Subgroups — #18011359

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558602	C₂	1	2	yes	maximal, complemented
2	#14558574	C₂	1	2	yes	maximal, complemented
3	#14558573	C₂	1	2	yes	maximal, complemented
4	SD₁₆×#3866	C₂	1	4	yes	maximal, complemented
5	C₂×#11924604	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558524	C₂	1	2	yes	maximal, complemented
7	#14558496	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₃₇	C₂	1	1	yes	maximal, characteristic
9	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
10	Q₈×#3866	C₂²	1	4	yes	—
11	D₄×#3866	—	2	4	no	—
12	D₄×#3866	C₂²	1	4	yes	—
13	#11924615	—	2	2	no	—
14	#11924615	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#3866	—	2	2	no	—
16	C₂×C₄⋊#3866	C₂²	1	2	yes	—
17	#11924604	C₂²	1	2	yes	—
18	#11924604	C₂²	1	2	yes	—
19	#11924599	C₂²	1	4	yes	—
20	#11924598	C₂²	1	4	yes	—
21	#11924590	C₂²	1	2	yes	—
22	C₈×#3866	C₂²	1	2	yes	—
23	#41×D₃₇	C₂²	1	2	yes	—
24	#11924559	C₄	1	1	yes	characteristic, complemented
25	#11924558	C₄	1	1	yes	characteristic, complemented
26	#11924557	C₄	1	2	yes	complemented
27	#11924556	C₄	1	2	yes	complemented
28	#11924555	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₃₇	C₂²	1	1	yes	characteristic
30	SD₁₆×D₃₇	C₂²	1	2	yes	—
31	D₈×D₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
33	#135×C₃₇	C₄	1	1	yes	characteristic, complemented
34	C₂²×#3866	—	2	2	no	—
35	C₂²×#3866	—	2	4	no	—
36	#41×C₃₇	C₂×C₄	1	2	yes	complemented
37	#10596966	C₂×C₄	1	2	yes	complemented
38	Q₈×D₃₇	C₂³	1	2	yes	—
39	#10596964	—	2	2	no	—
40	#10596964	C₂×C₄	1	2	yes	complemented
41	D₄×D₃₇	—	2	2	no	—
42	D₄×D₃₇	C₂³	1	2	yes	—
43	#10596962	—	2	2	no	—
44	#10596962	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₃₇	—	2	2	no	—
46	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
47	C₂²⋊#3866	—	2	4	no	—
48	C₂²⋊#3866	D₄	1	2	yes	—
49	#10596957	—	2	1	no	—
50	C₄⋊#3866	—	2	4	no	—
51	C₄⋊#3866	—	2	2	no	—
52	C₄⋊#3866	D₄	1	2	yes	—
53	C₄⋊#3866	C₂³	1	4	yes	—
54	C₄×#3866	—	2	4	no	—
55	C₄×#3866	C₂³	1	2	yes	—
56	#10596953	—	2	1	no	—
57	D₃₇⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
59	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
60	D₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
62	#10596942	C₂×C₄	1	2	yes	complemented
63	#10596941	C₂×C₄	1	2	yes	complemented
64	#10596940	C₂×C₄	1	2	yes	complemented
65	#10596939	C₂×C₄	1	2	yes	complemented
66	C₂×#64436	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₁₄₈	C₂×C₄	1	1	yes	characteristic, complemented
68	D₂₉₆	C₂×C₄	1	1	yes	characteristic, complemented
69	#10596930	C₂×C₄	1	2	yes	complemented
70	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic, complemented
71	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₃₇	C₂³	1	1	yes	characteristic
73	C₈×D₃₇	C₂³	1	1	yes	characteristic
74	C₂²×D₃₇	—	2	2	no	—
75	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂×#3866	—	2	4	no	—
77	C₂×#3866	—	8	4	no	—
78	C₂×#3866	—	8	2	no	—
79	C₂×#3866	C₂×D₄	1	4	yes	—
80	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
81	D₄×C₃₇	—	2	2	no	—
