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
73	#10596941	C₂×C₄	1	2	yes	complemented
74	#10596941	C₂×C₄	1	2	yes	complemented
75	#10596940	C₂×C₄	1	2	yes	complemented
76	#10596940	C₂×C₄	1	2	yes	complemented
77	C₂×#64436	C₂×C₄	1	1	yes	characteristic, complemented
78	#10596930	C₂×C₄	1	2	yes	complemented
79	#10596930	C₂×C₄	1	2	yes	complemented
80	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic, complemented
81	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic, complemented
82	C₈×D₃₇	C₂³	1	1	yes	characteristic
83	C₈×D₃₇	C₂³	1	1	yes	characteristic
84	C₂³×C₃₇	—	2	1	no	—
85	C₂²×D₃₇	—	2	1	no	—
86	C₂²×D₃₇	—	2	2	no	—
87	C₂²×D₃₇	—	2	2	no	—
88	C₂²×D₃₇	—	8	2	no	—
89	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×#3866	—	2	2	no	—
91	C₂×#3866	—	2	2	no	—
92	C₂×#3866	—	8	4	no	—
93	C₂×#3866	C₂×D₄	1	4	yes	—
94	Q₈×C₃₇	—	2	1	no	—
95	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
96	D₄×C₃₇	—	2	1	no	—
97	D₄×C₃₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₃₇	—	2	1	no	—
99	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₃₇	—	2	2	no	—
101	C₂²⋊D₃₇	—	2	2	no	—
102	C₂×Dic₃₇	—	2	1	no	—
103	C₂×Dic₃₇	#41	1	1	yes	characteristic
104	D₁₄₈	—	2	1	no	—
105	D₁₄₈	C₂²×C₄	1	2	yes	—
106	C₄×D₃₇	—	2	2	no	—
107	C₄×D₃₇	—	2	2	no	—
108	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
110	C₄×D₃₇	#41	1	1	yes	characteristic
111	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
112	Dic₇₄	—	2	1	no	—
113	Dic₇₄	—	2	2	no	—
114	Dic₇₄	—	2	2	no	—
115	Dic₇₄	C₂²×C₄	1	2	yes	—
116	#64438	—	2	2	no	—
117	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
119	#64436	C₂²×C₄	1	1	yes	characteristic
120	#64436	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₃₇	—	2	2	no	—
122	C₂²×C₃₇	—	8	1	no	—
123	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂×D₃₇	—	2	2	no	—
125	C₂×D₃₇	—	2	2	no	—
126	C₂×D₃₇	—	4	4	no	—
127	C₂×D₃₇	—	8	2	no	—
128	C₂×D₃₇	—	8	1	no	—
129	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
130	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
131	#3866	—	2	4	no	—
132	#3866	—	4	2	no	—
133	#3866	—	4	2	no	—
134	C₄×C₃₇	—	2	2	no	—
135	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
136	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
137	Dic₃₇	—	2	2	no	—
138	Dic₃₇	C₄×D₄	1	1	yes	characteristic
139	Dic₃₇	C₂×#41	1	1	yes	characteristic
140	#2949	—	37	1	no	maximal, Sylow
141	C₂×C₃₇	—	2	1	no	—
142	C₂×C₃₇	—	4	2	no	—
143	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
144	D₃₇	—	2	1	no	—
145	D₃₇	—	4	2	no	—
146	D₃₇	#577	1	2	yes	—
147	C₂²×SD₁₆	—	37	1	no	—
148	#519	—	37	1	no	—
