Subgroups — #18003342

Order 4736.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 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, complemented
2	#14558083	C₂	1	1	yes	maximal, characteristic
3	#14557612	C₂	1	2	yes	maximal, complemented
4	#14557606	C₂	1	4	yes	maximal, complemented
5	C₈×D₁₄₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557501	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10596962	C₂²	1	2	yes	—
10	#11924544	—	2	8	no	—
11	C₂×#10596929	C₂²	1	4	yes	—
12	C₂²×#64436	C₂²	1	2	yes	—
13	C₂×C₈×D₃₇	C₂²	1	4	yes	—
14	#11924536	C₄	1	1	yes	characteristic
15	#11924533	C₄	1	1	yes	characteristic
16	#11924524	C₄	1	2	yes	—
17	#11924510	C₄	1	2	yes	—
18	#11924510	C₄	1	2	yes	—
19	C₄×D₁₄₈	C₂²	1	2	yes	—
20	C₄×D₁₄₈	C₂²	1	2	yes	—
21	C₄⋊#64436	C₂²	1	1	yes	characteristic
22	C₄⋊#64436	C₂²	1	2	yes	—
23	C₄⋊#64436	C₂²	1	1	yes	characteristic
24	#11924448	C₂²	1	4	yes	—
25	#11924448	C₂²	1	4	yes	—
26	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
29	#97×C₃₇	C₂²	1	2	yes	—
30	C₄×C₈×C₃₇	C₂²	1	2	yes	—
31	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₃₇	C₂×C₄	1	4	yes	—
35	#10596962	—	2	4	no	—
36	#10596962	—	2	4	no	—
37	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
38	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
39	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₃₇	C₂³	1	4	yes	—
41	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
42	M₁₆×C₃₇	—	2	2	no	—
43	M₁₆×C₃₇	D₄	1	4	yes	complemented
44	C₂×C₈×C₃₇	C₂³	1	2	yes	—
45	C₂×C₈×C₃₇	C₂³	1	2	yes	—
46	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₃₇	C₂³	1	1	yes	characteristic
49	C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	#10596938	C₂×C₄	1	4	yes	—
51	#10596938	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
54	C₂×#64436	—	2	4	no	—
55	C₂×#64436	C₂³	1	2	yes	—
56	C₂×#64436	C₂³	1	2	yes	—
57	#10596929	—	2	8	no	—
58	C₈×D₃₇	—	2	8	no	—
59	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₃₇	—	2	2	no	—
62	C₂×C₄×C₃₇	—	4	1	no	—
63	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₃₇	#41	1	2	yes	—
70	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₃₇	—	2	8	no	—
72	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
73	D₁₄₈	—	2	4	no	—
74	D₁₄₈	—	2	4	no	—
75	D₁₄₈	—	4	2	no	—
76	C₄×D₃₇	—	2	4	no	—
77	C₄×D₃₇	—	2	4	no	—
78	Dic₇₄	—	2	4	no	—
79	C₈×C₃₇	—	2	2	no	—
80	C₈×C₃₇	C₂×D₄	1	4	yes	—
81	#64436	—	2	2	no	—
82	#64436	—	2	2	no	—
83	C₂²×C₃₇	—	2	1	no	—
84	C₂²×C₃₇	—	2	1	no	—
85	C₂²×C₃₇	C₄×D₄	1	2	yes	—
86	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₃₇	—	2	4	no	—
88	C₂×D₃₇	—	2	4	no	—
89	C₂×D₃₇	—	2	4	no	—
90	C₄×C₃₇	—	2	1	no	—
91	C₄×C₃₇	—	2	1	no	—
92	C₄×C₃₇	—	2	1	no	—
93	C₄×C₃₇	C₄×D₄	1	2	yes	—
94	C₄×C₃₇	#131	1	2	yes	complemented
95	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
97	Dic₃₇	—	2	4	no	—
98	#2935	—	37	1	no	maximal, Sylow
99	C₂×C₃₇	—	2	2	no	—
100	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
102	C₂×C₃₇	#568	1	1	yes	characteristic
103	D₃₇	—	4	4	no	—
104	C₂×#131	—	37	2	no	—
105	#532	—	37	1	no	—
106	#436	—	37	2	no	—
107	#435	—	37	4	no	—
108	C₈×D₄	—	37	2	no	—
109	C₂×C₄⋊C₈	—	37	1	no	—
