Finite Group Databasebeta

Subgroups — #14558231

Order 2368.

238 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 99 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	#11924531	C₂	1	2	yes	maximal, complemented
2	#11924527	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924521	C₂	1	2	yes	maximal, complemented
4	#11924511	C₂	1	1	yes	maximal, characteristic, complemented
5	#11924510	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924510	C₂	1	2	yes	maximal, complemented
7	#11924498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊D₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#11924494	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₃₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₃₇	C₂²	1	1	yes	characteristic
16	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
17	C₂×D₁₄₈	C₂²	1	2	yes	—
18	C₂×C₄×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₃₇	C₂²	1	2	yes	—
20	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
23	C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
24	#10596938	C₂²	1	2	yes	—
25	#10596938	C₂²	1	2	yes	—
26	#10596938	C₂²	1	1	yes	characteristic
27	#10596938	C₂²	1	2	yes	—
28	#10596938	C₂²	1	2	yes	—
29	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₃₇	C₂²	1	2	yes	—
31	#10596936	C₂²	1	2	yes	—
32	#10596936	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₇	C₂²	1	2	yes	—
34	C₄×Dic₃₇	C₂²	1	1	yes	characteristic
35	C₂³×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×D₃₇	C₂³	1	1	yes	characteristic
37	C₂²×D₃₇	C₂³	1	2	yes	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	—	4	2	no	—
40	C₂×C₄×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄×C₃₇	C₂³	1	2	yes	—
42	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
44	C₂²⋊D₃₇	—	2	4	no	—
45	C₂²⋊D₃₇	—	2	2	no	—
46	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₇	C₂³	1	2	yes	—
49	C₂×Dic₃₇	C₂³	1	1	yes	characteristic
50	D₁₄₈	—	2	2	no	—
51	D₁₄₈	—	4	2	no	—
52	C₄×D₃₇	—	2	2	no	—
53	C₄×D₃₇	—	4	2	no	—
54	C₂²×C₃₇	—	2	1	no	—
55	C₂²×C₃₇	—	2	1	no	—
56	C₂²×C₃₇	#41	1	2	yes	—
57	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
58	C₂×D₃₇	—	2	1	no	—
59	C₂×D₃₇	—	2	2	no	—
60	C₂×D₃₇	—	2	1	no	—
61	C₂×D₃₇	—	2	2	no	—
62	C₂×D₃₇	—	2	2	no	—
63	C₂×D₃₇	—	2	1	no	—
64	C₄×C₃₇	—	2	1	no	—
65	C₄×C₃₇	—	2	2	no	—
66	C₄×C₃₇	—	2	2	no	—
67	C₄×C₃₇	#41	1	2	yes	—
68	Dic₃₇	—	2	1	no	—
69	Dic₃₇	—	2	2	no	—
70	Dic₃₇	—	2	1	no	—
71	Dic₃₇	—	2	1	no	—
72	C₂×C₃₇	—	2	2	no	—
73	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
74	C₂×C₃₇	C₂×#41	1	1	yes	characteristic
75	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
76	D₃₇	—	4	1	no	—
77	D₃₇	—	4	2	no	—
78	#553	—	37	1	no	maximal, Sylow
79	C₃₇	#553	1	1	yes	characteristic, complemented, Sylow
80	#126	—	37	2	no	—
81	#125	—	37	1	no	—
82	#123	—	37	2	no	—
83	#121	—	37	2	no	—
84	#121	—	37	1	no	—
85	#121	—	37	1	no	—
86	C₄×D₄	—	37	2	no	—
87	C₄×D₄	—	37	1	no	—
88	C₄×D₄	—	37	1	no	—
89	#117	—	37	1	no	—
90	C₂×C₄⋊C₄	D₃₇	1	1	yes	characteristic, complemented
91	C₂×D₄	—	37	2	no	—
92	C₂×D₄	—	37	1	no	—
93	C₂×D₄	—	37	1	no	—
94	C₂×D₄	—	37	2	no	—
95	C₂²×C₄	—	37	1	no	—
96	C₂²×C₄	—	37	2	no	—
97	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₃₇	1	2	yes	—
99	C₄⋊C₄	—	37	2	no	—
100	C₄⋊C₄	—	37	1	no	—
101	C₄⋊C₄	—	37	1	no	—
102	C₄⋊C₄	—	37	2	no	—
103	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
104	C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
105	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
106	C₂²⋊C₄	—	37	1	no	—
107	C₂²⋊C₄	—	37	2	no	—
108	C₂²⋊C₄	—	37	2	no	—
109	C₂²⋊C₄	—	37	2	no	—
110	C₂²⋊C₄	—	37	1	no	—
111	C₂²⋊C₄	—	37	2	no	—
112	C₄²	—	37	2	no	—
113	C₄²	—	37	1	no	—
114	C₂³	—	37	1	no	—
115	C₂³	—	37	2	no	—
116	C₂³	C₂²×D₃₇	1	1	yes	characteristic
117	D₄	—	74	4	no	—
118	D₄	—	74	2	no	—
119	D₄	—	74	2	no	—
120	D₄	—	148	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	37	1	no	—
124	C₂×C₄	—	37	2	no	—
125	C₂×C₄	—	37	1	no	—
126	C₂×C₄	—	37	1	no	—
127	C₂×C₄	—	74	2	no	—
128	C₂×C₄	—	148	2	no	—
129	C₂×C₄	C₂²×D₃₇	1	2	yes	—
130	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₃₇	1	2	yes	—
132	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	74	1	no	—
136	C₂²	—	74	2	no	—
137	C₂²	—	74	1	no	—
138	C₂²	—	74	2	no	—
139	C₂²	—	74	2	no	—
140	C₂²	—	74	1	no	—
141	C₂²	#10596966	1	2	yes	—
142	C₂²	C₂³×D₃₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	74	1	no	—
147	C₄	—	74	2	no	—
148	C₄	—	74	1	no	—
149	C₄	—	74	1	no	—
150	C₄	#10596962	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	148	1	no	—
153	C₂	—	148	2	no	—
154	C₂	C₂×#10596962	1	1	yes	characteristic
155	C₂	C₂×#10596966	1	1	yes	characteristic
156	C₂	#11924581	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₃₇	C₂
hypercenter	C₂×C₄⋊C₄	D₃₇
hypocenter	C₃₇	#553
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #11924531	C₂	semidirect
2. #11924527	C₂	semidirect
3. #11924521	C₂	semidirect
4. #11924511	C₂	semidirect
5. #11924510	C₂	semidirect
6. #11924510	C₂	semidirect
7. #11924498	C₂	semidirect
8. C₄×C₂²⋊D₃₇	C₂	semidirect
9. C₄×C₂²⋊D₃₇	C₂	semidirect
10. #11924494	C₂	semidirect
11. C₂×C₄⋊C₄×C₃₇	C₂	semidirect
79. C₃₇	#553	semidirect
90. C₂×C₄⋊C₄	D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924531	2
#11924527	2
#11924521	2
#11924511	2
#11924510	2
#11924498	2
C₄×C₂²⋊D₃₇	2
#11924494	2
C₂×C₄⋊C₄×C₃₇	2
#553	37