149	#518	—	37	1	no	—
150	#440	—	37	4	no	—
151	C₄×SD₁₆	—	37	4	no	—
152	#427	—	37	1	no	—
153	#419	—	37	1	no	—
154	#418	—	37	1	no	—
155	#405	—	37	1	no	—
156	C₃₇	#2949	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	37	1	no	—
158	C₂²×D₄	—	37	1	no	—
159	C₂×SD₁₆	—	37	1	no	—
160	C₂×SD₁₆	—	37	1	no	—
161	C₂×SD₁₆	—	37	2	no	—
162	C₂×SD₁₆	—	37	2	no	—
163	C₂×SD₁₆	—	37	2	no	—
164	C₂×SD₁₆	—	37	2	no	—
165	C₂×SD₁₆	—	37	1	no	—
166	C₂×SD₁₆	#3866	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	37	1	no	—
168	C₂²×C₈	—	37	1	no	—
169	C₄×Q₈	—	37	4	no	—
170	C₄×Q₈	—	74	2	no	—
171	C₄×D₄	—	37	4	no	—
172	C₄×D₄	—	74	2	no	—
173	#117	—	37	1	no	—
174	#117	—	37	1	no	—
175	#117	—	37	1	no	—
176	#117	—	74	2	no	—
177	C₂×C₂²⋊C₄	—	74	2	no	—
178	#107	—	37	2	no	—
179	#106	—	37	2	no	—
180	#103	—	37	4	no	—
181	#102	—	37	4	no	—
182	#97	—	37	2	no	—
183	C₄×C₈	—	37	2	no	—
184	C₂⁴	—	74	1	no	—
185	C₂×Q₈	—	37	2	no	—
186	C₂×Q₈	—	37	2	no	—
187	C₂×Q₈	—	37	1	no	—
188	C₂×Q₈	—	74	1	no	—
189	C₂×Q₈	—	74	1	no	—
190	C₂×Q₈	—	74	1	no	—
191	C₂×Q₈	C₂×#3866	1	1	yes	characteristic, complemented
192	C₂×D₄	—	37	2	no	—
193	C₂×D₄	—	37	2	no	—
194	C₂×D₄	—	37	1	no	—
195	C₂×D₄	—	74	1	no	—
196	C₂×D₄	—	74	1	no	—
197	C₂×D₄	—	74	1	no	—
198	C₂×D₄	C₂×#3866	1	1	yes	characteristic
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	74	1	no	—
201	C₂²×C₄	—	74	4	no	—
202	SD₁₆	—	37	2	no	—
203	SD₁₆	—	37	2	no	—
204	SD₁₆	—	37	2	no	—
205	SD₁₆	—	37	2	no	—
206	SD₁₆	—	37	2	no	—
207	SD₁₆	—	37	2	no	—
208	SD₁₆	C₂×#3866	1	2	yes	complemented
209	SD₁₆	C₂×#3866	1	2	yes	complemented
210	M₁₆	—	74	1	no	—
211	M₁₆	—	74	1	no	—
212	C₂×C₈	—	37	1	no	—
213	C₂×C₈	—	37	1	no	—
214	C₂×C₈	—	37	1	no	—
215	C₂×C₈	—	37	1	no	—
216	C₂×C₈	—	37	1	no	—
217	C₂×C₈	—	37	2	no	—
218	C₂×C₈	C₂×#3866	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	37	2	no	—
220	C₄⋊C₄	—	37	2	no	—
221	C₄⋊C₄	—	37	2	no	—
222	C₄⋊C₄	—	74	4	no	—
223	C₂²⋊C₄	—	37	2	no	—
224	C₂²⋊C₄	—	74	4	no	—
225	C₂²⋊C₄	—	74	1	no	—
226	C₂²⋊C₄	—	74	1	no	—
227	C₄²	—	37	2	no	—
228	C₄²	—	74	1	no	—
229	C₄²	—	74	1	no	—
230	C₄²	—	74	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	37	1	no	—
233	C₂³	—	296	2	no	—
234	C₂³	—	74	2	no	—
235	C₂³	—	74	2	no	—
236	C₂³	—	74	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	37	2	no	—
239	Q₈	—	74	2	no	—
240	Q₈	—	74	2	no	—
241	Q₈	—	74	1	no	—
242	Q₈	C₂²×#3866	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	37	2	no	—
245	D₄	—	74	2	no	—
246	D₄	—	74	2	no	—
247	D₄	—	74	1	no	—
248	D₄	C₂²×#3866	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	37	1	no	—
251	C₂×C₄	—	37	1	no	—
252	C₂×C₄	—	37	1	no	—
