Finite Group Databasebeta

Subgroups — #18010870

Order 4736.

258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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	#14558514	C₂	1	2	yes	maximal, complemented
2	#14558439	C₂	1	2	yes	maximal, complemented
3	#14558438	C₂	1	2	yes	maximal, complemented
4	#14558084	C₂	1	1	yes	maximal, characteristic
5	#11924615	—	2	2	no	—
6	#11924595	—	2	2	no	—
7	#11924595	C₂²	1	4	yes	—
8	C₂×#10596966	C₄	1	1	yes	characteristic, complemented
9	C₂×#10596966	C₂²	1	1	yes	characteristic
10	#11924533	C₂²	1	1	yes	characteristic
11	#11924525	C₄	1	1	yes	characteristic, complemented
12	#11924525	C₂²	1	1	yes	characteristic
13	#11924511	—	2	2	no	—
14	#11924498	—	2	2	no	—
15	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
16	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
19	#10596966	—	4	1	no	—
20	#10596966	—	4	1	no	—
21	#10596966	—	4	1	no	—
22	#10596966	—	4	1	no	—
23	C₂×D₁₄₈	—	2	1	no	—
24	C₂×D₁₄₈	—	2	2	no	—
25	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₄₈	D₄	1	2	yes	—
27	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₃₇	—	2	1	no	—
29	C₂×C₄×D₃₇	—	2	1	no	—
30	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₃₇	D₄	1	2	yes	—
32	C₂²⋊#3866	—	2	4	no	—
33	C₄⋊#3866	—	4	2	no	—
34	C₄×#3866	—	4	2	no	—
35	C₄⋊C₄×C₃₇	—	2	2	no	—
36	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
37	#10596938	—	2	2	no	—
38	#10596938	—	2	2	no	—
39	C₄×Dic₃₇	—	2	1	no	—
40	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₃₇	—	2	1	no	—
42	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₃₇	C₂×D₄	1	2	yes	—
44	C₂×#3866	—	4	4	no	—
45	Q₈×C₃₇	—	4	1	no	—
46	Q₈×C₃₇	—	4	1	no	—
47	C₂×C₄×C₃₇	—	2	1	no	—
48	C₂×C₄×C₃₇	—	2	1	no	—
49	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₃₇	—	2	1	no	—
52	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
53	D₁₄₈	—	4	2	no	—
54	D₁₄₈	—	4	1	no	—
55	D₁₄₈	—	4	1	no	—
56	D₁₄₈	—	4	4	no	—
57	D₁₄₈	—	4	2	no	—
58	C₄×D₃₇	—	2	2	no	—
59	C₄×D₃₇	—	2	2	no	—
60	C₄×D₃₇	—	2	1	no	—
61	C₄×D₃₇	—	2	1	no	—
62	C₄×D₃₇	—	4	2	no	—
63	C₄×D₃₇	—	4	1	no	—
64	C₄×D₃₇	—	4	1	no	—
65	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₃₇	—	2	1	no	—
67	C₂×D₃₇	—	2	2	no	—
68	C₂×D₃₇	—	4	1	no	—
69	C₂×D₃₇	—	4	2	no	—
70	C₂×D₃₇	—	4	2	no	—
71	C₂×D₃₇	—	4	1	no	—
72	#3866	—	8	4	no	—
73	C₄×C₃₇	—	2	2	no	—
74	C₄×C₃₇	—	2	1	no	—
75	C₄×C₃₇	—	2	2	no	—
76	C₄×C₃₇	—	4	1	no	—
77	Dic₃₇	—	4	1	no	—
78	Dic₃₇	#99	1	1	yes	characteristic
79	Dic₃₇	#99	1	1	yes	characteristic
80	#2126	—	37	1	no	maximal, Sylow
81	C₂×C₃₇	—	2	1	no	—
82	C₂×C₃₇	C₂×#99	1	1	yes	characteristic
83	D₃₇	—	4	2	no	—
84	D₃₇	—	4	1	no	—
85	D₃₇	—	8	1	no	—
86	#532	—	37	1	no	—
87	#410	—	37	2	no	—
88	#354	—	37	2	no	—
89	#353	—	37	2	no	—
90	C₃₇	#2126	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	37	1	no	—
92	C₂×#41	—	37	1	no	—
93	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
94	#127	—	37	1	no	—
95	#124	—	37	1	no	—
96	#124	—	37	1	no	—
97	#121	—	74	2	no	—
98	C₄×D₄	—	74	2	no	—
99	#117	—	74	2	no	—
100	C₂×C₄²	—	37	1	no	—
101	#99	—	37	4	no	—
102	#99	—	74	2	no	—
103	#41	—	148	1	no	—
104	#41	—	148	1	no	—
105	#41	—	148	1	no	—
106	#41	—	148	1	no	—
107	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
108	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
109	C₂×D₄	—	37	2	no	—
110	C₂×D₄	—	37	1	no	—
111	C₂×D₄	—	37	1	no	—
112	C₂×D₄	—	74	2	no	—
113	C₂×D₄	—	74	1	no	—
114	C₂²×C₄	—	37	2	no	—
115	C₂²×C₄	—	37	1	no	—
116	C₂²×C₄	—	74	1	no	—
117	C₂²×C₄	—	74	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	148	2	no	—
120	C₂²⋊C₄	—	74	2	no	—
121	C₂²⋊C₄	—	74	2	no	—
122	C₂²⋊C₄	—	74	4	no	—
123	C₄²	—	37	1	no	—
124	C₄²	—	74	1	no	—
125	C₄²	—	148	2	no	—
126	C₄²	C₂×#3866	1	1	yes	characteristic
127	C₂³	—	37	2	no	—
128	C₂³	—	37	1	no	—
129	C₂³	—	74	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	148	2	no	—
133	D₄	—	148	1	no	—
134	D₄	—	148	1	no	—
135	D₄	—	148	4	no	—
136	D₄	—	148	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	37	1	no	—
140	C₂×C₄	—	74	2	no	—
141	C₂×C₄	—	74	2	no	—
142	C₂×C₄	—	74	1	no	—
143	C₂×C₄	—	74	1	no	—
144	C₂×C₄	—	74	1	no	—
145	C₂×C₄	—	148	2	no	—
146	C₂×C₄	—	148	1	no	—
147	C₂×C₄	—	148	1	no	—
148	C₂×C₄	—	148	4	no	—
149	C₂×C₄	C₂²⋊#3866	1	2	yes	—
150	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
151	C₂²	—	74	2	no	—
152	C₂²	—	74	1	no	—
153	C₂²	—	148	2	no	—
154	C₂²	—	148	1	no	—
155	C₂²	—	148	2	no	—
156	C₂²	—	148	1	no	—
157	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	37	1	no	—
163	C₄	—	37	1	no	—
164	C₄	—	296	4	no	—
165	C₄	—	148	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	296	1	no	—
168	C₂	—	148	2	no	—
169	C₂	—	148	1	no	—
170	C₂	D₃₇⋊#99	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₇⋊#99
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂×C₃₇	C₂×#99
prime radical	#14558084	C₂
cosocle	C₂×D₁₄₈	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#2126
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14558514	C₂	semidirect
2. #14558439	C₂	semidirect
3. #14558438	C₂	semidirect
8. C₂×#10596966	C₄	semidirect
11. #11924525	C₄	semidirect
15. C₄²×D₃₇	C₄	semidirect
16. C₄⋊Q₈×C₃₇	C₄	semidirect
90. C₃₇	#2126	semidirect
93. 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
#14558514	2
#14558439	2
#14558438	2
#14558084	2
#2126	37