Subgroups — #18003304

Order 4736.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 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	1	yes	maximal, characteristic, complemented
2	C₂×#11924543	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558079	C₂	1	1	yes	maximal, characteristic
4	#14557608	C₂	1	4	yes	maximal, complemented
5	#14557596	C₂	1	4	yes	maximal, complemented
6	#14557556	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557499	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596962	C₂²	1	1	yes	characteristic
11	#11924544	—	2	4	no	—
12	#11924543	—	2	4	no	—
13	C₂×#10596934	C₂²	1	1	yes	characteristic
14	C₂×#10596934	C₂²	1	2	yes	—
15	C₂×#10596929	C₂²	1	2	yes	—
16	C₂²×#64436	C₂²	1	2	yes	—
17	C₂²×#64436	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₃₇	C₂²	1	2	yes	—
19	#11924534	C₄	1	1	yes	characteristic
20	#11924532	C₄	1	1	yes	characteristic
21	#11924525	C₄	1	1	yes	characteristic
22	#11924523	C₄	1	1	yes	characteristic
23	#11924508	C₄	1	4	yes	—
24	C₄×C₂²⋊D₃₇	C₂²	1	4	yes	—
25	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
26	#11924465	C₂²	1	4	yes	—
27	#11924448	C₂²	1	4	yes	—
28	C₈×Dic₃₇	C₂²	1	2	yes	—
29	C₈×Dic₃₇	C₂²	1	2	yes	—
30	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈×C₃₇	C₂²	1	4	yes	—
34	#41×C₃₇	—	2	2	no	—
35	#41×C₃₇	—	2	2	no	—
36	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₃₇	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₃₇	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
43	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
44	#10596962	—	2	2	no	—
45	#10596962	—	2	2	no	—
46	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃₇	C₂³	1	2	yes	—
48	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₃₇	—	2	2	no	—
50	C₂×C₈×C₃₇	—	2	2	no	—
51	C₂×C₈×C₃₇	C₂³	1	2	yes	—
52	C₂×C₈×C₃₇	C₂³	1	2	yes	—
53	C₂²⋊Dic₃₇	C₂×C₄	1	4	yes	—
54	#10596938	C₂×C₄	1	4	yes	—
55	#10596936	C₂×C₄	1	4	yes	—
56	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₃₇	C₂³	1	2	yes	—
59	#10596934	—	2	2	no	—
60	#10596934	—	2	4	no	—
61	C₂×#64436	—	2	4	no	—
62	C₂×#64436	D₄	1	4	yes	complemented
63	C₂×#64436	C₂³	1	2	yes	—
64	C₂×#64436	C₂³	1	1	yes	characteristic
65	C₂×#64436	C₂³	1	1	yes	characteristic
66	#10596929	—	2	4	no	—
67	C₈×D₃₇	—	2	4	no	—
68	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
69	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
71	Q₈×C₃₇	—	2	2	no	—
72	D₄×C₃₇	—	2	4	no	—
73	D₄×C₃₇	—	2	2	no	—
74	C₂×C₄×C₃₇	—	2	2	no	—
75	C₂×C₄×C₃₇	—	2	2	no	—
76	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₃₇	#41	1	2	yes	—
80	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₃₇	—	2	4	no	—
82	C₂²⋊D₃₇	—	2	8	no	—
83	C₂×Dic₃₇	—	2	4	no	—
84	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
86	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
87	D₁₄₈	—	2	2	no	—
88	C₄×D₃₇	—	2	2	no	—
89	C₄×D₃₇	—	2	2	no	—
90	Dic₇₄	—	2	2	no	—
91	C₈×C₃₇	—	2	2	no	—
92	C₈×C₃₇	—	2	2	no	—
93	#64436	—	2	2	no	—
94	#64436	C₂×D₄	1	2	yes	—
95	#64436	C₂×D₄	1	2	yes	—
96	C₂²×C₃₇	—	2	2	no	—
97	C₂²×C₃₇	—	2	2	no	—
98	C₂²×C₃₇	—	2	4	no	—
99	C₂²×C₃₇	C₄×D₄	1	2	yes	—
100	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₃₇	—	2	2	no	—
102	C₂×D₃₇	—	2	4	no	—
103	C₄×C₃₇	—	2	2	no	—
104	C₄×C₃₇	C₄×D₄	1	2	yes	—
105	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
107	Dic₃₇	—	2	2	no	—
108	Dic₃₇	—	2	2	no	—
109	Dic₃₇	#131	1	4	yes	complemented
110	#2931	—	37	1	no	maximal, Sylow
111	C₂×C₃₇	—	2	2	no	—
112	C₂×C₃₇	—	4	2	no	—
113	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂×C₃₇	C₂×#131	1	2	yes	—
115	D₃₇	—	4	2	no	—
116	C₂×#131	—	37	1	no	—
