Finite Group Databasebeta

Subgroups — #18010790

Order 4736.

228 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14558536	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596957	C₂	1	1	yes	maximal, characteristic
3	#14558413	C₂	1	2	yes	maximal
4	#14558413	C₂	1	2	yes	maximal
5	C₂×#11924526	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596957	C₂²	1	2	yes	—
7	C₂×#10596957	C₂²	1	2	yes	—
8	#11924603	—	2	1	no	—
9	C₄⋊#64438	—	2	1	no	—
10	#11924591	—	2	1	no	—
11	C₄×#64438	—	2	1	no	—
12	#11924526	—	2	1	no	—
13	#11924526	—	2	1	no	—
14	#11924526	—	2	1	no	—
15	#11924526	—	2	1	no	—
16	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
17	C₂×#10596936	C₄	1	1	yes	characteristic
18	C₂×#10596936	C₄	1	1	yes	characteristic
19	C₂×#10596936	C₂²	1	1	yes	characteristic
20	C₂×#10596936	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₃₇	C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
30	#10596957	—	2	2	no	—
31	#10596957	—	2	4	no	—
32	#10596957	Q₈	1	4	yes	—
33	C₂×#64438	—	2	2	no	—
34	C₂×#64438	—	2	2	no	—
35	C₄⋊C₄×C₃₇	—	2	1	no	—
36	C₄⋊C₄×C₃₇	—	2	1	no	—
37	C₄⋊Dic₃₇	—	2	1	no	—
38	C₄⋊Dic₃₇	—	2	1	no	—
39	#10596936	—	2	1	no	—
40	#10596936	—	2	1	no	—
41	#10596936	—	2	1	no	—
42	#10596936	—	2	1	no	—
43	#10596936	—	2	1	no	—
44	#10596936	—	2	1	no	—
45	#10596936	—	2	1	no	—
46	#10596936	—	2	1	no	—
47	C₄×Dic₃₇	D₄	1	1	yes	characteristic
48	C₄×Dic₃₇	D₄	1	1	yes	characteristic
49	C₄×Dic₃₇	D₄	1	1	yes	characteristic
50	C₄×Dic₃₇	D₄	1	1	yes	characteristic
51	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	—	2	1	no	—
53	C₂×C₄×C₃₇	—	2	1	no	—
54	C₂×C₄×C₃₇	—	4	1	no	—
55	C₂×C₄×C₃₇	—	4	1	no	—
56	C₂×C₄×C₃₇	—	4	1	no	—
57	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₃₇	—	2	1	no	—
60	C₂×Dic₃₇	—	2	1	no	—
61	C₂×Dic₃₇	—	4	1	no	—
62	C₂×Dic₃₇	—	4	1	no	—
63	C₂×Dic₃₇	—	4	1	no	—
64	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₃₇	C₂×Q₈	1	2	yes	—
69	C₂×Dic₃₇	#41	1	2	yes	—
70	#64438	—	2	4	no	—
71	#64438	—	4	2	no	—
72	C₂²×C₃₇	—	2	1	no	—
73	C₂²×C₃₇	—	2	1	no	—
74	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
76	C₄×C₃₇	—	2	1	no	—
77	C₄×C₃₇	—	2	1	no	—
78	C₄×C₃₇	—	4	1	no	—
79	C₄×C₃₇	—	4	1	no	—
80	Dic₃₇	—	2	1	no	—
81	Dic₃₇	—	2	1	no	—
82	Dic₃₇	—	4	1	no	—
83	Dic₃₇	—	4	1	no	—
84	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
85	Dic₃₇	#124	1	1	yes	characteristic
86	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
87	#1998	—	37	1	no	maximal, Sylow
88	C₂×C₃₇	—	2	2	no	—
89	C₂×C₃₇	#391	1	1	yes	characteristic
90	C₂×C₃₇	#413	1	1	yes	characteristic
91	C₂×C₃₇	#413	1	1	yes	characteristic
