Subgroups — #18011401

Order 4736.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558582	C₂	1	2	yes	maximal, complemented
2	#14558560	C₂	1	4	yes	maximal, complemented
3	D₄×#64438	C₂	1	2	yes	maximal, complemented
4	#14558552	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558550	C₂	1	2	yes	maximal, complemented
6	C₄×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#64438	C₂	1	2	yes	maximal, complemented
8	C₄×#10596964	C₂	1	1	yes	maximal, characteristic
9	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
10	C₂²×#64438	C₂²	1	4	yes	—
11	#11924603	C₂²	1	4	yes	—
12	C₄⋊#64438	C₂²	1	2	yes	—
13	#11924593	C₂²	1	2	yes	—
14	C₂²⋊#64438	C₂²	1	4	yes	—
15	C₄×#64438	C₂²	1	1	yes	characteristic
16	C₄×#64438	C₂²	1	2	yes	—
17	C₄×#64438	C₂²	1	4	yes	—
18	C₄×#64438	C₂²	1	1	yes	characteristic
19	C₄×#64438	C₂²	1	2	yes	—
20	C₂×#10596964	C₄	1	1	yes	characteristic
21	C₄×Dic₇₄	C₄	1	1	yes	characteristic
22	#11924501	C₂²	1	1	yes	characteristic
23	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₃₇	C₄	1	2	yes	—
25	#11924496	C₂²	1	2	yes	—
26	#11924494	C₄	1	1	yes	characteristic
27	#11924492	C₄	1	2	yes	—
28	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
29	C₄²×D₃₇	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₃₇	C₄	1	1	yes	characteristic
31	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
34	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
35	C₂²×Dic₃₇	C₂³	1	2	yes	—
36	#10596964	C₈	1	8	yes	complemented
37	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
41	C₂×#64438	—	2	8	no	—
42	C₂×#64438	C₂³	1	2	yes	—
43	C₂×#64438	C₂³	1	4	yes	—
44	C₂×#64438	C₂³	1	2	yes	—
45	D₃₇⋊C₈	—	2	4	no	—
46	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
48	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
50	#10596938	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
52	#10596936	C₂×C₄	1	2	yes	—
53	#10596936	C₂×C₄	1	2	yes	—
54	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
56	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
57	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
58	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
59	C₄×Dic₃₇	C₂³	1	2	yes	—
60	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
61	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
62	D₄×C₃₇	C₂×C₈	1	4	yes	complemented
63	C₂×C₄×C₃₇	—	2	4	no	—
64	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²⋊D₃₇	C₂×C₈	1	8	yes	complemented
69	C₂×Dic₃₇	—	2	4	no	—
70	C₂×Dic₃₇	C₂×C₈	1	8	yes	complemented
71	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
76	C₄×D₃₇	—	2	2	no	—
77	C₄×D₃₇	—	2	2	no	—
78	C₄×D₃₇	C₂×C₈	1	4	yes	complemented
79	Dic₇₄	C₂×C₈	1	4	yes	complemented
80	#64438	—	2	2	no	—
81	#64438	—	2	4	no	—
82	#64438	#41	1	4	yes	complemented
83	C₂²×C₃₇	—	2	2	no	—
84	C₂²×C₃₇	—	2	2	no	—
85	C₂²×C₃₇	C₂²×C₈	1	4	yes	—
86	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₃₇	—	2	1	no	—
88	C₂×D₃₇	—	2	1	no	—
89	C₂×D₃₇	C₂²×C₈	1	2	yes	—
90	C₄×C₃₇	—	2	1	no	—
91	C₄×C₃₇	—	2	2	no	—
92	C₄×C₃₇	C₂²×C₈	1	2	yes	—
93	C₄×C₃₇	#131	1	2	yes	—
94	Dic₃₇	—	2	2	no	—
95	Dic₃₇	—	2	1	no	—
96	Dic₃₇	C₂²×C₈	1	4	yes	—
97	Dic₃₇	C₂²×C₈	1	2	yes	—
98	Dic₃₇	C₂×#41	1	2	yes	—
99	C₈×#41	—	37	1	no	maximal, Sylow
100	C₂×C₃₇	—	2	4	no	—
101	C₂×C₃₇	C₄×#41	1	1	yes	characteristic
102	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
103	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
104	D₃₇	—	2	2	no	—
105	C₄×#41	—	37	1	no	—
106	C₈×Q₈	—	37	2	no	—
107	C₈×D₄	—	37	2	no	—
108	C₈×D₄	—	37	4	no	—