117	C₂×#131	—	37	1	no	—
118	#532	—	37	1	no	—
119	#436	—	37	4	no	—
120	C₈×D₄	—	37	4	no	—
121	#423	—	37	1	no	—
122	#408	—	37	1	no	—
123	#408	D₃₇	1	1	yes	characteristic, complemented
124	C₂×C₄×C₈	—	37	1	no	—
125	C₃₇	#2931	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	37	1	no	—
127	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
128	#131	—	74	4	no	—
129	#131	—	74	4	no	—
130	C₂×M₁₆	—	37	1	no	—
131	C₂×M₁₆	—	37	2	no	—
132	C₂×M₁₆	—	37	2	no	—
133	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
134	C₂²×C₈	—	37	1	no	—
135	C₂²×C₈	—	37	2	no	—
136	C₂²×C₈	—	37	2	no	—
137	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	37	1	no	—
139	#127	—	37	1	no	—
140	#124	—	37	1	no	—
141	#124	—	37	1	no	—
142	#121	—	37	4	no	—
143	C₄×D₄	—	37	4	no	—
144	C₂×C₄²	—	37	1	no	—
145	C₄⋊C₈	—	37	4	no	—
146	C₂²⋊C₈	—	37	4	no	—
147	C₂²⋊C₈	C₂×D₃₇	1	4	yes	—
148	C₄×C₈	—	37	2	no	—
149	C₄×C₈	—	37	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	74	2	no	—
153	#41	—	74	2	no	—
154	C₂×Q₈	—	37	1	no	—
155	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
156	C₂×D₄	—	37	4	no	—
157	C₂×D₄	—	37	2	no	—
158	C₂×D₄	—	37	1	no	—
159	C₂×D₄	C₄×D₃₇	1	2	yes	—
160	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
161	C₂²×C₄	—	37	2	no	—
162	C₂²×C₄	—	37	2	no	—
163	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
164	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	74	4	no	—
167	M₁₆	—	74	4	no	—
168	M₁₆	—	74	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	37	4	no	—
171	C₂×C₈	—	37	1	no	—
172	C₂×C₈	—	37	1	no	—
173	C₂×C₈	—	37	2	no	—
174	C₂×C₈	—	74	4	no	—
175	C₂×C₈	—	74	4	no	—
176	C₂×C₈	C₂²×D₃₇	1	2	yes	—
177	C₂×C₈	C₂²×D₃₇	1	2	yes	—
178	C₄⋊C₄	—	37	4	no	—
179	C₂²⋊C₄	—	37	4	no	—
180	C₂²⋊C₄	—	37	4	no	—
181	C₄²	—	37	1	no	—
182	C₄²	—	37	1	no	—
183	C₄²	—	37	2	no	—
184	C₂³	—	37	2	no	—
185	C₂³	C₂×C₄×D₃₇	1	2	yes	—
186	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	74	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	74	8	no	—
192	D₄	—	74	4	no	—
193	D₄	—	74	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	37	2	no	—
197	C₂×C₄	—	37	2	no	—
198	C₂×C₄	—	37	2	no	—
199	C₂×C₄	—	74	4	no	—
200	C₂×C₄	—	74	2	no	—
201	C₂×C₄	—	74	2	no	—
202	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
204	C₂×C₄	D₄×D₃₇	1	2	yes	—
205	C₂×C₄	#10596964	1	2	yes	—
206	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	37	2	no	—
210	C₈	—	37	2	no	—
211	C₈	—	74	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	74	2	no	—
216	C₂²	—	74	4	no	—
217	C₂²	#11924496	1	2	yes	—
218	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	37	4	no	—
221	C₄	—	74	2	no	—
222	C₄	—	74	2	no	—
223	C₄	#11924496	1	2	yes	—
224	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
225	C₄	C₂×#10596964	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	148	2	no	—
229	C₂	C₂×#11924496	1	1	yes	characteristic
230	C₂	#131×D₃₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#408	C₂
prime radical	#14558079	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#408	D₃₇
hypocenter	C₃₇	#2931
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#11924544	C₂	semidirect
2. C₂×#11924543	C₂	semidirect
4. #14557608	C₂	semidirect
5. #14557596	C₂	semidirect
6. #14557556	C₂	semidirect
7. #14557499	C₂	semidirect
8. C₂×C₈×Dic₃₇	C₂	semidirect
9. C₃₇×#408	C₂	semidirect
62. C₂×#64436	D₄	semidirect
109. Dic₃₇	#131	semidirect
123. #408	D₃₇	semidirect
125. C₃₇	#2931	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
C₂×#11924543	2
#14558079	2
#14557608	2
#14557596	2
#14557556	2
#14557499	2
C₂×C₈×Dic₃₇	2
C₃₇×#408	2
#2931	37