Subgroups — #18007698

Order 4736.

360 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924542	C₂	1	1	yes	maximal, characteristic
2	#14557951	C₂	1	4	yes	maximal, complemented
3	#14557949	C₂	1	4	yes	maximal, complemented
4	C₂×#11924476	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×D₃₇	C₂	1	4	yes	maximal, complemented
6	C₂×M₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10596962	C₄	1	1	yes	characteristic
8	#11924542	C₄	1	2	yes	—
9	#11924542	C₄	1	2	yes	—
10	#11924542	C₂²	1	2	yes	—
11	#11924542	C₂²	1	2	yes	—
12	C₂×#10596934	C₄	1	1	yes	characteristic
13	C₂×#10596929	C₄	1	2	yes	—
14	C₂×C₈×D₃₇	C₂²	1	2	yes	—
15	#11924476	C₂²	1	2	yes	—
16	#11924476	C₂²	1	2	yes	complemented
17	#11924475	C₂²	1	8	yes	complemented
18	C₂×#10596925	C₂²	1	2	yes	complemented
19	C₁₆×D₃₇	C₂²	1	8	yes	complemented
20	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	M₃₂×C₃₇	C₂²	1	2	yes	—
22	M₃₂×C₃₇	C₂²	1	2	yes	complemented
23	C₂×C₁₆×C₃₇	C₂²	1	2	yes	complemented
24	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₃₇	C₈	1	2	yes	—
26	#10596962	C₈	1	2	yes	—
27	#10596962	C₈	1	2	yes	—
28	#10596962	C₂×C₄	1	2	yes	—
29	#10596962	C₂×C₄	1	2	yes	—
30	C₂×D₁₄₈	C₈	1	1	yes	characteristic
31	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
32	C₂×Dic₇₄	C₈	1	1	yes	characteristic
33	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	#10596934	C₂×C₄	1	2	yes	—
40	#10596934	C₂×C₄	1	2	yes	—
41	C₂×#64436	C₂×C₄	1	2	yes	—
42	#10596929	C₂×C₄	1	4	yes	—
43	#10596929	C₂×C₄	1	4	yes	—
44	C₈×D₃₇	C₂×C₄	1	4	yes	—
45	C₈×D₃₇	C₂³	1	4	yes	—
46	C₁₆×C₃₇	C₂³	1	4	yes	—
47	#10596925	C₂³	1	4	yes	—
48	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₃₇	C₂×C₈	1	2	yes	—
50	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊D₃₇	C₂×C₈	1	4	yes	—
57	C₂²⋊D₃₇	C₂×C₈	1	4	yes	—
58	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
59	D₁₄₈	C₂×C₈	1	2	yes	—
60	D₁₄₈	C₂×C₈	1	2	yes	—
61	C₄×D₃₇	C₂×C₈	1	4	yes	—
62	C₄×D₃₇	C₂²×C₄	1	4	yes	—
63	Dic₇₄	C₂×C₈	1	2	yes	—
64	Dic₇₄	C₂×C₈	1	2	yes	—
65	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₈×C₃₇	C₂⁴	1	1	yes	characteristic
69	#64436	C₂²×C₄	1	4	yes	—
70	C₂²×C₃₇	—	4	1	no	—
71	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
72	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×D₃₇	—	4	2	no	—
75	C₂×D₃₇	C₂²×C₈	1	4	yes	—
76	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
77	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
78	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
80	Dic₃₇	C₂²×C₈	1	4	yes	—
81	#3407	—	37	1	no	maximal, Sylow
82	C₂×C₃₇	—	2	1	no	—
83	C₂×C₃₇	—	2	1	no	—
84	C₂×C₃₇	—	2	1	no	—
85	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
86	D₃₇	—	2	4	no	—
87	C₂×#131	—	37	1	no	—
88	#504	—	37	4	no	—
89	#504	—	37	4	no	—
90	C₂×M₃₂	—	37	1	no	—
91	C₂×M₃₂	—	37	4	no	—
92	C₂×M₃₂	D₃₇	1	1	yes	characteristic, complemented
93	C₃₇	#3407	1	1	yes	characteristic, complemented, Sylow
94	C₂×#41	—	37	1	no	—
95	#131	—	37	2	no	—
96	#131	—	37	2	no	—
97	#131	—	37	2	no	—
98	#131	—	37	2	no	—