82	D₄×C₃₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₃₇	—	2	2	no	—
84	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₃₇	—	2	2	no	—
86	C₂²⋊D₃₇	—	2	2	no	—
87	C₂²⋊D₃₇	—	2	2	no	—
88	C₂×Dic₃₇	—	2	2	no	—
89	C₂×Dic₃₇	#41	1	1	yes	characteristic
90	D₁₄₈	—	2	2	no	—
91	D₁₄₈	C₂²×C₄	1	2	yes	—
92	C₄×D₃₇	—	2	2	no	—
93	C₄×D₃₇	—	2	2	no	—
94	C₄×D₃₇	—	2	2	no	—
95	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
97	C₄×D₃₇	#41	1	1	yes	characteristic
98	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
99	Dic₇₄	—	2	2	no	—
100	Dic₇₄	C₂²×C₄	1	2	yes	—
101	#64438	—	2	2	no	—
102	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
104	#64436	C₂²×C₄	1	1	yes	characteristic
105	#64436	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₃₇	—	2	2	no	—
107	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
108	C₂×D₃₇	—	2	2	no	—
109	C₂×D₃₇	—	2	2	no	—
110	C₂×D₃₇	—	4	2	no	—
111	C₂×D₃₇	—	4	2	no	—
112	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
113	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
114	#3866	—	2	4	no	—
115	#3866	—	4	4	no	—
116	C₄×C₃₇	—	2	2	no	—
117	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
118	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
119	Dic₃₇	—	2	2	no	—
120	Dic₃₇	C₄×D₄	1	1	yes	characteristic
121	Dic₃₇	C₂×#41	1	1	yes	characteristic
122	#2952	—	37	1	no	maximal, Sylow
123	C₂×C₃₇	—	2	1	no	—
124	C₂×C₃₇	—	4	2	no	—
125	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
126	D₃₇	—	2	1	no	—
127	D₃₇	—	4	2	no	—
128	D₃₇	#577	1	2	yes	—
129	C₂×#135	—	37	1	no	—
130	#520	—	37	2	no	—
131	#442	—	37	2	no	—
132	#441	—	37	2	no	—
133	C₄×SD₁₆	—	37	4	no	—
134	C₂×#106	—	37	1	no	—
135	#417	—	37	2	no	—
136	#405	—	37	1	no	—
137	C₃₇	#2952	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	37	2	no	—
139	#135	—	37	2	no	—
140	#135	—	37	1	no	—
141	#135	—	37	2	no	—
142	#135	—	37	1	no	—
143	#135	—	37	1	no	—
144	#135	#3866	1	1	yes	characteristic, complemented
145	C₂×Q₁₆	—	37	1	no	—
146	C₂×SD₁₆	—	37	2	no	—
147	C₂×D₈	—	37	1	no	—
148	C₂×M₁₆	—	37	1	no	—
149	C₂²×C₈	—	37	1	no	—
150	C₄×Q₈	—	37	4	no	—
151	C₄×D₄	—	37	4	no	—
152	C₄×D₄	—	74	4	no	—
153	#117	—	37	1	no	—
154	#117	—	74	2	no	—
155	C₂×C₄⋊C₄	—	37	2	no	—
156	C₂×C₄⋊C₄	—	74	2	no	—
157	#106	—	37	2	no	—
158	#106	—	37	2	no	—
159	#103	—	37	4	no	—
160	#102	—	37	4	no	—
161	#97	—	37	2	no	—
162	C₄×C₈	—	37	2	no	—
163	#41	—	37	2	no	—
164	#41	—	37	2	no	—
165	#41	—	37	2	no	—
166	#41	—	74	2	no	—
167	#41	—	74	2	no	—
168	#41	C₂×#3866	1	2	yes	complemented
169	C₂×Q₈	—	37	2	no	—
170	C₂×D₄	—	37	2	no	—
171	C₂×D₄	—	74	2	no	—
172	C₂²×C₄	—	37	1	no	—
173	C₂²×C₄	—	74	2	no	—
174	C₂²×C₄	—	74	4	no	—
175	C₂²×C₄	—	74	2	no	—
176	Q₁₆	—	37	2	no	—
177	Q₁₆	—	37	1	no	—
178	Q₁₆	C₂×#3866	1	1	yes	characteristic, complemented
179	SD₁₆	—	37	2	no	—
180	SD₁₆	—	37	2	no	—
181	SD₁₆	—	37	2	no	—
182	SD₁₆	C₂×#3866	1	2	yes	complemented
183	D₈	—	37	2	no	—
184	D₈	—	37	1	no	—
185	D₈	C₂×#3866	1	1	yes	characteristic, complemented
186	M₁₆	—	74	1	no	—
187	M₁₆	—	74	1	no	—
188	C₂×C₈	—	37	1	no	—
189	C₂×C₈	—	37	1	no	—
190	C₂×C₈	—	37	1	no	—
