Subgroups — #17994294

Order 4736.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#64436	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924495	C₂	1	1	yes	maximal, characteristic
3	#14557287	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	—	2	2	no	—
6	C₂²×#64436	—	2	2	no	—
7	C₂²×#64436	C₄	1	4	yes	complemented
8	C₂²×#64436	C₂²	1	2	yes	—
9	#11924495	—	2	2	no	—
10	#11924495	—	2	2	no	—
11	C₂×C₄⋊Dic₃₇	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₃₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₃₇	—	2	2	no	—
16	C₂×M₁₆×C₃₇	—	2	2	no	—
17	C₂×M₁₆×C₃₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
19	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₇	D₄	1	2	yes	—
24	C₂²×C₄×C₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₃₇	D₄	1	2	yes	—
26	C₂²×C₄×C₃₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃₇	C₂×C₄	1	4	yes	—
29	M₁₆×C₃₇	—	2	8	no	—
30	C₂×C₈×C₃₇	—	2	2	no	—
31	C₂×C₈×C₃₇	—	2	2	no	—
32	C₂×C₈×C₃₇	C₂×C₄	1	4	yes	—
33	C₂²⋊Dic₃₇	—	2	4	no	—
34	C₄⋊Dic₃₇	—	2	4	no	—
35	C₄×Dic₃₇	—	2	2	no	—
36	C₄×Dic₃₇	—	2	2	no	—
37	C₂×#64436	—	2	8	no	—
38	C₂×#64436	—	2	2	no	—
39	C₂×#64436	—	2	2	no	—
40	C₂×#64436	C₂×C₄	1	4	yes	—
41	C₂³×C₃₇	—	2	4	no	—
42	C₂³×C₃₇	C₄²	1	2	yes	—
43	C₂³×C₃₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
45	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	C₄²	1	2	yes	—
47	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₃₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×Dic₃₇	—	2	4	no	—
60	C₂×Dic₃₇	—	2	8	no	—
61	C₈×C₃₇	—	2	4	no	—
62	#64436	—	2	4	no	—
63	C₂²×C₃₇	—	2	4	no	—
64	C₂²×C₃₇	—	2	4	no	—
65	C₂²×C₃₇	—	2	4	no	—
66	C₂²×C₃₇	#95	1	4	yes	—
67	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₃₇	#131	1	4	yes	—
71	C₄×C₃₇	#95	1	4	yes	—
72	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	Dic₃₇	—	4	4	no	—
77	#1729	—	37	1	no	maximal, Sylow
78	C₂×C₃₇	—	2	4	no	—
79	C₂×C₃₇	C₂×#95	1	1	yes	characteristic
80	C₂×C₃₇	#405	1	2	yes	—
81	C₂×C₃₇	#408	1	2	yes	—
82	C₂×C₃₇	#424	1	2	yes	—
83	C₂²×M₁₆	D₃₇	1	1	yes	characteristic
84	C₂³×C₈	—	37	1	no	—
85	C₂×#117	—	37	1	no	—
86	#336	—	37	4	no	—
87	C₃₇	#1729	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	Dic₃₇	1	4	yes	complemented
92	C₂²×C₈	—	37	4	no	—
93	C₂²×C₈	—	37	2	no	—
94	C₂²×C₈	—	74	2	no	—
95	C₂²×C₈	—	74	2	no	—
96	C₂²×C₈	C₂×D₃₇	1	2	yes	—
97	#117	—	74	2	no	—
98	#117	—	74	2	no	—
99	C₂×C₄⋊C₄	—	37	2	no	—
100	C₂×C₂²⋊C₄	—	37	2	no	—
101	C₂×C₄²	—	37	2	no	—
102	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
103	C₂²×C₄	—	37	4	no	—
104	C₂²×C₄	Dic₇₄	1	2	yes	—
105	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₃₇	1	2	yes	—
107	C₂²×C₄	D₁₄₈	1	2	yes	—
108	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
109	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	M₁₆	—	2	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	37	4	no	—
116	C₂×C₈	—	74	8	no	—
117	C₂×C₈	—	74	2	no	—
118	C₂×C₈	—	74	2	no	—
119	C₂×C₈	C₂×Dic₃₇	1	4	yes	—
120	C₄⋊C₄	—	74	4	no	—
121	C₂²⋊C₄	—	74	4	no	—
122	C₄²	—	74	2	no	—
123	C₄²	—	74	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₃₇	1	2	yes	—
126	C₂³	#10596936	1	2	yes	—
127	C₂³	#10596938	1	2	yes	—
128	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
129	C₂×C₄	—	74	4	no	—
130	C₂×C₄	—	74	8	no	—
131	C₂×C₄	C₄×Dic₃₇	1	2	yes	—
132	C₂×C₄	#10596936	1	4	yes	—
133	C₂×C₄	#10596936	1	2	yes	—
134	C₂×C₄	C₄⋊Dic₃₇	1	4	yes	—
135	C₂×C₄	#10596938	1	4	yes	—
136	C₂×C₄	#10596938	1	2	yes	—
137	C₂×C₄	C₂²⋊Dic₃₇	1	4	yes	—
138	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
140	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
141	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	74	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	#11924443	1	4	yes	—
150	C₂²	C₂×C₄×Dic₃₇	1	1	yes	characteristic
151	C₂²	C₂×#10596938	1	1	yes	characteristic
152	C₂²	C₂×#10596936	1	1	yes	characteristic
153	C₂²	#11924544	1	4	yes	—
154	C₄	—	148	4	no	—
155	C₄	#11924443	1	4	yes	—
156	C₄	C₂×#10596938	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
158	C₄	C₂×#10596936	1	1	yes	characteristic
159	C₄	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×#11924443	1	1	yes	characteristic
162	C₂	#14557491	1	2	yes	—
163	C₂	#14557501	1	2	yes	—
164	C₂	#14557561	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂×C₄²
Fitting	C₂²×M₁₆×C₃₇	C₂
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂⁴×C₃₇	C₂×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂²×M₁₆	D₃₇
hypocenter	C₃₇	#1729
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14557287	C₂	semidirect
7. C₂²×#64436	C₄	semidirect
17. C₂×M₁₆×C₃₇	C₄	semidirect
87. C₃₇	#1729	semidirect
91. C₂×M₁₆	Dic₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#64436	2
C₂×#11924495	2
#14557287	2
C₂²×M₁₆×C₃₇	2
#1729	37