Subgroups — #18003286

Order 4736.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#64436	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558044	C₂	1	1	yes	maximal, characteristic
4	#14557601	C₂	1	4	yes	maximal, complemented
5	#14557596	C₂	1	4	yes	maximal, complemented
6	#14557561	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557501	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557491	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596962	C₂²	1	1	yes	characteristic
11	#11924544	—	2	4	no	—
12	C₂×#10596929	C₂²	1	2	yes	complemented
13	C₂²×#64436	—	2	4	no	—
14	C₂²×#64436	C₂²	1	1	yes	characteristic
15	C₂²×#64436	C₂²	1	2	yes	—
16	C₂²×#64436	C₂²	1	2	yes	—
17	C₂²×#64436	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₃₇	C₂²	1	2	yes	complemented
19	#11924519	C₄	1	2	yes	—
20	#11924515	C₄	1	2	yes	—
21	#11924510	C₄	1	2	yes	—
22	#11924505	C₄	1	2	yes	—
23	C₄×C₂²⋊D₃₇	C₂²	1	4	yes	—
24	#11924495	C₂²	1	1	yes	characteristic
25	#11924465	C₂²	1	4	yes	complemented
26	#11924448	C₂²	1	4	yes	complemented
27	#11924446	C₂²	1	2	yes	complemented
28	C₈×Dic₃₇	C₂²	1	2	yes	complemented
29	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
32	C₂²⋊C₈×C₃₇	C₂²	1	4	yes	—
33	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	—	2	2	no	—
35	C₂²×C₄×C₃₇	—	2	2	no	—
36	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
42	#10596962	—	2	2	no	—
43	#10596962	—	2	2	no	—
44	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₃₇	C₂³	1	2	yes	—
46	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
47	M₁₆×C₃₇	—	2	2	no	—
48	M₁₆×C₃₇	—	2	2	no	—
49	C₂×C₈×C₃₇	C₂³	1	2	yes	—
50	C₂×C₈×C₃₇	C₂³	1	2	yes	—
51	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
52	C₂²⋊Dic₃₇	C₂×C₄	1	4	yes	—
53	#10596938	C₂×C₄	1	4	yes	—
54	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
55	#10596936	C₂×C₄	1	4	yes	—
56	C₄×Dic₃₇	C₂³	1	2	yes	—
57	C₂×#64436	—	2	2	no	—
58	C₂×#64436	—	2	4	no	—
59	C₂×#64436	—	2	4	no	—
60	C₂×#64436	D₄	1	4	yes	complemented
61	C₂×#64436	C₂³	1	2	yes	—
62	C₂×#64436	C₂³	1	1	yes	characteristic
63	C₂×#64436	C₂³	1	1	yes	characteristic
64	#10596929	—	2	4	no	—
65	C₈×D₃₇	—	2	4	no	—
66	C₂³×C₃₇	—	2	2	no	—
67	C₂³×C₃₇	—	2	4	no	—
68	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
70	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₃₇	—	2	2	no	—
72	C₂×C₄×C₃₇	—	2	4	no	—
73	C₂×C₄×C₃₇	—	2	2	no	—
74	C₂×C₄×C₃₇	—	2	2	no	—
75	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₃₇	#41	1	2	yes	—
79	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₃₇	—	2	4	no	—
81	C₂²⋊D₃₇	—	2	8	no	—
82	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
83	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
84	D₁₄₈	—	2	2	no	—
85	C₄×D₃₇	—	2	2	no	—
86	C₄×D₃₇	—	2	2	no	—
87	Dic₇₄	—	2	2	no	—
88	C₈×C₃₇	—	2	2	no	—
89	C₈×C₃₇	—	2	2	no	—
90	#64436	—	2	2	no	—
91	#64436	C₂×D₄	1	2	yes	—
92	#64436	C₂×D₄	1	2	yes	—
93	C₂²×C₃₇	—	2	8	no	—
94	C₂²×C₃₇	—	2	2	no	—
95	C₂²×C₃₇	—	2	2	no	—
96	C₂²×C₃₇	—	2	2	no	—
97	C₂²×C₃₇	C₄×D₄	1	2	yes	—
98	C₂²×C₃₇	#131	1	4	yes	complemented
99	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₃₇	—	2	2	no	—
101	C₂×D₃₇	—	2	4	no	—
102	C₄×C₃₇	—	2	2	no	—
103	C₄×C₃₇	C₄×D₄	1	2	yes	—
104	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
106	Dic₃₇	—	2	2	no	—
107	Dic₃₇	—	2	4	no	—
108	#2929	—	37	1	no	maximal, Sylow
109	C₂×C₃₇	—	2	2	no	—
110	C₂×C₃₇	—	2	4	no	—
111	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₃₇	C₂×#131	1	2	yes	—
113	D₃₇	—	4	2	no	—
114	C₂×#131	—	37	1	no	—
115	C₂³×C₈	—	37	1	no	—
116	#525	—	37	1	no	—
