Finite Group Databasebeta

Subgroups — #18010755

Order 4736.

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14558543	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558532	C₂	1	2	yes	maximal, complemented
3	#14558514	C₂	1	2	yes	maximal, complemented
4	#14558496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11924559	C₂	1	1	yes	maximal, characteristic
6	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
7	#11924615	C₂²	1	2	yes	—
8	#11924615	C₂²	1	1	yes	characteristic
9	#11924605	—	2	1	no	—
10	#11924604	—	2	1	no	—
11	#11924601	—	2	4	no	—
12	#11924595	—	2	4	no	—
13	#11924590	—	2	1	no	—
14	C₈×#3866	—	2	1	no	—
15	C₂×#10596966	C₂²	1	2	yes	—
16	#11924559	—	2	1	no	—
17	#11924559	—	2	1	no	—
18	#11924559	—	2	1	no	—
19	#11924559	—	2	1	no	—
20	C₂×#10596941	C₄	1	2	yes	complemented
21	C₂×D₂₉₆	C₄	1	1	yes	characteristic, complemented
22	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₃₇	C₄	1	1	yes	characteristic, complemented
24	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
25	#10596966	—	2	2	no	—
26	#10596966	—	2	2	no	—
27	#10596966	—	2	2	no	—
28	#10596966	—	2	2	no	—
29	C₂×D₁₄₈	—	2	2	no	—
30	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₃₇	—	2	2	no	—
32	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
33	C₂²⋊#3866	—	2	2	no	—
34	C₂²⋊#3866	D₄	1	2	yes	—
35	#10596957	—	2	1	no	—
36	C₄⋊#3866	—	2	2	no	—
37	C₄⋊#3866	—	2	1	no	—
38	C₄×#3866	—	2	1	no	—
39	C₄×#3866	—	2	2	no	—
40	#10596953	D₄	1	2	yes	—
41	D₃₇⋊C₈	—	2	1	no	—
42	Q₁₆×C₃₇	—	2	1	no	—
43	Q₁₆×C₃₇	—	2	1	no	—
44	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
45	#10596941	—	2	2	no	—
46	#10596941	—	2	2	no	—
47	C₂×#64436	C₂×C₄	1	1	yes	characteristic
48	D₂₉₆	—	2	1	no	—
49	D₂₉₆	—	2	1	no	—
50	C₈×D₃₇	D₄	1	1	yes	characteristic
51	C₈×D₃₇	D₄	1	1	yes	characteristic
52	C₈×D₃₇	D₄	1	1	yes	characteristic
53	C₈×D₃₇	D₄	1	1	yes	characteristic
54	C₂²×D₃₇	—	2	2	no	—
55	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×#3866	—	2	2	no	—
57	C₂×#3866	—	4	2	no	—
58	Q₈×C₃₇	—	2	2	no	—
59	Q₈×C₃₇	—	2	2	no	—
60	C₂×C₄×C₃₇	—	2	2	no	—
61	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₃₇	#41	1	1	yes	characteristic
63	D₁₄₈	—	2	2	no	—
64	D₁₄₈	—	2	2	no	—
65	D₁₄₈	—	4	2	no	—
66	D₁₄₈	—	4	2	no	—
67	C₄×D₃₇	—	4	2	no	—
68	C₄×D₃₇	—	4	2	no	—
69	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
70	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
71	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
72	C₄×D₃₇	#41	1	1	yes	characteristic
73	#64438	—	2	2	no	—
74	C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
76	#64436	C₂²⋊C₄	1	1	yes	characteristic
77	#64436	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
79	C₂×D₃₇	—	4	1	no	—
80	C₂×D₃₇	—	4	2	no	—
81	C₂×D₃₇	—	8	2	no	—
82	C₂×D₃₇	#121	1	1	yes	characteristic
83	C₂×D₃₇	#127	1	1	yes	characteristic
84	#3866	—	4	2	no	—
85	#3866	—	8	2	no	—
86	C₄×C₃₇	—	4	2	no	—
87	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
89	Dic₃₇	#121	1	1	yes	characteristic
90	Dic₃₇	#124	1	1	yes	characteristic
91	#1976	—	37	1	no	maximal, Sylow
92	C₂×C₃₇	—	2	1	no	—