92	C₂×#125	—	37	1	no	—
93	#423	—	37	1	no	—
94	C₄×M₁₆	—	37	1	no	—
95	#343	—	37	2	no	—
96	#343	—	37	2	no	—
97	C₃₇	#1998	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	37	2	no	—
99	C₂×M₁₆	—	37	2	no	—
100	#125	—	74	1	no	—
101	#125	—	74	1	no	—
102	#125	—	74	1	no	—
103	#125	—	74	1	no	—
104	C₂×C₄⋊C₄	—	37	1	no	—
105	C₂×C₄⋊C₄	—	37	1	no	—
106	C₂×C₄⋊C₄	—	37	1	no	—
107	C₂×C₄⋊C₄	—	37	1	no	—
108	C₂×C₄⋊C₄	—	37	1	no	—
109	C₂×C₄⋊C₄	#3866	1	1	yes	characteristic
110	C₂×C₄²	—	37	1	no	—
111	C₄⋊C₈	—	74	1	no	—
112	C₄⋊C₈	—	74	1	no	—
113	#97	—	74	1	no	—
114	C₄×C₈	—	74	1	no	—
115	C₂²×C₄	—	37	1	no	—
116	C₂²×C₄	—	37	1	no	—
117	C₂²×C₄	—	37	1	no	—
118	C₂²×C₄	—	37	1	no	—
119	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
120	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
121	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
122	M₁₆	—	37	4	no	—
123	M₁₆	—	74	2	no	—
124	M₁₆	—	74	4	no	—
125	C₂×C₈	—	74	2	no	—
126	C₂×C₈	—	74	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	74	1	no	—
130	C₄⋊C₄	—	74	1	no	—
131	C₄⋊C₄	—	74	1	no	—
132	C₄⋊C₄	—	74	1	no	—
133	C₄⋊C₄	—	74	1	no	—
134	C₄⋊C₄	—	74	1	no	—
135	C₄⋊C₄	—	74	1	no	—
136	C₄⋊C₄	—	74	1	no	—
137	C₄⋊C₄	—	74	1	no	—
138	C₄⋊C₄	—	74	1	no	—
139	C₄²	—	37	1	no	—
140	C₄²	—	37	1	no	—
141	C₄²	—	37	1	no	—
142	C₄²	—	37	1	no	—
143	C₂³	C₂²×#3866	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	37	1	no	—
150	C₂×C₄	—	37	1	no	—
151	C₂×C₄	—	37	2	no	—
152	C₂×C₄	—	37	2	no	—
153	C₂×C₄	—	37	1	no	—
154	C₂×C₄	—	37	1	no	—
155	C₂×C₄	—	74	1	no	—
156	C₂×C₄	—	74	1	no	—
157	C₂×C₄	—	148	1	no	—
158	C₂×C₄	—	148	1	no	—
159	C₂×C₄	—	148	1	no	—
160	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
162	C₈	—	74	4	no	—
163	C₈	—	148	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
167	C₂²	Q₈×#3866	1	2	yes	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	37	2	no	—
173	C₄	—	37	1	no	—
174	C₄	—	37	1	no	—
175	C₄	—	74	1	no	—
176	C₄	—	74	1	no	—
177	C₄	—	148	1	no	—
178	C₄	—	148	1	no	—
179	C₂	—	2	2	no	—
180	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
181	C₂	#14558519	1	1	yes	characteristic
182	C₂	#14558520	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator, prime radical	C₂³×C₃₇	C₂²×C₄
Frattini	C₂³	C₂²×#3866
Fitting	C₂×C₄⋊C₄×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	C₂×C₄⋊C₄	#3866
hypocenter	C₃₇	#1998
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
97. C₃₇	#1998	semidirect

Maximal subgroups

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

subgroup	index
#14558536	2
C₄×#10596957	2
#14558413	2
C₂×#11924526	2
#1998	37