Finite Group Databasebeta

Subgroups — #18010602

Order 4736.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558601	C₂	1	1	yes	maximal, characteristic
2	#14558526	C₂	1	2	yes	maximal
3	D₃₇⋊#101	C₂	1	1	yes	maximal, characteristic
4	#14558404	C₂	1	2	yes	maximal
5	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
6	D₃₇⋊M₁₆	C₂²	1	2	yes	—
7	Q₈×#3866	—	2	4	no	—
8	#11924615	—	2	2	no	—
9	#11924615	C₂²	1	2	yes	—
10	#11924599	—	2	4	no	—
11	#11924597	—	2	2	no	—
12	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
14	#11924536	C₄	1	1	yes	characteristic, complemented
15	#11924535	—	2	2	no	—
16	#11924534	C₄	1	1	yes	characteristic, complemented
17	#11924534	C₄	1	1	yes	characteristic, complemented
18	C₄⋊C₄×D₃₇	—	2	2	no	—
19	C₄²×D₃₇	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
23	Q₈×D₃₇	—	2	2	no	—
24	Q₈×D₃₇	—	2	2	no	—
25	Q₈×D₃₇	—	2	2	no	—
26	Q₈×D₃₇	—	2	2	no	—
27	C₂×C₄×D₃₇	—	2	1	no	—
28	C₂×C₄×D₃₇	—	2	1	no	—
29	C₂×C₄×D₃₇	D₄	1	2	yes	—
30	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₇₄	—	2	1	no	—
32	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₇₄	D₄	1	2	yes	—
35	C₂²⋊#3866	—	2	2	no	—
36	#10596957	—	2	2	no	—
37	C₄⋊#3866	—	2	4	no	—
38	C₄⋊#3866	D₄	1	4	yes	—
39	C₄×#3866	—	2	4	no	—
40	C₄×#3866	—	2	2	no	—
41	#10596953	—	2	2	no	—
42	D₃₇⋊C₈	—	2	2	no	—
43	C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
45	C₄⋊Dic₃₇	—	2	2	no	—
46	#10596936	—	2	2	no	—
47	#10596936	—	2	2	no	—
48	C₄×Dic₃₇	—	2	1	no	—
49	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×#3866	—	2	4	no	—
52	Q₈×C₃₇	—	2	2	no	—
53	Q₈×C₃₇	—	2	2	no	—
54	C₂×C₄×C₃₇	—	2	1	no	—
55	C₂×C₄×C₃₇	—	2	1	no	—
56	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₃₇	—	2	1	no	—
59	C₂×Dic₃₇	—	2	1	no	—
60	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
61	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
62	C₄×D₃₇	—	2	4	no	—
63	C₄×D₃₇	—	2	2	no	—
64	C₄×D₃₇	—	2	2	no	—
65	C₄×D₃₇	—	4	1	no	—
66	C₄×D₃₇	—	4	1	no	—
67	C₄×D₃₇	C₂×D₄	1	2	yes	—
68	C₄×D₃₇	#41	1	2	yes	—
69	Dic₇₄	—	2	2	no	—
70	Dic₇₄	—	2	4	no	—
71	Dic₇₄	—	2	2	no	—
72	Dic₇₄	—	4	2	no	—
73	#64438	—	4	2	no	—
74	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×D₃₇	—	2	2	no	—
76	C₂×D₃₇	#120	1	1	yes	characteristic
77	C₂×D₃₇	#123	1	1	yes	characteristic
78	#3866	—	4	4	no	—
79	C₄×C₃₇	—	2	2	no	—
80	C₄×C₃₇	—	2	2	no	—
81	C₄×C₃₇	—	4	1	no	—
82	C₄×C₃₇	C₄×D₄	1	2	yes	—
83	Dic₃₇	—	2	2	no	—
84	Dic₃₇	—	2	2	no	—
85	Dic₃₇	—	4	1	no	—
86	Dic₃₇	#121	1	2	yes	—
87	#1886	—	37	1	no	maximal, Sylow
88	C₂×C₃₇	—	2	1	no	—
89	C₂×C₃₇	#386	1	1	yes	characteristic
90	D₃₇	—	2	1	no	—
91	D₃₇	#456	1	2	yes	—
92	C₂×C₄⋊Q₈	—	37	1	no	—
93	#519	—	37	1	no	—
94	#419	—	37	2	no	—
95	C₂×#101	—	37	1	no	—
