Finite Group Databasebeta

Subgroups — #18010731

Order 4736.

360 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14558538	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924601	C₂	1	2	yes	maximal, complemented
3	C₂×#11924600	C₂	1	2	yes	maximal, complemented
4	C₄²×#3866	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558082	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	C₂²	1	2	yes	—
7	C₂×C₄×#3866	—	2	2	no	—
8	C₂×C₄×#3866	C₂²	1	2	yes	—
9	#11924603	—	2	2	no	—
10	#11924601	—	2	4	no	—
11	#11924600	—	2	4	no	—
12	C₂×#10596966	C₂²	1	1	yes	characteristic
13	C₂×#10596964	C₂²	1	1	yes	characteristic
14	#11924534	C₄	1	1	yes	characteristic, complemented
15	#11924532	C₄	1	1	yes	characteristic, complemented
16	#11924524	C₄	1	1	yes	characteristic, complemented
17	#11924509	—	2	2	no	—
18	#11924496	—	2	2	no	—
19	C₄²×D₃₇	C₂²	1	1	yes	characteristic
20	#124×C₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂²×#3866	—	2	2	no	—
22	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₂²⋊D₃₇	—	2	1	no	—
25	C₂×C₂²⋊D₃₇	—	2	2	no	—
26	C₂²×Dic₃₇	—	2	1	no	—
27	#10596966	—	2	2	no	—
28	#10596966	—	2	2	no	—
29	#10596964	—	2	2	no	—
30	#10596964	—	2	2	no	—
31	C₂×D₁₄₈	—	2	1	no	—
32	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₃₇	—	2	1	no	—
34	C₂×C₄×D₃₇	D₄	1	2	yes	—
35	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
36	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
37	C₂×#64438	—	2	2	no	—
38	C₄×#3866	—	2	8	no	—
39	C₄×#3866	—	2	2	no	—
40	C₄×#3866	D₄	1	4	yes	complemented
41	#10596953	—	2	2	no	—
42	#10596953	—	2	2	no	—
43	C₂²⋊C₄×C₃₇	—	2	2	no	—
44	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
45	#10596938	—	2	2	no	—
46	#10596936	—	2	2	no	—
47	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₃₇	D₄	1	2	yes	—
49	C₂³×C₃₇	—	2	1	no	—
50	C₂²×D₃₇	—	2	1	no	—
51	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×#3866	—	2	8	no	—
53	C₂×#3866	—	2	4	no	—
54	Q₈×C₃₇	—	2	2	no	—
55	D₄×C₃₇	—	2	2	no	—
56	C₂×C₄×C₃₇	—	2	1	no	—
57	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²⋊D₃₇	—	4	2	no	—
60	C₂²⋊D₃₇	—	4	4	no	—
61	C₂×Dic₃₇	—	2	1	no	—
62	C₂×Dic₃₇	—	4	1	no	—
63	C₂×Dic₃₇	—	4	1	no	—
64	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
65	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
66	D₁₄₈	—	2	2	no	—
67	D₁₄₈	—	4	2	no	—
68	C₄×D₃₇	—	2	4	no	—
69	C₄×D₃₇	—	4	1	no	—
70	C₄×D₃₇	—	4	1	no	—
71	C₄×D₃₇	C₂×D₄	1	2	yes	—
72	C₄×D₃₇	#41	1	2	yes	—
73	Dic₇₄	—	2	2	no	—
74	#64438	—	4	2	no	—
75	C₂²×C₃₇	—	4	1	no	—
76	C₂²×C₃₇	—	4	2	no	—
77	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×D₃₇	—	2	2	no	—
79	C₂×D₃₇	—	4	1	no	—
80	C₂×D₃₇	—	4	2	no	—
81	C₂×D₃₇	#121	1	2	yes	—
82	#3866	—	2	8	no	—
83	C₄×C₃₇	—	2	2	no	—
84	C₄×C₃₇	—	4	1	no	—
85	C₄×C₃₇	C₄×D₄	1	2	yes	—
86	Dic₃₇	—	4	1	no	—
87	Dic₃₇	#104	1	4	yes	complemented
88	Dic₃₇	#124	1	1	yes	characteristic
89	Dic₃₇	#127	1	1	yes	characteristic
90	#1962	—	37	1	no	maximal, Sylow
91	C₂×C₃₇	—	8	1	no	—
92	C₂×C₃₇	#390	1	1	yes	characteristic
93	C₂×C₃₇	C₂×#104	1	2	yes	—
94	D₃₇	—	2	2	no	—
