Finite Group Databasebeta

Subgroups — #18010712

Order 4736.

200 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 72 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×#11924607	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924606	C₂	1	1	yes	maximal, characteristic
3	#14558538	C₂	1	1	yes	maximal, characteristic
4	C₄×#10596953	C₂	1	1	yes	maximal, characteristic
5	#14558409	C₂	1	1	yes	maximal, characteristic
6	#14558409	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596953	C₂²	1	1	yes	characteristic
9	C₂×#10596953	C₂²	1	1	yes	characteristic
10	C₂×#10596953	C₂²	1	1	yes	characteristic
11	C₂×#10596953	C₂²	1	1	yes	characteristic
12	#11924607	—	2	2	no	—
13	#11924606	—	2	2	no	—
14	#11924603	—	2	1	no	—
15	#11924603	—	2	1	no	—
16	#11924591	—	2	1	no	—
17	C₄×#64438	—	2	1	no	—
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
20	C₄²×D₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊#64436	C₄	1	1	yes	characteristic
22	#11924465	C₄	1	1	yes	characteristic
23	#11924465	C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₃₇	C₄	1	1	yes	characteristic
25	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
26	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
27	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
28	C₂×#64438	—	2	1	no	—
29	C₂×#64438	—	2	1	no	—
30	C₂×#64438	—	2	1	no	—
31	C₂×#64438	—	2	1	no	—
32	#10596953	—	2	1	no	—
33	#10596953	—	2	1	no	—
34	#10596953	—	2	1	no	—
35	#10596953	—	2	1	no	—
36	#10596953	—	2	1	no	—
37	#10596953	—	2	1	no	—
38	#10596953	D₄	1	2	yes	—
39	#10596953	Q₈	1	2	yes	—
40	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₃₇	D₄	1	1	yes	characteristic
45	C₄×Dic₃₇	D₄	1	1	yes	characteristic
46	C₂×#64436	C₂×C₄	1	1	yes	characteristic
47	C₂×#64436	C₂×C₄	1	1	yes	characteristic
48	C₈×D₃₇	—	2	1	no	—
49	C₈×D₃₇	—	2	1	no	—
50	C₈×D₃₇	—	2	1	no	—
51	C₈×D₃₇	—	2	1	no	—
52	C₂²×D₃₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
59	C₄×D₃₇	—	2	2	no	—
60	C₄×D₃₇	—	4	1	no	—
61	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
62	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
63	C₄×D₃₇	#41	1	1	yes	characteristic
64	C₄×D₃₇	#41	1	1	yes	characteristic
65	#64438	—	2	2	no	—
66	#64438	—	2	2	no	—
67	#64438	—	4	1	no	—
68	#64438	—	4	1	no	—
69	C₈×C₃₇	—	2	1	no	—
70	C₈×C₃₇	—	2	1	no	—
71	#64436	—	2	1	no	—
72	#64436	—	2	1	no	—
73	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×D₃₇	—	2	1	no	—
75	C₂×D₃₇	—	2	1	no	—
76	C₂×D₃₇	#125	1	1	yes	characteristic
77	C₂×D₃₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₄×C₃₇	—	2	1	no	—
79	C₄×C₃₇	—	2	1	no	—
80	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
81	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
82	Dic₃₇	—	2	1	no	—
83	Dic₃₇	#108	1	2	yes	complemented
84	Dic₃₇	#121	1	1	yes	characteristic
85	Dic₃₇	C₂²⋊Q₈	1	1	yes	characteristic