96	#339	—	37	2	no	—
97	C₃₇	#1886	1	1	yes	characteristic, complemented, Sylow
98	C₂²×Q₈	—	37	1	no	—
99	C₂²×Q₈	—	37	1	no	—
100	C₂×M₁₆	—	37	2	no	—
101	C₄⋊Q₈	—	37	1	no	—
102	C₄⋊Q₈	—	37	1	no	—
103	C₄⋊Q₈	—	37	1	no	—
104	C₄⋊Q₈	—	74	2	no	—
105	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
106	C₄×Q₈	—	74	4	no	—
107	#117	—	37	2	no	—
108	#117	—	74	2	no	—
109	C₂×C₄⋊C₄	—	74	2	no	—
110	C₂×C₄²	—	37	1	no	—
111	#103	—	74	4	no	—
112	#101	—	74	2	no	—
113	C₂×Q₈	—	37	2	no	—
114	C₂×Q₈	—	37	1	no	—
115	C₂×Q₈	—	37	1	no	—
116	C₂×Q₈	—	74	1	no	—
117	C₂×Q₈	—	74	2	no	—
118	C₂×Q₈	—	74	2	no	—
119	C₂×Q₈	—	74	2	no	—
120	C₂×Q₈	—	74	2	no	—
121	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
122	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
123	C₂²×C₄	—	37	1	no	—
124	C₂²×C₄	—	37	2	no	—
125	C₂²×C₄	—	74	1	no	—
126	C₂²×C₄	—	74	1	no	—
127	M₁₆	—	74	2	no	—
128	M₁₆	—	74	2	no	—
129	C₂×C₈	—	74	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	37	4	no	—
132	C₄⋊C₄	—	74	2	no	—
133	C₄⋊C₄	—	74	2	no	—
134	C₄⋊C₄	—	74	4	no	—
135	C₄⋊C₄	—	74	2	no	—
136	C₂²⋊C₄	—	74	2	no	—
137	C₄²	—	37	1	no	—
138	C₄²	—	74	1	no	—
139	C₄²	—	74	2	no	—
140	C₄²	—	74	4	no	—
141	C₄²	C₂×#3866	1	1	yes	characteristic
142	C₂³	—	37	1	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	74	4	no	—
146	Q₈	—	74	2	no	—
147	Q₈	—	74	2	no	—
148	Q₈	—	148	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	37	1	no	—
152	C₂×C₄	—	37	2	no	—
153	C₂×C₄	—	37	2	no	—
154	C₂×C₄	—	37	2	no	—
155	C₂×C₄	—	74	2	no	—
156	C₂×C₄	—	74	2	no	—
157	C₂×C₄	—	74	4	no	—
158	C₂×C₄	—	74	1	no	—
159	C₂×C₄	—	74	4	no	—
160	C₂×C₄	—	74	1	no	—
161	C₂×C₄	—	148	1	no	—
162	C₂×C₄	—	148	1	no	—
163	C₂×C₄	C₂²⋊#3866	1	2	yes	—
164	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
165	C₈	—	148	2	no	—
166	C₂²	—	37	1	no	—
167	C₂²	—	37	1	no	—
168	C₂²	—	74	2	no	—
169	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	2	2	no	—
172	C₄	—	4	1	no	—
173	C₄	—	37	2	no	—
174	C₄	—	74	2	no	—
175	C₄	—	74	2	no	—
176	C₄	—	148	4	no	—
177	C₄	—	148	1	no	—
178	C₄	D₄×#3866	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	37	2	no	—
181	C₂	—	74	1	no	—
182	C₂	#14558481	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14558481
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂×C₃₇	#386
prime radical	C₂²×D₃₇	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1886
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
14. #11924536	C₄	semidirect
16. #11924534	C₄	semidirect
17. #11924534	C₄	semidirect
20. C₄⋊Q₈×C₃₇	C₄	semidirect
97. C₃₇	#1886	semidirect
105. C₄⋊Q₈	#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
#14558526	2
D₃₇⋊#101	2
#14558404	2
C₄⋊Q₈×D₃₇	2
#1886	37