109	#431	—	37	2	no	—
110	#431	—	37	1	no	—
111	C₂×C₄×C₈	—	37	2	no	—
112	C₂×C₄×C₈	—	37	1	no	—
113	C₃₇	C₈×#41	1	1	yes	characteristic, complemented, Sylow
114	C₂×#41	—	37	1	no	—
115	C₂²×C₈	—	37	4	no	—
116	C₂²×C₈	—	37	2	no	—
117	C₄×Q₈	—	37	1	no	—
118	C₄×Q₈	—	37	1	no	—
119	C₄×D₄	—	37	1	no	—
120	C₄×D₄	—	37	2	no	—
121	C₄×D₄	—	37	2	no	—
122	C₄×D₄	#3866	1	1	yes	characteristic
123	#117	—	37	1	no	—
124	#117	—	37	2	no	—
125	C₂×C₄²	—	37	2	no	—
126	C₂×C₄²	—	37	1	no	—
127	C₄⋊C₈	—	37	2	no	—
128	C₄⋊C₈	—	37	4	no	—
129	C₂²⋊C₈	—	37	4	no	—
130	C₂²⋊C₈	—	37	2	no	—
131	C₄×C₈	—	37	1	no	—
132	C₄×C₈	—	37	2	no	—
133	C₄×C₈	—	37	1	no	—
134	C₄×C₈	—	37	4	no	—
135	C₄×C₈	—	37	2	no	—
136	#41	—	37	8	no	—
137	C₂×Q₈	—	37	1	no	—
138	C₂×D₄	—	37	2	no	—
139	C₂×D₄	C₂×#3866	1	1	yes	characteristic
140	C₂²×C₄	—	37	2	no	—
141	C₂²×C₄	—	37	1	no	—
142	C₂²×C₄	—	37	2	no	—
143	C₂²×C₄	—	37	1	no	—
144	C₂²×C₄	—	37	1	no	—
145	C₂²×C₄	C₂×#3866	1	2	yes	—
146	C₂×C₈	—	37	2	no	—
147	C₂×C₈	—	37	2	no	—
148	C₂×C₈	—	37	4	no	—
149	C₂×C₈	—	74	4	no	—
150	C₂×C₈	—	74	8	no	—
151	C₄⋊C₄	—	37	2	no	—
152	C₄⋊C₄	—	37	2	no	—
153	C₄⋊C₄	—	37	1	no	—
154	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
155	C₂²⋊C₄	—	37	2	no	—
156	C₂²⋊C₄	—	37	2	no	—
157	C₂²⋊C₄	C₂×#3866	1	2	yes	—
158	C₄²	—	37	2	no	—
159	C₄²	—	37	1	no	—
160	C₄²	—	37	2	no	—
161	C₄²	—	37	1	no	—
162	C₄²	—	37	2	no	—
163	C₄²	—	37	1	no	—
164	C₄²	C₂×#3866	1	1	yes	characteristic
165	C₂³	—	37	1	no	—
166	C₂³	C₂²×#3866	1	2	yes	—
167	Q₈	—	37	4	no	—
168	D₄	—	37	8	no	—
169	D₄	D₃₇⋊C₈	1	4	yes	complemented
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	37	8	no	—
172	C₂×C₄	—	37	1	no	—
173	C₂×C₄	—	37	1	no	—
174	C₂×C₄	—	37	2	no	—
175	C₂×C₄	—	37	4	no	—
176	C₂×C₄	—	37	2	no	—
177	C₂×C₄	—	37	1	no	—
178	C₂×C₄	—	74	2	no	—
179	C₂×C₄	—	74	4	no	—
180	C₂×C₄	—	74	2	no	—
181	C₂×C₄	C₂²×#3866	1	2	yes	—
182	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
183	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
184	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
185	C₈	—	37	4	no	—
186	C₈	—	74	2	no	—
187	C₈	—	74	4	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	37	2	no	—
191	C₂²	—	74	1	no	—
192	C₂²	—	74	1	no	—
193	C₂²	C₂×D₃₇⋊C₈	1	4	yes	—
194	C₂²	C₂³×#3866	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	37	2	no	—
198	C₄	—	37	2	no	—
199	C₄	—	37	4	no	—
200	C₄	—	74	2	no	—
201	C₄	—	74	1	no	—
202	C₄	C₂×D₃₇⋊C₈	1	2	yes	—
203	C₄	#11924623	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	74	2	no	—
206	C₂	#41×#3866	1	1	yes	characteristic
207	C₂	C₂²×D₃₇⋊C₈	1	1	yes	characteristic
208	C₂	C₂×#11924623	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator	C₂×C₃₇	C₂³×C₈
Frattini	C₂²	C₂³×#3866
Fitting	C₄×D₄×C₃₇	C₄
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂×#10596964	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×D₄	#3866
hypocenter	C₃₇	C₈×#41
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14558582	C₂	semidirect
2. #14558560	C₂	semidirect
3. D₄×#64438	C₂	semidirect
4. #14558552	C₂	semidirect
5. #14558550	C₂	semidirect
6. C₄×D₃₇⋊C₈	C₂	semidirect
7. C₂×C₄×#64438	C₂	semidirect
36. #10596964	C₈	semidirect
62. D₄×C₃₇	C₂×C₈	semidirect
68. C₂²⋊D₃₇	C₂×C₈	semidirect
70. C₂×Dic₃₇	C₂×C₈	semidirect
78. C₄×D₃₇	C₂×C₈	semidirect
79. Dic₇₄	C₂×C₈	semidirect
82. #64438	#41	semidirect
113. C₃₇	C₈×#41	semidirect
169. D₄	D₃₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#14558582	2
#14558560	2
D₄×#64438	2
#14558552	2
#14558550	2
C₄×D₃₇⋊C₈	2
C₂×C₄×#64438	2
C₄×#10596964	2
C₈×#41	37