253	C₂×C₄	—	37	1	no	—
254	C₂×C₄	—	37	1	no	—
255	C₂×C₄	—	37	4	no	—
256	C₂×C₄	—	296	4	no	—
257	C₂×C₄	—	74	2	no	—
258	C₂×C₄	—	74	2	no	—
259	C₂×C₄	—	74	1	no	—
260	C₂×C₄	—	74	2	no	—
261	C₂×C₄	—	74	2	no	—
262	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
263	C₈	—	37	1	no	—
264	C₈	—	37	1	no	—
265	C₈	—	74	2	no	—
266	C₈	C₂²×#3866	1	1	yes	characteristic
267	C₈	C₂²×#3866	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	37	1	no	—
271	C₂²	—	37	1	no	—
272	C₂²	—	296	1	no	—
273	C₂²	—	296	2	no	—
274	C₂²	—	74	2	no	—
275	C₂²	—	74	2	no	—
276	C₂²	—	148	4	no	—
277	C₂²	D₄×#3866	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	37	1	no	—
280	C₄	—	37	1	no	—
281	C₄	—	74	4	no	—
282	C₄	—	74	2	no	—
283	C₄	—	148	2	no	—
284	C₄	—	148	2	no	—
285	C₄	D₄×#3866	1	1	yes	characteristic
286	C₄	C₂³×#3866	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	37	2	no	—
290	C₂	—	74	1	no	—
291	C₂	—	148	2	no	—
292	C₂	C₂×D₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#3866
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₄	C₂³×#3866
Fitting	C₂×SD₁₆×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	C₂×D₄×D₃₇	C₂²
cosocle	C₄×D₃₇	C₂⁴
hypercenter	C₂×SD₁₆	#3866
hypocenter	C₃₇	#2949
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #14558601	C₂	semidirect
3. #14558572	C₂	semidirect
4. SD₁₆×#3866	C₂	semidirect
5. #14558543	C₂	semidirect
6. #14558526	C₂	semidirect
8. #14558496	C₂	semidirect
12. Q₈×#3866	C₂²	semidirect
20. #11924605	C₂²	semidirect
21. #11924604	C₂²	semidirect
22. #11924599	C₂²	semidirect
24. #11924590	C₂²	semidirect
25. C₈×#3866	C₂²	semidirect
28. C₂×#10596930	C₄	semidirect
29. C₂×#10596940	C₄	semidirect
30. C₂×#10596941	C₄	semidirect
31. SD₁₆×D₃₇	C₄	semidirect
32. SD₁₆×D₃₇	C₄	semidirect
36. C₂×SD₁₆×C₃₇	C₄	semidirect
39. C₂×Q₈×C₃₇	C₂×C₄	semidirect
44. Q₈×D₃₇	C₂×C₄	semidirect
54. C₂×Dic₇₄	C₂×C₄	semidirect
70. SD₁₆×C₃₇	C₂×C₄	semidirect
71. SD₁₆×C₃₇	C₂×C₄	semidirect
72. C₂×C₈×C₃₇	C₂×C₄	semidirect
73. #10596941	C₂×C₄	semidirect
74. #10596941	C₂×C₄	semidirect
75. #10596940	C₂×C₄	semidirect
76. #10596940	C₂×C₄	semidirect
77. C₂×#64436	C₂×C₄	semidirect
78. #10596930	C₂×C₄	semidirect
79. #10596930	C₂×C₄	semidirect
80. C₈×D₃₇	C₂×C₄	semidirect
81. C₈×D₃₇	C₂×C₄	semidirect
156. C₃₇	#2949	semidirect
166. C₂×SD₁₆	#3866	semidirect
191. C₂×Q₈	C₂×#3866	semidirect
208. SD₁₆	C₂×#3866	semidirect
209. SD₁₆	C₂×#3866	semidirect
218. C₂×C₈	C₂×#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558601	2
#14558600	2
#14558572	2
SD₁₆×#3866	2
#14558543	2
#14558526	2
#14558525	2
#14558496	2
C₂×SD₁₆×D₃₇	2
#2949	37