Subgroups — #18003025

Order 4736.

556 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×#11924544	C₂	1	2	yes	maximal
2	C₄×#10596962	C₂	1	1	yes	maximal, characteristic
3	#14557491	C₂	1	2	yes	maximal, complemented
4	#14557487	C₂	1	4	yes	maximal, complemented
5	#14557485	C₂	1	2	yes	maximal, complemented
6	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×D₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10596962	C₂²	1	1	yes	characteristic
10	#11924544	C₄	1	16	yes	complemented
11	C₂×#10596929	C₂²	1	4	yes	—
12	C₂²×#64436	C₂²	1	2	yes	—
13	C₂×C₈×D₃₇	C₂²	1	4	yes	—
14	C₄×Dic₇₄	C₄	1	2	yes	—
15	C₄×D₁₄₈	C₄	1	2	yes	—
16	C₄×C₂²⋊D₃₇	C₄	1	4	yes	—
17	#11924495	C₂²	1	1	yes	characteristic
18	#11924493	C₂²	1	2	yes	—
19	C₄²×D₃₇	C₂²	1	2	yes	—
20	#11924447	C₂²	1	1	yes	characteristic
21	#11924447	C₂²	1	1	yes	characteristic
22	#11924447	C₂²	1	2	yes	—
23	#11924446	C₂²	1	4	yes	—
24	C₈×Dic₃₇	C₂²	1	4	yes	—
25	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
27	#97×C₃₇	C₂²	1	2	yes	—
28	C₄×C₈×C₃₇	C₂²	1	2	yes	—
29	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
32	#10596962	C₂×C₄	1	8	yes	—
33	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₃₇	C₂³	1	2	yes	—
36	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
37	M₁₆×C₃₇	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₃₇	C₂³	1	4	yes	—
39	C₄²×C₃₇	C₂×C₄	1	2	yes	—
40	C₄²×C₃₇	C₂³	1	1	yes	characteristic
41	C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
43	#10596938	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
45	#10596936	C₂×C₄	1	4	yes	—
46	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
47	C₄×Dic₃₇	C₂³	1	2	yes	—
48	C₂×#64436	C₂×C₄	1	8	yes	complemented
49	C₂×#64436	C₂³	1	2	yes	—
50	C₂×#64436	C₂³	1	2	yes	—
51	#10596929	C₂×C₄	1	16	yes	complemented
52	C₈×D₃₇	C₂×C₄	1	16	yes	complemented
53	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₇	—	4	2	no	—
56	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₃₇	C₄²	1	8	yes	—
62	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
63	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
64	D₁₄₈	C₄²	1	4	yes	—
65	C₄×D₃₇	—	4	4	no	—
66	C₄×D₃₇	C₂²×C₄	1	8	yes	—
67	Dic₇₄	C₄²	1	4	yes	—
68	C₈×C₃₇	C₂²×C₄	1	8	yes	—
69	#64436	C₂²×C₄	1	4	yes	—
70	#64436	C₂²×C₄	1	4	yes	—
71	C₂²×C₃₇	—	2	2	no	—
72	C₂²×C₃₇	C₂×C₄²	1	2	yes	—
73	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₃₇	—	2	4	no	—
75	C₂×D₃₇	C₂×C₄²	1	4	yes	—
76	C₄×C₃₇	—	2	2	no	—
77	C₄×C₃₇	—	2	2	no	—
78	C₄×C₃₇	C₂×C₄²	1	2	yes	—
79	C₄×C₃₇	C₂³×C₄	1	2	yes	—
80	Dic₃₇	—	2	4	no	—
81	Dic₃₇	C₂×C₄²	1	4	yes	—
82	#2875	—	37	1	no	maximal, Sylow
83	C₂×C₃₇	—	2	2	no	—
84	C₂×C₃₇	C₂²×C₄²	1	1	yes	characteristic
85	C₂×C₃₇	#568	1	2	yes	—
86	D₃₇	—	2	4	no	—
87	C₂×#131	—	37	2	no	—
88	C₄×#41	—	37	1	no	—
89	#405	—	37	2	no	—
90	#405	—	37	4	no	—
91	C₄×M₁₆	—	37	2	no	—
92	C₄×M₁₆	D₃₇	1	1	yes	characteristic, complemented
