Finite Group Databasebeta

Subgroups — #17994479

Order 4736.

224 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#11924471	C₂	1	1	yes	maximal, characteristic
2	#14557569	C₂	1	1	yes	maximal, characteristic
3	#14557491	C₂	1	1	yes	maximal, characteristic
4	#14557489	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924447	C₂	1	1	yes	maximal, characteristic
6	#14557293	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596934	C₂²	1	1	yes	characteristic
9	C₂²×#64436	C₂²	1	1	yes	characteristic
10	C₂²×#64436	C₂²	1	1	yes	characteristic
11	#11924495	C₂²	1	1	yes	characteristic
12	#11924471	C₄	1	4	yes	complemented
13	#11924469	C₄	1	2	yes	complemented
14	#11924467	C₄	1	2	yes	complemented
15	#11924447	—	2	1	no	—
16	#11924447	C₄	1	4	yes	complemented
17	#11924446	—	2	1	no	—
18	C₄×#64436	—	2	1	no	—
19	C₈×Dic₃₇	—	2	1	no	—
20	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
21	#117×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	—	2	1	no	—
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₃₇	—	2	1	no	—
26	M₁₆×C₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₃₇	—	2	1	no	—
28	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
29	C₂²⋊C₄×C₃₇	—	2	1	no	—
30	C₄²×C₃₇	—	2	1	no	—
31	C₄²×C₃₇	—	2	1	no	—
32	C₄²×C₃₇	C₂×C₄	1	2	yes	—
33	C₂²⋊Dic₃₇	—	2	1	no	—
34	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
35	C₄×Dic₃₇	—	2	1	no	—
36	#10596934	—	2	1	no	—
37	#10596934	C₂×C₄	1	2	yes	—
38	C₂×#64436	—	2	1	no	—
39	C₂×#64436	C₂×C₄	1	2	yes	—
40	C₂×#64436	C₂×C₄	1	1	yes	characteristic
41	C₂×#64436	C₂×C₄	1	1	yes	characteristic
42	C₂×#64436	C₂×C₄	1	2	yes	—
43	C₂×#64436	C₂×C₄	1	2	yes	—
44	C₂×#64436	D₄	1	1	yes	characteristic
45	C₂×#64436	D₄	1	1	yes	characteristic
46	C₂×#64436	Q₈	1	2	yes	—
47	C₂³×C₃₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₃₇	—	2	2	no	—
49	C₂×C₄×C₃₇	—	2	2	no	—
50	C₂×C₄×C₃₇	—	4	2	no	—
51	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
57	C₂×Dic₃₇	—	2	2	no	—
58	C₈×C₃₇	—	2	2	no	—
59	#64436	—	2	2	no	—
60	#64436	C₄²	1	2	yes	—
61	#64436	C₄²	1	2	yes	—
62	#64436	C₄⋊C₄	1	2	yes	—
63	#64436	C₄⋊C₄	1	2	yes	—
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	#117	1	1	yes	characteristic
66	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
67	C₂²×C₃₇	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₃₇	—	4	2	no	—
69	C₄×C₃₇	—	4	2	no	—
70	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
71	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
73	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
74	Dic₃₇	—	4	2	no	—
75	#1773	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	1	no	—
77	C₂×C₃₇	—	2	1	no	—
78	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃₇	#444	1	2	yes	—
80	C₂×#108	—	37	1	no	—
81	#427	—	37	1	no	—
82	#405	—	37	1	no	—
83	#405	—	37	1	no	—
84	C₂×#97	—	37	1	no	—
85	#339	—	37	1	no	—
86	#339	D₃₇	1	1	yes	characteristic
87	C₃₇	#1773	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₁₆	—	37	1	no	—
89	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
90	C₂²×C₈	—	37	1	no	—
91	C₂²×C₈	—	37	1	no	—
92	#117	—	37	1	no	—
93	#117	C₂×D₃₇	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
95	#108	—	37	4	no	—
96	#107	—	37	2	no	—
97	#106	—	37	2	no	—
98	#97	—	37	4	no	—
99	#97	—	74	1	no	—
100	#97	—	74	1	no	—
101	C₄×C₈	—	74	1	no	—
102	C₄×C₈	—	74	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
105	M₁₆	—	2	1	no	—
106	M₁₆	—	37	2	no	—
107	M₁₆	—	74	1	no	—
108	M₁₆	C₄×D₃₇	1	2	yes	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	37	2	no	—
111	C₂×C₈	—	37	2	no	—
112	C₂×C₈	—	37	1	no	—
113	C₂×C₈	—	37	1	no	—
114	C₂×C₈	—	37	2	no	—
115	C₂×C₈	—	37	1	no	—
116	C₂×C₈	—	37	1	no	—
117	C₂×C₈	—	37	2	no	—
118	C₂×C₈	—	74	1	no	—
119	C₄⋊C₄	—	37	2	no	—
120	C₄⋊C₄	C₄×D₃₇	1	2	yes	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	74	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	74	1	no	—
126	C₄²	C₄×D₃₇	1	2	yes	—
127	C₂³	#10596964	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	74	2	no	—
132	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
135	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
136	C₂×C₄	#10596964	1	1	yes	characteristic
137	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	37	2	no	—
140	C₈	—	37	2	no	—
141	C₈	—	37	2	no	—
142	C₈	—	37	2	no	—
143	C₈	—	74	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	#11924492	1	1	yes	characteristic
146	C₂²	#11924496	1	1	yes	characteristic
147	C₂²	#11924501	1	1	yes	characteristic
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	148	2	no	—
151	C₄	C₄²×D₃₇	1	1	yes	characteristic
152	C₄	C₄⋊C₄×D₃₇	1	1	yes	characteristic
153	C₄	#11924496	1	1	yes	characteristic
154	C₄	#11924501	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	1	no	—
157	C₂	#14557384	1	1	yes	characteristic
158	C₂	#14557652	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#339	C₂
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂³×C₃₇	#41
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#339	D₃₇
hypocenter	C₃₇	#1773
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
12. #11924471	C₄	semidirect
13. #11924469	C₄	semidirect
14. #11924467	C₄	semidirect
16. #11924447	C₄	semidirect
87. C₃₇	#1773	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924471	2
#14557569	2
#14557491	2
#14557489	2
C₂×#11924447	2
#14557293	2
C₃₇×#339	2
#1773	37