Finite Group Databasebeta

Subgroups — #14558349

Order 2368.

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

normal subgroups only

This group has 248 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	#11924584	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924583	C₂	1	1	yes	maximal, characteristic, complemented
3	#11924569	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924568	C₂	1	2	yes	maximal, complemented
5	#11924565	C₂	1	2	yes	maximal, complemented
6	#11924557	C₂	1	2	yes	maximal, complemented
7	#11924556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596942	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₁₆×D₃₇	C₂	1	2	yes	maximal, complemented
10	#11924544	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	#10596966	—	2	1	no	—
15	#10596966	C₂²	1	2	yes	—
16	Q₈×D₃₇	—	2	1	no	—
17	Q₈×D₃₇	C₂²	1	2	yes	—
18	Q₈×D₃₇	C₂²	1	2	yes	—
19	#10596964	—	2	2	no	—
20	#10596964	C₂²	1	2	yes	—
21	#10596962	—	2	2	no	—
22	#10596962	—	2	1	no	—
23	#10596962	C₂²	1	1	yes	characteristic
24	#10596962	C₂²	1	1	yes	characteristic
25	C₂×Dic₇₄	—	2	1	no	—
26	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
27	Q₁₆×C₃₇	C₂²	1	2	yes	—
28	SD₁₆×C₃₇	C₂²	1	2	yes	—
29	M₁₆×C₃₇	C₂²	1	1	yes	characteristic
30	#10596942	C₂²	1	1	yes	characteristic
31	#10596942	C₂²	1	2	yes	—
32	#10596942	C₂²	1	2	yes	—
33	#10596941	C₂²	1	1	yes	characteristic
34	#10596940	C₂²	1	1	yes	characteristic
35	#10596939	C₂²	1	1	yes	characteristic
36	C₂×#64436	C₂²	1	1	yes	characteristic
37	Dic₁₄₈	C₂²	1	2	yes	—
38	#10596930	C₂²	1	2	yes	—
39	#10596929	C₂²	1	2	yes	—
40	C₈×D₃₇	C₂²	1	2	yes	—
41	Q₈×C₃₇	—	2	1	no	—
42	Q₈×C₃₇	C₂³	1	1	yes	characteristic
43	Q₈×C₃₇	C₂³	1	2	yes	—
44	D₄×C₃₇	—	2	1	no	—
45	D₄×C₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₃₇	—	2	1	no	—
47	C₂×C₄×C₃₇	—	2	1	no	—
48	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
49	C₂²⋊D₃₇	—	2	1	no	—
50	C₂²⋊D₃₇	—	2	2	no	—
51	C₂²⋊D₃₇	D₄	1	2	yes	—
52	C₂×Dic₃₇	—	2	1	no	—
53	C₂×Dic₃₇	—	2	2	no	—
54	D₁₄₈	—	2	2	no	—
55	D₁₄₈	D₄	1	1	yes	characteristic
56	D₁₄₈	C₂³	1	1	yes	characteristic
57	C₄×D₃₇	—	2	2	no	—
58	C₄×D₃₇	—	2	2	no	—
59	C₄×D₃₇	—	2	1	no	—
60	C₄×D₃₇	—	2	2	no	—
61	C₄×D₃₇	C₂³	1	2	yes	—
62	Dic₇₄	—	2	1	no	—
63	Dic₇₄	—	2	2	no	—
64	Dic₇₄	—	2	2	no	—
65	Dic₇₄	D₄	1	1	yes	characteristic
66	Dic₇₄	C₂³	1	2	yes	—
67	Dic₇₄	C₂³	1	1	yes	characteristic
68	C₈×C₃₇	C₂³	1	2	yes	—
69	#64436	C₂³	1	1	yes	characteristic
70	#64436	C₂³	1	1	yes	characteristic
71	C₂²×C₃₇	—	2	1	no	—
72	C₂²×C₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂×D₃₇	—	2	1	no	—
74	C₂×D₃₇	C₂×D₄	1	2	yes	—
75	C₄×C₃₇	—	2	1	no	—
76	C₄×C₃₇	—	2	2	no	—
77	C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
78	C₄×C₃₇	C₂⁴	1	1	yes	characteristic
79	Dic₃₇	—	2	1	no	—
80	Dic₃₇	—	2	2	no	—
81	Dic₃₇	C₂×D₄	1	2	yes	—
82	C₂×C₃₇	—	2	1	no	—
83	C₂×C₃₇	—	4	1	no	—
84	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
85	D₃₇	—	2	2	no	—
