Subgroups — #18003534

Order 4736.

348 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558086	C₂	1	1	yes	maximal, characteristic
2	#14557663	C₂	1	4	yes	maximal, complemented
3	#14557661	C₂	1	4	yes	maximal, complemented
4	#14557561	C₂	1	1	yes	maximal, characteristic
5	#14557560	C₂	1	1	yes	maximal, characteristic
6	#14557556	C₂	1	1	yes	maximal, characteristic
7	#14557488	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924446	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×#10596934	C₂²	1	1	yes	characteristic
11	C₂²×#64436	C₂²	1	1	yes	characteristic
12	#11924535	C₄	1	2	yes	—
13	#11924527	C₄	1	2	yes	—
14	#11924513	C₄	1	4	yes	—
15	C₄×Dic₇₄	C₂²	1	4	yes	—
16	#11924495	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊#64436	C₂²	1	4	yes	—
19	#11924465	C₂²	1	4	yes	—
20	#11924465	C₂²	1	4	yes	—
21	#11924446	C₂²	1	2	yes	—
22	#11924446	C₂²	1	2	yes	—
23	#11924446	C₂²	1	2	yes	—
24	C₈×Dic₃₇	C₂²	1	2	yes	—
25	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
27	#117×C₃₇	C₂²	1	1	yes	characteristic
28	#97×C₃₇	C₂²	1	2	yes	—
29	C₄×C₈×C₃₇	C₂²	1	2	yes	—
30	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
32	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
33	M₁₆×C₃₇	—	2	2	no	—
34	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₃₇	C₂³	1	2	yes	—
36	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
40	C₄²×C₃₇	C₂³	1	2	yes	—
41	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
44	#10596936	C₂×C₄	1	4	yes	—
45	#10596936	C₂×C₄	1	4	yes	—
46	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₃₇	C₂³	1	2	yes	—
49	C₄×Dic₃₇	C₂³	1	2	yes	—
50	#10596934	—	2	2	no	—
51	C₂×#64436	—	2	2	no	—
52	C₂×#64436	C₂³	1	2	yes	—
53	C₂×#64436	C₂³	1	2	yes	—
54	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₃₇	#41	1	2	yes	—
59	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₃₇	—	4	2	no	—
61	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
63	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
64	Dic₇₄	—	4	4	no	—
65	C₈×C₃₇	—	2	2	no	—
66	C₈×C₃₇	C₂×Q₈	1	2	yes	—
67	C₈×C₃₇	C₂×Q₈	1	2	yes	—
68	#64436	—	2	2	no	—
69	#64436	—	2	2	no	—
70	C₂²×C₃₇	—	2	2	no	—
71	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
72	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₃₇	—	2	4	no	—
74	C₄×C₃₇	C₄×Q₈	1	2	yes	—
75	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
77	Dic₃₇	—	2	2	no	—
78	Dic₃₇	—	2	4	no	—
79	Dic₃₇	—	2	2	no	—
80	#2961	—	37	1	no	maximal, Sylow
81	C₂×C₃₇	—	2	2	no	—
82	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
83	C₂×C₃₇	#568	1	2	yes	—
84	#533	—	37	1	no	—
85	#446	—	37	4	no	—
86	#446	—	37	4	no	—
87	#424	—	37	1	no	—
88	#424	—	37	1	no	—
89	#423	—	37	1	no	—
90	#405	—	37	1	no	—
91	#405	D₃₇	1	1	yes	characteristic
92	C₂×#97	—	37	1	no	—
93	C₃₇	#2961	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₁₆	—	37	1	no	—
95	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
96	C₂²×C₈	—	37	1	no	—
97	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
98	C₄⋊Q₈	—	37	2	no	—
99	#125	—	37	2	no	—
100	C₂²⋊Q₈	—	37	4	no	—
101	C₄×Q₈	—	37	4	no	—
102	#117	—	37	1	no	—
103	#117	C₂×D₃₇	1	1	yes	characteristic
104	C₂×C₄²	—	37	1	no	—
105	C₄⋊C₈	—	37	4	no	—
106	C₄⋊C₈	—	37	4	no	—
107	C₄⋊C₈	—	37	4	no	—
108	#97	—	37	2	no	—
109	#97	—	37	2	no	—
110	#97	—	37	2	no	—
111	#97	C₂×D₃₇	1	2	yes	—
112	C₄×C₈	—	37	2	no	—
113	C₄×C₈	C₂×D₃₇	1	2	yes	—
114	C₂×Q₈	—	37	4	no	—
115	C₂²×C₄	—	37	2	no	—
116	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	74	2	no	—
119	C₂×C₈	—	37	2	no	—
120	C₂×C₈	—	37	2	no	—
121	C₂×C₈	—	74	2	no	—
122	C₂×C₈	Dic₇₄	1	4	yes	complemented
123	C₂×C₈	C₂²×D₃₇	1	2	yes	—
124	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
126	C₄⋊C₄	—	37	4	no	—
127	C₄⋊C₄	—	37	4	no	—
128	C₄⋊C₄	—	37	2	no	—
129	C₄⋊C₄	—	37	4	no	—
130	C₄⋊C₄	C₄×D₃₇	1	2	yes	—
131	C₂²⋊C₄	—	37	2	no	—
132	C₂²⋊C₄	C₄×D₃₇	1	2	yes	—
133	C₄²	—	37	1	no	—
134	C₄²	—	37	1	no	—
135	C₄²	—	37	2	no	—
136	C₄²	—	37	2	no	—
137	C₄²	C₂²×D₃₇	1	2	yes	—
138	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
139	Q₈	—	148	4	no	—
140	C₂×C₄	—	37	2	no	—
141	C₂×C₄	—	37	2	no	—
142	C₂×C₄	—	37	4	no	—
143	C₂×C₄	—	148	2	no	—
144	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
147	C₂×C₄	#10596962	1	2	yes	—
148	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	74	2	no	—
151	C₈	—	74	2	no	—
152	C₈	C₂×Dic₇₄	1	2	yes	—
153	C₈	C₂×Dic₇₄	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₇₄	1	2	yes	—
156	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	74	2	no	—
159	C₄	—	74	2	no	—
160	C₄	—	74	4	no	—
161	C₄	C₄×Dic₇₄	1	2	yes	—
162	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
163	C₄	C₂×#10596962	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₇₄	1	1	yes	characteristic
166	C₂	#14558298	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#405	C₂
prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#405	D₃₇
hypocenter	C₃₇	#2961
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #14557663	C₂	semidirect
3. #14557661	C₂	semidirect
34. C₂×C₈×C₃₇	Q₈	semidirect
93. C₃₇	#2961	semidirect
122. C₂×C₈	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#14558086	2
#14557663	2
#14557661	2
#14557561	2
#14557560	2
#14557556	2
#14557488	2
C₂×#11924446	2
C₃₇×#405	2
#2961	37