95	D₃₇	—	8	1	no	—
96	#532	—	37	1	no	—
97	#423	—	37	1	no	—
98	C₂×#104	—	37	2	no	—
99	C₂×#104	—	37	2	no	—
100	C₄³	—	37	1	no	—
101	C₃₇	#1962	1	1	yes	characteristic, complemented, Sylow
102	C₂×#41	—	37	1	no	—
103	C₂×#41	—	37	1	no	—
104	C₂×M₁₆	—	37	2	no	—
105	C₄⋊Q₈	—	37	1	no	—
106	#127	—	37	1	no	—
107	#124	—	37	1	no	—
108	#124	#3866	1	1	yes	characteristic, complemented
109	#121	—	74	2	no	—
110	C₄×D₄	—	74	2	no	—
111	C₂×C₄²	—	37	1	no	—
112	C₂×C₄²	—	37	2	no	—
113	C₂×C₄²	—	74	2	no	—
114	C₄⋊C₈	—	74	2	no	—
115	#104	—	74	4	no	—
116	#104	—	74	4	no	—
117	#41	—	74	2	no	—
118	#41	—	74	2	no	—
119	#41	—	74	2	no	—
120	#41	—	74	2	no	—
121	C₂×Q₈	—	37	1	no	—
122	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
123	C₂×D₄	—	37	1	no	—
124	C₂×D₄	—	74	1	no	—
125	C₂×D₄	—	74	2	no	—
126	C₂×D₄	—	74	1	no	—
127	C₂×D₄	C₂×#3866	1	1	yes	characteristic
128	C₂²×C₄	—	37	1	no	—
129	C₂²×C₄	—	37	2	no	—
130	C₂²×C₄	—	74	1	no	—
131	C₂²×C₄	—	74	1	no	—
132	C₂²×C₄	—	74	2	no	—
133	M₁₆	—	74	2	no	—
134	M₁₆	—	74	2	no	—
135	C₂×C₈	—	74	2	no	—
136	C₄⋊C₄	—	74	2	no	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	74	2	no	—
139	C₄²	—	37	2	no	—
140	C₄²	—	37	1	no	—
141	C₄²	—	37	4	no	—
142	C₄²	—	74	2	no	—
143	C₄²	—	74	8	no	—
144	C₄²	C₂×#3866	1	1	yes	characteristic
145	C₂³	—	2	1	no	—
146	C₂³	—	37	1	no	—
147	C₂³	—	74	1	no	—
148	Q₈	—	2	2	no	—
149	Q₈	—	74	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	74	2	no	—
152	D₄	—	148	2	no	—
153	D₄	—	148	4	no	—
154	D₄	—	148	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	37	2	no	—
157	C₂×C₄	—	37	1	no	—
158	C₂×C₄	—	37	2	no	—
159	C₂×C₄	—	37	2	no	—
160	C₂×C₄	—	74	4	no	—
161	C₂×C₄	—	74	4	no	—
162	C₂×C₄	—	74	8	no	—
163	C₂×C₄	—	74	1	no	—
164	C₂×C₄	—	148	1	no	—
165	C₂×C₄	—	148	1	no	—
166	C₂×C₄	—	148	1	no	—
167	C₂×C₄	—	148	1	no	—
168	C₂×C₄	C₂²⋊#3866	1	2	yes	—
169	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
170	C₈	—	148	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	37	2	no	—
174	C₂²	—	74	2	no	—
175	C₂²	—	148	1	no	—
176	C₂²	—	148	2	no	—
177	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	37	1	no	—
181	C₄	—	37	1	no	—
182	C₄	—	37	4	no	—
183	C₄	—	74	8	no	—
184	C₄	—	148	1	no	—
185	C₄	D₄×#3866	1	2	yes	—
186	C₂	—	8	1	no	—
187	C₂	—	296	1	no	—
188	C₂	—	74	2	no	—
189	C₂	#14558485	1	1	yes	characteristic
190	C₂	D₃₇⋊#104	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#124×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	#14558082	C₂
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#124	#3866
hypocenter	C₃₇	#1962
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #14558538	C₂	semidirect
2. C₂×#11924601	C₂	semidirect
3. C₂×#11924600	C₂	semidirect
4. C₄²×#3866	C₂	semidirect
14. #11924534	C₄	semidirect
15. #11924532	C₄	semidirect
16. #11924524	C₄	semidirect
20. #124×C₃₇	C₄	semidirect
40. C₄×#3866	D₄	semidirect
87. Dic₃₇	#104	semidirect
101. C₃₇	#1962	semidirect
108. #124	#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558538	2
C₂×#11924601	2
C₂×#11924600	2
C₄²×#3866	2
#14558082	2
#1962	37