191	C₂×C₈	—	37	1	no	—
192	C₂×C₈	—	37	1	no	—
193	C₂×C₈	—	37	2	no	—
194	C₂×C₈	C₂×#3866	1	1	yes	characteristic, complemented
195	C₄⋊C₄	—	37	2	no	—
196	C₄⋊C₄	—	37	4	no	—
197	C₄⋊C₄	—	74	2	no	—
198	C₄⋊C₄	—	74	4	no	—
199	C₂²⋊C₄	—	37	2	no	—
200	C₂²⋊C₄	—	74	4	no	—
201	C₄²	—	37	2	no	—
202	C₄²	—	74	4	no	—
203	C₂³	—	37	1	no	—
204	C₂³	—	74	2	no	—
205	Q₈	—	37	2	no	—
206	Q₈	—	74	2	no	—
207	Q₈	C₂²×#3866	1	2	yes	—
208	D₄	—	2	2	no	—
209	D₄	—	37	2	no	—
210	D₄	—	74	2	no	—
211	D₄	—	74	2	no	—
212	D₄	—	74	2	no	—
213	D₄	—	74	2	no	—
214	D₄	C₂²×#3866	1	2	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	37	1	no	—
217	C₂×C₄	—	37	1	no	—
218	C₂×C₄	—	37	1	no	—
219	C₂×C₄	—	37	1	no	—
220	C₂×C₄	—	37	1	no	—
221	C₂×C₄	—	37	4	no	—
222	C₂×C₄	—	296	4	no	—
223	C₂×C₄	—	296	2	no	—
224	C₂×C₄	—	74	2	no	—
225	C₂×C₄	—	74	2	no	—
226	C₂×C₄	—	74	2	no	—
227	C₂×C₄	—	74	2	no	—
228	C₂×C₄	—	74	4	no	—
229	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
230	C₈	—	37	1	no	—
231	C₈	—	37	1	no	—
232	C₈	—	74	2	no	—
233	C₈	C₂²×#3866	1	1	yes	characteristic
234	C₈	C₂²×#3866	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	37	1	no	—
237	C₂²	—	37	1	no	—
238	C₂²	—	74	2	no	—
239	C₂²	—	74	2	no	—
240	C₂²	—	148	2	no	—
241	C₂²	—	148	2	no	—
242	C₂²	D₄×#3866	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	37	1	no	—
245	C₄	—	37	1	no	—
246	C₄	—	74	4	no	—
247	C₄	—	74	2	no	—
248	C₄	—	148	4	no	—
249	C₄	D₄×#3866	1	1	yes	characteristic
250	C₄	C₂³×#3866	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	37	2	no	—
254	C₂	—	74	1	no	—
255	C₂	—	148	2	no	—
256	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	#135×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#135×D₃₇	C₂
cosocle	C₄×D₃₇	C₂⁴
hypercenter	#135	#3866
hypocenter	C₃₇	#2952
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #14558602	C₂	semidirect
2. #14558574	C₂	semidirect
3. #14558573	C₂	semidirect
4. SD₁₆×#3866	C₂	semidirect
5. C₂×#11924604	C₂	semidirect
6. #14558524	C₂	semidirect
7. #14558496	C₂	semidirect
24. #11924559	C₄	semidirect
25. #11924558	C₄	semidirect
26. #11924557	C₄	semidirect
27. #11924556	C₄	semidirect
28. #11924555	C₄	semidirect
33. #135×C₃₇	C₄	semidirect
36. #41×C₃₇	C₂×C₄	semidirect
37. #10596966	C₂×C₄	semidirect
40. #10596964	C₂×C₄	semidirect
44. #10596962	C₂×C₄	semidirect
58. Q₁₆×C₃₇	C₂×C₄	semidirect
59. SD₁₆×C₃₇	C₂×C₄	semidirect
60. D₈×C₃₇	C₂×C₄	semidirect
61. C₂×C₈×C₃₇	C₂×C₄	semidirect
62. #10596942	C₂×C₄	semidirect
63. #10596941	C₂×C₄	semidirect
64. #10596940	C₂×C₄	semidirect
65. #10596939	C₂×C₄	semidirect
66. C₂×#64436	C₂×C₄	semidirect
67. Dic₁₄₈	C₂×C₄	semidirect
68. D₂₉₆	C₂×C₄	semidirect
69. #10596930	C₂×C₄	semidirect
70. C₈×D₃₇	C₂×C₄	semidirect
71. C₈×D₃₇	C₂×C₄	semidirect
137. C₃₇	#2952	semidirect
144. #135	#3866	semidirect
168. #41	C₂×#3866	semidirect
178. Q₁₆	C₂×#3866	semidirect
182. SD₁₆	C₂×#3866	semidirect
185. D₈	C₂×#3866	semidirect
194. 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
#14558602	2
#14558574	2
#14558573	2
SD₁₆×#3866	2
C₂×#11924604	2
#14558524	2
#14558496	2
#135×D₃₇	2
#2952	37