117	#435	—	37	4	no	—
118	C₈×D₄	—	37	4	no	—
119	#424	—	37	1	no	—
120	#408	—	37	1	no	—
121	#407	D₃₇	1	1	yes	characteristic, complemented
122	#405	—	37	1	no	—
123	C₃₇	#2929	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	37	1	no	—
125	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
126	#131	—	74	4	no	—
127	C₂×M₁₆	—	37	2	no	—
128	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
129	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
130	C₂²×C₈	—	37	1	no	—
131	C₂²×C₈	—	37	1	no	—
132	C₂²×C₈	—	37	2	no	—
133	C₂²×C₈	—	37	2	no	—
134	C₂²×C₈	—	37	2	no	—
135	C₂²×C₈	—	74	4	no	—
136	#123	—	37	2	no	—
137	C₂²⋊Q₈	—	37	2	no	—
138	#121	—	37	2	no	—
139	#120	—	37	2	no	—
140	C₄×D₄	—	37	4	no	—
141	#117	—	37	1	no	—
142	C₄⋊C₈	—	37	4	no	—
143	C₂²⋊C₈	—	37	4	no	—
144	C₂²⋊C₈	C₂×D₃₇	1	4	yes	—
145	#97	—	37	2	no	—
146	C₄×C₈	—	37	2	no	—
147	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
148	#41	—	74	2	no	—
149	#41	—	74	2	no	—
150	C₂×Q₈	—	37	1	no	—
151	C₂×D₄	—	37	4	no	—
152	C₂×D₄	—	37	2	no	—
153	C₂×D₄	—	37	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	37	2	no	—
157	C₂²×C₄	C₄×D₃₇	1	2	yes	—
158	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	74	4	no	—
164	C₂×C₈	—	37	4	no	—
165	C₂×C₈	—	37	1	no	—
166	C₂×C₈	—	37	1	no	—
167	C₂×C₈	—	37	2	no	—
168	C₂×C₈	—	74	4	no	—
169	C₂×C₈	—	74	4	no	—
170	C₂×C₈	—	74	4	no	—
171	C₂×C₈	—	74	2	no	—
172	C₂×C₈	C₂²×D₃₇	1	2	yes	—
173	C₂×C₈	C₂²×D₃₇	1	2	yes	—
174	C₄⋊C₄	—	37	4	no	—
175	C₄⋊C₄	—	37	2	no	—
176	C₂²⋊C₄	—	37	2	no	—
177	C₂²⋊C₄	—	37	4	no	—
178	C₂²⋊C₄	—	37	4	no	—
179	C₄²	—	37	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	37	2	no	—
183	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
184	C₂³	C₂×C₄×D₃₇	1	2	yes	—
185	Q₈	—	74	2	no	—
186	D₄	—	74	4	no	—
187	D₄	—	74	8	no	—
188	D₄	—	74	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	37	4	no	—
194	C₂×C₄	—	37	2	no	—
195	C₂×C₄	—	74	2	no	—
196	C₂×C₄	—	74	2	no	—
197	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
199	C₂×C₄	D₄×D₃₇	1	2	yes	—
200	C₂×C₄	#10596964	1	2	yes	—
201	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	37	2	no	—
205	C₈	—	37	2	no	—
206	C₈	—	74	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	74	2	no	—
212	C₂²	—	74	4	no	—
213	C₂²	#11924496	1	2	yes	—
214	C₂²	#11924544	1	4	yes	complemented
215	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	74	2	no	—
218	C₄	—	74	4	no	—
219	C₄	#11924496	1	2	yes	—
220	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
221	C₄	C₂×#10596964	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	148	2	no	—
225	C₂	C₂×#11924496	1	1	yes	characteristic
226	C₂	C₂×#11924544	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#407	C₂
prime radical	#14558044	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#407	D₃₇
hypocenter	C₃₇	#2929
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
1. C₂×#11924544	C₂	semidirect
2. C₂³×#64436	C₂	semidirect
4. #14557601	C₂	semidirect
5. #14557596	C₂	semidirect
6. #14557561	C₂	semidirect
7. #14557501	C₂	semidirect
8. #14557491	C₂	semidirect
9. C₃₇×#407	C₂	semidirect
12. C₂×#10596929	C₂²	semidirect
18. C₂×C₈×D₃₇	C₂²	semidirect
25. #11924465	C₂²	semidirect
26. #11924448	C₂²	semidirect
27. #11924446	C₂²	semidirect
28. C₈×Dic₃₇	C₂²	semidirect
60. C₂×#64436	D₄	semidirect
98. C₂²×C₃₇	#131	semidirect
121. #407	D₃₇	semidirect
123. C₃₇	#2929	semidirect
214. C₂²	#11924544	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924544	2
C₂³×#64436	2
#14558044	2
#14557601	2
#14557596	2
#14557561	2
#14557501	2
#14557491	2
C₃₇×#407	2
#2929	37