86	D₃₇	—	4	1	no	—
87	#578	—	37	1	no	maximal, Sylow
88	C₃₇	#578	1	1	yes	characteristic, complemented, Sylow
89	2₋¹⁺⁴	—	37	1	no	—
90	2₋¹⁺⁴	—	37	1	no	—
91	#137	—	37	2	no	—
92	#137	—	37	2	no	—
93	#137	—	37	1	no	—
94	#137	D₃₇	1	1	yes	characteristic, complemented
95	#135	—	37	1	no	—
96	#135	—	37	2	no	—
97	C₂×Q₁₆	—	37	1	no	—
98	C₂×Q₁₆	—	37	2	no	—
99	#131	—	37	1	no	—
100	#41	—	37	1	no	—
101	#41	—	37	2	no	—
102	#41	—	37	2	no	—
103	#41	—	37	1	no	—
104	#41	—	74	2	no	—
105	#41	—	74	1	no	—
106	#41	—	74	2	no	—
107	#41	—	74	1	no	—
108	#41	C₂×D₃₇	1	1	yes	characteristic
109	C₂×Q₈	—	37	2	no	—
110	C₂×Q₈	—	37	2	no	—
111	C₂×Q₈	—	37	1	no	—
112	C₂×Q₈	—	74	1	no	—
113	C₂×Q₈	—	74	1	no	—
114	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic
115	Q₁₆	—	37	1	no	—
116	Q₁₆	—	37	2	no	—
117	Q₁₆	—	37	2	no	—
118	Q₁₆	—	37	2	no	—
119	Q₁₆	C₂×D₃₇	1	2	yes	—
120	SD₁₆	—	37	1	no	—
121	SD₁₆	—	37	1	no	—
122	SD₁₆	—	37	2	no	—
123	SD₁₆	C₂×D₃₇	1	2	yes	—
124	D₈	—	37	1	no	—
125	M₁₆	—	37	2	no	—
126	M₁₆	C₂×D₃₇	1	1	yes	characteristic
127	C₂×C₈	—	37	1	no	—
128	C₂×C₈	—	37	2	no	—
129	Q₈	—	2	1	no	—
130	Q₈	—	37	1	no	—
131	Q₈	—	37	1	no	—
132	Q₈	—	37	2	no	—
133	Q₈	—	74	2	no	—
134	Q₈	—	74	2	no	—
135	Q₈	—	74	1	no	—
136	Q₈	C₂²×D₃₇	1	1	yes	characteristic
137	Q₈	C₂²×D₃₇	1	2	yes	—
138	D₄	—	2	1	no	—
139	D₄	—	37	2	no	—
140	D₄	—	37	1	no	—
141	D₄	—	37	1	no	—
142	D₄	—	74	2	no	—
143	D₄	—	74	2	no	—
144	D₄	—	74	1	no	—
145	D₄	C₂²×D₃₇	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	37	2	no	—
149	C₂×C₄	—	74	2	no	—
150	C₂×C₄	—	74	2	no	—
151	C₂×C₄	—	74	2	no	—
152	C₂×C₄	—	74	2	no	—
153	C₂×C₄	—	74	1	no	—
154	C₂×C₄	—	74	1	no	—
155	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
156	C₈	—	37	1	no	—
157	C₈	—	37	1	no	—
158	C₈	C₂²×D₃₇	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	37	2	no	—
161	C₂²	—	74	1	no	—
162	C₂²	D₄×D₃₇	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	2	no	—
165	C₄	—	37	2	no	—
166	C₄	—	74	1	no	—
167	C₄	—	74	2	no	—
168	C₄	D₄×D₃₇	1	1	yes	characteristic
169	C₄	C₂³×D₃₇	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	4	1	no	—
172	C₂	—	74	2	no	—
173	C₂	—	148	1	no	—
174	C₂	C₂×D₄×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₃₇
commutator, cosocle	C₄×C₃₇	C₂⁴
Frattini	C₄	C₂³×D₃₇
Fitting	#137×C₃₇	C₂
socle	C₂×C₃₇	C₂²×D₄
hypercenter	#137	D₃₇
hypocenter	C₃₇	#578
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #11924584	C₂	semidirect
2. #11924583	C₂	semidirect
3. #11924569	C₂	semidirect
4. #11924568	C₂	semidirect
5. #11924565	C₂	semidirect
6. #11924557	C₂	semidirect
7. #11924556	C₂	semidirect
8. C₂×#10596942	C₂	semidirect
9. Q₁₆×D₃₇	C₂	semidirect
10. #11924544	C₂	semidirect
11. #137×C₃₇	C₂	semidirect
88. C₃₇	#578	semidirect
94. #137	D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924584	2
#11924583	2
#11924569	2
#11924568	2
#11924565	2
#11924557	2
#11924556	2
C₂×#10596942	2
Q₁₆×D₃₇	2
#11924544	2
#137×C₃₇	2
#578	37