99	C₂×M₁₆	—	37	1	no	—
100	C₂×M₁₆	—	37	2	no	—
101	C₂²×C₈	—	37	2	no	—
102	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
103	M₃₂	—	37	2	no	—
104	M₃₂	—	37	2	no	—
105	M₃₂	—	37	8	no	—
106	M₃₂	C₂×D₃₇	1	2	yes	—
107	M₃₂	C₂×D₃₇	1	2	yes	complemented
108	C₂×C₁₆	—	37	2	no	—
109	C₂×C₁₆	—	37	8	no	—
110	C₂×C₁₆	C₂×D₃₇	1	2	yes	complemented
111	#41	—	37	2	no	—
112	#41	—	37	2	no	—
113	#41	—	37	2	no	—
114	#41	—	37	2	no	—
115	C₂×Q₈	—	37	1	no	—
116	C₂×D₄	—	37	2	no	—
117	C₂×D₄	—	37	1	no	—
118	C₂²×C₄	—	37	2	no	—
119	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
120	M₁₆	—	37	2	no	—
121	M₁₆	—	37	2	no	—
122	M₁₆	—	37	4	no	—
123	M₁₆	—	37	4	no	—
124	C₂×C₈	—	37	2	no	—
125	C₂×C₈	—	37	4	no	—
126	C₂×C₈	—	37	4	no	—
127	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
128	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
129	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
131	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
132	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
133	C₁₆	—	37	4	no	—
134	C₁₆	C₂²×D₃₇	1	4	yes	—
135	C₂³	—	37	2	no	—
136	C₂³	C₈×D₃₇	1	1	yes	characteristic
137	Q₈	—	37	2	no	—
138	Q₈	—	37	2	no	—
139	D₄	—	37	4	no	—
140	D₄	—	37	4	no	—
141	D₄	—	37	2	no	—
142	D₄	—	37	2	no	—
143	C₂×C₄	—	37	2	no	—
144	C₂×C₄	—	37	4	no	—
145	C₂×C₄	—	37	4	no	—
146	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
147	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
148	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
152	C₈	—	37	4	no	—
153	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
154	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
155	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
156	C₈	C₂³×D₃₇	1	1	yes	characteristic
157	C₂²	—	4	1	no	—
158	C₂²	—	37	4	no	—
159	C₂²	—	148	2	no	—
160	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
161	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
162	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
163	C₄	—	37	4	no	—
164	C₄	C₂×C₈×D₃₇	1	1	yes	characteristic
165	C₄	C₂×C₈×D₃₇	1	1	yes	characteristic
166	C₄	C₂×C₈×D₃₇	1	1	yes	characteristic
167	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	74	4	no	—
172	C₂	C₂²×C₈×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₃₇
commutator, socle	C₂×C₃₇	C₂³×C₈
Fitting	C₂×M₃₂×C₃₇	C₂
prime radical	C₂×#10596962	C₄
cosocle	C₈×C₃₇	C₂⁴
hypercenter	C₂×M₃₂	D₃₇
hypocenter	C₃₇	#3407
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #14557951	C₂	semidirect
3. #14557949	C₂	semidirect
4. C₂×#11924476	C₂	semidirect
5. M₃₂×D₃₇	C₂	semidirect
6. C₂×M₃₂×C₃₇	C₂	semidirect
16. #11924476	C₂²	semidirect
17. #11924475	C₂²	semidirect
18. C₂×#10596925	C₂²	semidirect
19. C₁₆×D₃₇	C₂²	semidirect
22. M₃₂×C₃₇	C₂²	semidirect
23. C₂×C₁₆×C₃₇	C₂²	semidirect
92. C₂×M₃₂	D₃₇	semidirect
93. C₃₇	#3407	semidirect
107. M₃₂	C₂×D₃₇	semidirect
110. C₂×C₁₆	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×#11924542	2
#14557951	2
#14557949	2
C₂×#11924476	2
M₃₂×D₃₇	2
C₂×M₃₂×C₃₇	2
#3407	37