93	C₂×C₃₇	#390	1	1	yes	characteristic
94	D₃₇	—	2	1	no	—
95	D₃₇	—	2	1	no	—
96	D₃₇	—	8	2	no	—
97	C₂×#135	—	37	1	no	—
98	#427	—	37	1	no	—
99	#421	—	37	2	no	—
100	#410	—	37	2	no	—
101	#405	—	37	1	no	—
102	C₃₇	#1976	1	1	yes	characteristic, complemented, Sylow
103	C₂×#41	—	37	2	no	—
104	#135	—	74	1	no	—
105	#135	—	74	1	no	—
106	#135	—	74	1	no	—
107	#135	—	74	1	no	—
108	C₂×Q₁₆	#3866	1	1	yes	characteristic, complemented
109	C₂×SD₁₆	—	37	2	no	—
110	C₂×D₈	—	37	1	no	—
111	C₂×M₁₆	—	37	1	no	—
112	C₂²×C₈	—	37	1	no	—
113	#117	—	37	1	no	—
114	#117	—	37	2	no	—
115	#107	—	74	1	no	—
116	#106	—	74	1	no	—
117	#104	—	74	4	no	—
118	#99	—	74	4	no	—
119	#97	—	74	1	no	—
120	C₄×C₈	—	74	1	no	—
121	#41	—	74	2	no	—
122	#41	—	74	2	no	—
123	#41	—	74	2	no	—
124	#41	—	74	2	no	—
125	C₂×Q₈	C₂×#3866	1	2	yes	—
126	C₂×D₄	—	37	2	no	—
127	C₂×D₄	—	74	2	no	—
128	C₂²×C₄	—	37	1	no	—
129	C₂²×C₄	—	74	2	no	—
130	Q₁₆	—	2	1	no	—
131	Q₁₆	—	2	1	no	—
132	SD₁₆	—	74	2	no	—
133	SD₁₆	—	74	2	no	—
134	D₈	—	74	1	no	—
135	D₈	—	74	1	no	—
136	M₁₆	—	37	2	no	—
137	M₁₆	—	74	1	no	—
138	C₂×C₈	—	37	1	no	—
139	C₂×C₈	—	37	1	no	—
140	C₂×C₈	—	37	1	no	—
141	C₂×C₈	—	37	1	no	—
142	C₂×C₈	—	37	1	no	—
143	C₂×C₈	—	74	1	no	—
144	C₂×C₈	C₂×#3866	1	1	yes	characteristic
145	C₄⋊C₄	—	74	1	no	—
146	C₄⋊C₄	—	74	2	no	—
147	C₂²⋊C₄	—	37	2	no	—
148	C₂²⋊C₄	—	74	2	no	—
149	C₄²	—	74	1	no	—
150	C₄²	—	74	2	no	—
151	C₂³	—	37	1	no	—
152	C₂³	—	74	2	no	—
153	Q₈	—	2	2	no	—
154	Q₈	—	2	2	no	—
155	D₄	—	74	2	no	—
156	D₄	—	74	2	no	—
157	D₄	—	148	2	no	—
158	D₄	—	148	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	37	1	no	—
161	C₂×C₄	—	37	1	no	—
162	C₂×C₄	—	37	1	no	—
163	C₂×C₄	—	37	1	no	—
164	C₂×C₄	—	37	1	no	—
165	C₂×C₄	—	74	2	no	—
166	C₂×C₄	—	148	2	no	—
167	C₂×C₄	—	148	2	no	—
168	C₂×C₄	—	148	2	no	—
169	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
170	C₈	—	37	1	no	—
171	C₈	—	37	1	no	—
172	C₈	—	74	2	no	—
173	C₈	C₂²⋊#3866	1	1	yes	characteristic
174	C₈	C₂²⋊#3866	1	1	yes	characteristic
175	C₂²	—	37	1	no	—
176	C₂²	—	37	1	no	—
177	C₂²	—	296	2	no	—
178	C₂²	—	148	1	no	—
179	C₂²	—	148	2	no	—
180	C₂²	D₄×#3866	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	37	1	no	—
183	C₄	—	37	1	no	—
184	C₄	—	296	2	no	—
185	C₄	—	148	2	no	—
186	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
187	C₄	D₄×#3866	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	296	2	no	—
190	C₂	—	74	1	no	—
191	C₂	—	74	1	no	—
192	C₂	#14558485	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14558485
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×Q₁₆×C₃₇	C₄
socle	C₂×C₃₇	#390
prime radical	C₂×#11924559	C₂
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₂×Q₁₆	#3866
hypocenter	C₃₇	#1976
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #14558543	C₂	semidirect
2. #14558532	C₂	semidirect
3. #14558514	C₂	semidirect
4. #14558496	C₂	semidirect
20. C₂×#10596941	C₄	semidirect
21. C₂×D₂₉₆	C₄	semidirect
23. C₂×Q₁₆×C₃₇	C₄	semidirect
102. C₃₇	#1976	semidirect
108. C₂×Q₁₆	#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558543	2
#14558532	2
#14558514	2
#14558496	2
C₂×#11924559	2
#1976	37