93	C₂×#97	—	37	1	no	—
94	C₂×#97	—	37	2	no	—
95	C₃₇	#2875	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	37	1	no	—
97	#131	—	37	16	no	—
98	C₂×M₁₆	—	37	4	no	—
99	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
100	C₂²×C₈	—	37	2	no	—
101	C₂²×C₈	—	37	4	no	—
102	C₄×Q₈	—	37	2	no	—
103	C₄×D₄	—	37	4	no	—
104	C₄×D₄	—	37	2	no	—
105	#117	—	37	1	no	—
106	#117	—	37	2	no	—
107	C₂×C₄²	—	37	2	no	—
108	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
109	#97	—	37	4	no	—
110	#97	—	37	1	no	—
111	#97	—	37	1	no	—
112	#97	—	37	2	no	—
113	#97	C₂×D₃₇	1	2	yes	—
114	C₄×C₈	—	37	4	no	—
115	C₄×C₈	C₂×D₃₇	1	2	yes	—
116	#41	—	37	8	no	—
117	C₂×Q₈	—	37	1	no	—
118	C₂×D₄	—	37	2	no	—
119	C₂×D₄	—	37	1	no	—
120	C₂²×C₄	—	37	2	no	—
121	C₂²×C₄	—	37	4	no	—
122	C₂²×C₄	C₄×D₃₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
124	M₁₆	—	37	16	no	—
125	M₁₆	C₄×D₃₇	1	8	yes	complemented
126	C₂×C₈	—	37	8	no	—
127	C₂×C₈	—	37	2	no	—
128	C₂×C₈	—	37	2	no	—
129	C₂×C₈	—	37	16	no	—
130	C₂×C₈	C₂²×D₃₇	1	4	yes	—
131	C₄⋊C₄	—	37	4	no	—
132	C₄⋊C₄	—	37	2	no	—
133	C₂²⋊C₄	—	37	2	no	—
134	C₂²⋊C₄	—	37	4	no	—
135	C₄²	—	37	4	no	—
136	C₄²	—	37	2	no	—
137	C₄²	C₄×D₃₇	1	2	yes	—
138	C₄²	C₂²×D₃₇	1	1	yes	characteristic
139	C₄²	C₂²×D₃₇	1	1	yes	characteristic
140	C₂³	—	37	2	no	—
141	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
142	Q₈	—	37	4	no	—
143	D₄	—	37	8	no	—
144	D₄	—	37	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	37	2	no	—
147	C₂×C₄	—	37	8	no	—
148	C₂×C₄	—	37	4	no	—
149	C₂×C₄	—	148	4	no	—
150	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
154	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
155	C₈	—	37	4	no	—
156	C₈	—	37	4	no	—
157	C₈	C₂×C₄×D₃₇	1	8	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	37	4	no	—
160	C₂²	—	74	4	no	—
161	C₂²	C₄²×D₃₇	1	2	yes	—
162	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	37	4	no	—
166	C₄	—	74	4	no	—
167	C₄	C₄²×D₃₇	1	2	yes	—
168	C₄	C₂²×C₄×D₃₇	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	74	4	no	—
171	C₂	C₂×C₄²×D₃₇	1	1	yes	characteristic
172	C₂	#14558299	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator	C₂×C₃₇	C₂²×C₄²
Fitting	C₄×M₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂×#10596962	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#2875
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14557491	C₂	semidirect
4. #14557487	C₂	semidirect
5. #14557485	C₂	semidirect
6. C₂×#11924447	C₂	semidirect
7. #97×D₃₇	C₂	semidirect
8. C₄×M₁₆×C₃₇	C₂	semidirect
10. #11924544	C₄	semidirect
37. M₁₆×C₃₇	C₂×C₄	semidirect
48. C₂×#64436	C₂×C₄	semidirect
51. #10596929	C₂×C₄	semidirect
52. C₈×D₃₇	C₂×C₄	semidirect
92. C₄×M₁₆	D₃₇	semidirect
95. C₃₇	#2875	semidirect
125. M₁₆	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924544	2
C₄×#10596962	2
#14557491	2
#14557487	2
#14557485	2
C₂×#11924447	2
#97×D₃₇	2
C₄×M₁₆×C₃₇	2
#2875	37