86	#1950	—	37	1	no	maximal, Sylow
87	C₂×C₃₇	#389	1	1	yes	characteristic
88	C₂×C₃₇	C₂×#108	1	1	yes	characteristic
89	C₂×C₃₇	#430	1	1	yes	characteristic
90	D₃₇	—	2	1	no	—
91	D₃₇	—	2	1	no	—
92	C₂×#108	—	37	1	no	—
93	C₂×#108	—	37	1	no	—
94	#423	—	37	1	no	—
95	C₂×C₄⋊C₈	—	37	1	no	—
96	C₄×M₁₆	—	37	1	no	—
97	#341	—	37	1	no	—
98	#341	—	37	1	no	—
99	C₃₇	#1950	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	37	1	no	—
101	C₂×M₁₆	—	37	1	no	—
102	C₂×M₁₆	—	37	1	no	—
103	C₂×M₁₆	—	37	1	no	—
104	C₂²×C₈	—	37	1	no	—
105	C₂²×C₈	—	37	1	no	—
106	C₂×C₄²	—	37	1	no	—
107	#108	—	74	2	no	—
108	#108	—	74	2	no	—
109	C₄⋊C₈	—	37	1	no	—
110	C₄⋊C₈	—	37	1	no	—
111	C₄⋊C₈	—	37	1	no	—
112	C₄⋊C₈	—	74	1	no	—
113	C₄⋊C₈	—	74	1	no	—
114	C₄⋊C₈	#3866	1	1	yes	characteristic
115	#97	—	74	1	no	—
116	C₄×C₈	—	74	1	no	—
117	C₂²×C₄	—	37	1	no	—
118	C₂²×C₄	—	37	1	no	—
119	C₂²×C₄	—	37	1	no	—
120	M₁₆	—	37	2	no	—
121	M₁₆	—	37	2	no	—
122	M₁₆	—	74	1	no	—
123	M₁₆	—	74	1	no	—
124	M₁₆	—	74	1	no	—
125	M₁₆	—	74	1	no	—
126	M₁₆	—	74	1	no	—
127	M₁₆	—	74	1	no	—
128	C₂×C₈	—	37	1	no	—
129	C₂×C₈	—	37	1	no	—
130	C₂×C₈	—	74	1	no	—
131	C₂×C₈	—	74	1	no	—
132	C₂×C₈	—	74	1	no	—
133	C₂×C₈	—	74	1	no	—
134	C₂×C₈	—	74	1	no	—
135	C₂×C₈	—	74	1	no	—
136	C₂×C₈	—	74	1	no	—
137	C₂×C₈	—	74	1	no	—
138	C₂×C₈	C₂×#3866	1	1	yes	characteristic
139	C₂×C₈	C₂×#3866	1	1	yes	characteristic
140	C₄²	—	37	1	no	—
141	C₄²	—	37	1	no	—
142	C₄²	—	37	1	no	—
143	C₄²	C₂×#3866	1	1	yes	characteristic
144	C₂³	—	37	1	no	—
145	C₂×C₄	—	37	1	no	—
146	C₂×C₄	—	37	1	no	—
147	C₂×C₄	—	37	1	no	—
148	C₂×C₄	—	37	1	no	—
149	C₂×C₄	—	37	1	no	—
150	C₂×C₄	—	37	1	no	—
151	C₂×C₄	—	37	1	no	—
152	C₂×C₄	—	74	2	no	—
153	C₂×C₄	—	148	1	no	—
154	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
155	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
156	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	74	2	no	—
160	C₈	—	74	2	no	—
161	C₈	—	74	1	no	—
162	C₈	—	74	1	no	—
163	C₈	—	148	1	no	—
164	C₈	—	148	1	no	—
165	C₂²	—	37	1	no	—
166	C₂²	—	37	1	no	—
167	C₂²	—	74	1	no	—
168	C₂²	—	74	1	no	—
169	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	37	1	no	—
173	C₄	—	37	1	no	—
174	C₄	—	37	2	no	—
175	C₄	—	74	1	no	—
176	C₄	D₄×#3866	1	1	yes	characteristic
177	C₄	Q₈×#3866	1	1	yes	characteristic
178	C₂	—	74	1	no	—
179	C₂	—	74	1	no	—
180	C₂	#14558484	1	1	yes	characteristic
181	C₂	D₃₇⋊#108	1	1	yes	characteristic
182	C₂	#14558548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊C₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₇	C₂×Q₈
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄⋊C₈	#3866
hypocenter	C₃₇	#1950
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
83. Dic₃₇	#108	semidirect
99. C₃₇	#1950	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924607	2
C₂×#11924606	2
#14558538	2
C₄×#10596953	2
#14558409	2
C₄⋊C₈×D₃₇	2
#1950	37