Finite Group Databasebeta

Subgroups — #17993309

Order 4736.

332 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 77 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×#11924533	C₂	1	1	yes	maximal, characteristic
2	#14557590	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557276	C₂	1	4	yes	maximal, complemented
4	C₃₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₄₈	—	2	1	no	—
6	C₂²×D₁₄₈	—	2	1	no	—
7	C₂²×D₁₄₈	C₄	1	2	yes	—
8	#11924533	—	2	1	no	—
9	#11924533	—	2	1	no	—
10	#11924533	C₄	1	2	yes	—
11	#11924533	C₂²	1	2	yes	—
12	C₄⋊#64436	C₂²	1	2	yes	complemented
13	C₂²⋊#64436	—	2	1	no	—
14	#11924447	—	2	1	no	—
15	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	C₂²	1	2	yes	complemented
17	C₂²⋊C₈×C₃₇	—	2	1	no	—
18	#97×C₃₇	—	2	1	no	—
19	C₂³×D₃₇	—	2	2	no	—
20	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
23	C₂×D₁₄₈	—	2	2	no	—
24	C₂×D₁₄₈	—	2	2	no	—
25	C₂×D₁₄₈	—	2	2	no	—
26	C₂×D₁₄₈	—	2	2	no	—
27	C₂×D₁₄₈	—	2	2	no	—
28	C₂×D₁₄₈	—	4	4	no	—
29	C₂×D₁₄₈	—	4	4	no	—
30	C₂×D₁₄₈	—	4	4	no	—
31	C₂×D₁₄₈	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₃₇	—	2	2	no	—
33	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₃₇	D₄	1	1	yes	characteristic
35	C₄²×C₃₇	D₄	1	1	yes	characteristic
36	C₄²×C₃₇	C₂³	1	1	yes	characteristic
37	C₂×#64436	—	2	2	no	—
38	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×D₃₇	—	2	4	no	—
40	C₂²×D₃₇	—	4	4	no	—
41	C₂²×D₃₇	—	4	4	no	—
42	C₂²×D₃₇	—	4	4	no	—
43	C₂×C₄×C₃₇	—	2	2	no	—
44	C₂×C₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	D₁₄₈	—	4	4	no	—
53	D₁₄₈	—	4	4	no	—
54	D₁₄₈	—	4	4	no	—
55	D₁₄₈	—	4	4	no	—
56	D₁₄₈	—	4	4	no	—
57	D₁₄₈	—	4	4	no	—
58	C₈×C₃₇	—	4	2	no	—
59	#64436	—	4	2	no	—
60	C₂²×C₃₇	—	2	1	no	—
61	C₂²×C₃₇	—	2	1	no	—
62	C₂²×C₃₇	#100	1	2	yes	—
63	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₃₇	—	4	4	no	—
65	C₂×D₃₇	—	4	4	no	—
66	C₂×D₃₇	—	4	4	no	—
67	C₂×D₃₇	—	8	8	no	—
68	C₄×C₃₇	—	2	1	no	—
69	C₄×C₃₇	—	2	1	no	—
70	C₄×C₃₇	—	2	1	no	—
71	C₄×C₃₇	—	2	1	no	—
72	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
74	#1555	—	37	1	no	maximal, Sylow
75	C₂×C₃₇	—	2	2	no	—
76	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
77	C₂×C₃₇	#418	1	1	yes	characteristic
78	C₂×C₃₇	#418	1	1	yes	characteristic
79	D₃₇	—	8	4	no	—
80	C₂×#127	—	37	1	no	—
81	#432	—	37	1	no	—
82	#432	D₃₇	1	1	yes	characteristic, complemented
83	#331	—	37	4	no	—
84	C₃₇	#1555	1	1	yes	characteristic, complemented, Sylow
85	C₂²×D₄	—	37	2	no	—
86	C₂²×D₄	—	74	1	no	—
87	C₂²×D₄	—	74	1	no	—
88	#127	—	37	2	no	—
89	#127	—	37	2	no	—
90	#127	—	74	1	no	—
91	#127	—	74	1	no	—
92	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
93	C₄⋊C₈	—	37	2	no	—
94	C₄⋊C₈	C₂×D₃₇	1	2	yes	complemented
95	C₂²⋊C₈	—	2	1	no	—
96	C₂²⋊C₈	—	74	1	no	—
97	#97	—	2	1	no	—
98	#97	—	74	1	no	—
99	C₂⁴	—	74	2	no	—
100	C₂×D₄	—	37	4	no	—
101	C₂×D₄	—	74	2	no	—
102	C₂×D₄	—	74	2	no	—
103	C₂×D₄	—	74	2	no	—
104	C₂×D₄	—	74	2	no	—
105	C₂×D₄	—	74	2	no	—
106	C₂×D₄	—	148	4	no	—
107	C₂×D₄	—	148	4	no	—
108	C₂×D₄	—	148	4	no	—
109	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
110	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	74	2	no	—
114	C₄²	C₄×D₃₇	1	1	yes	characteristic
115	C₄²	D₁₄₈	1	1	yes	characteristic
116	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
117	C₄²	C₂²×D₃₇	1	1	yes	characteristic
118	C₂³	—	74	4	no	—
119	C₂³	—	148	4	no	—
120	C₂³	—	148	4	no	—
121	C₂³	—	148	4	no	—
122	C₂³	#10596938	1	1	yes	characteristic
123	D₄	—	148	4	no	—
124	D₄	—	148	4	no	—
125	D₄	—	148	4	no	—
126	D₄	—	148	4	no	—
127	D₄	—	148	4	no	—
128	D₄	—	148	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	#10596938	1	1	yes	characteristic
133	C₂×C₄	#10596938	1	1	yes	characteristic
134	C₂×C₄	#10596938	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
136	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	148	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	296	8	no	—
143	C₂²	—	148	4	no	—
144	C₂²	—	148	4	no	—
145	C₂²	—	148	4	no	—
146	C₂²	#11924452	1	2	yes	—
147	C₂²	C₂×#10596938	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#11924561	1	2	yes	—
153	C₄	#11924564	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	—	296	4	no	—
156	C₂	C₂×#11924452	1	1	yes	characteristic
157	C₂	#14557537	1	1	yes	characteristic
158	C₂	#14557539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10596938
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#432	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂×#11924533	C₂
cosocle	C₄²×C₃₇	C₂³
hypercenter	#432	D₃₇
hypocenter	C₃₇	#1555
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
2. #14557590	C₂	semidirect
3. #14557276	C₂	semidirect
4. C₃₇×#432	C₂	semidirect
12. C₄⋊#64436	C₂²	semidirect
16. C₄⋊C₈×C₃₇	C₂²	semidirect
82. #432	D₃₇	semidirect
84. C₃₇	#1555	semidirect
94. C₄⋊C₈	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924533	2
#14557590	2
#14557276	2
C₃₇×#432	2
#1555	37