110	#408	—	37	2	no	—
111	C₄×M₁₆	D₃₇	1	1	yes	characteristic, complemented
112	C₃₇	#2935	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	37	2	no	—
114	#131	—	74	8	no	—
115	C₂×M₁₆	—	37	4	no	—
116	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
117	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
118	C₂²×C₈	—	37	2	no	—
119	C₂²×C₈	—	37	4	no	—
120	C₄⋊Q₈	—	37	1	no	—
121	#127	—	37	1	no	—
122	#124	—	37	2	no	—
123	#121	—	37	2	no	—
124	#121	—	37	2	no	—
125	C₄×D₄	—	37	2	no	—
126	C₄×D₄	—	37	2	no	—
127	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
128	C₄⋊C₈	—	37	1	no	—
129	C₄⋊C₈	—	37	2	no	—
130	C₄⋊C₈	—	37	1	no	—
131	C₂²⋊C₈	—	37	4	no	—
132	C₂²⋊C₈	—	37	4	no	—
133	#97	C₂×D₃₇	1	2	yes	—
134	C₄×C₈	C₂×D₃₇	1	2	yes	—
135	#41	—	74	4	no	—
136	#41	—	74	4	no	—
137	C₂×Q₈	—	37	2	no	—
138	C₂×D₄	—	37	4	no	—
139	C₂×D₄	—	37	2	no	—
140	C₂×D₄	—	37	2	no	—
141	C₂×D₄	—	37	2	no	—
142	C₂²×C₄	—	37	4	no	—
143	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	74	8	no	—
148	M₁₆	D₁₄₈	1	4	yes	complemented
149	C₂×C₈	—	37	2	no	—
150	C₂×C₈	—	37	2	no	—
151	C₂×C₈	—	74	4	no	—
152	C₂×C₈	—	74	8	no	—
153	C₂×C₈	C₂²×D₃₇	1	2	yes	—
154	C₂×C₈	C₂²×D₃₇	1	2	yes	—
155	C₄⋊C₄	—	37	2	no	—
156	C₄⋊C₄	—	37	2	no	—
157	C₂²⋊C₄	—	37	4	no	—
158	C₂²⋊C₄	—	37	4	no	—
159	C₄²	C₄×D₃₇	1	1	yes	characteristic
160	C₄²	C₄×D₃₇	1	1	yes	characteristic
161	C₄²	C₂²×D₃₇	1	1	yes	characteristic
162	C₄²	C₂²×D₃₇	1	1	yes	characteristic
163	C₂³	—	37	4	no	—
164	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
165	Q₈	—	74	4	no	—
166	D₄	—	74	8	no	—
167	D₄	—	74	4	no	—
168	D₄	—	74	4	no	—
169	D₄	—	148	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	37	4	no	—
173	C₂×C₄	—	74	4	no	—
174	C₂×C₄	—	74	4	no	—
175	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
180	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
181	C₂×C₄	#10596962	1	2	yes	—
182	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	74	2	no	—
185	C₈	—	74	2	no	—
186	C₈	C₂×D₁₄₈	1	4	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	74	4	no	—
190	C₂²	—	74	4	no	—
191	C₂²	—	74	4	no	—
192	C₂²	C₄×D₁₄₈	1	2	yes	—
193	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	74	4	no	—
198	C₄	C₄×D₁₄₈	1	2	yes	—
199	C₄	#11924544	1	2	yes	complemented
200	C₄	C₂²×D₁₄₈	1	1	yes	characteristic
201	C₄	C₂×#10596962	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	148	4	no	—
204	C₂	C₂×C₄×D₁₄₈	1	1	yes	characteristic
205	C₂	C₂×#11924544	1	1	yes	characteristic
206	C₂	#14558299	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×M₁₆×C₃₇	C₂
prime radical	#14558083	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#2935
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11924544	C₂	semidirect
3. #14557612	C₂	semidirect
4. #14557606	C₂	semidirect
5. C₈×D₁₄₈	C₂	semidirect
6. C₂×C₄⋊#64436	C₂	semidirect
7. #14557501	C₂	semidirect
8. C₄×M₁₆×C₃₇	C₂	semidirect
43. M₁₆×C₃₇	D₄	semidirect
94. C₄×C₃₇	#131	semidirect
111. C₄×M₁₆	D₃₇	semidirect
112. C₃₇	#2935	semidirect
148. M₁₆	D₁₄₈	semidirect
199. C₄	#11924544	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
#14558083	2
#14557612	2
#14557606	2
C₈×D₁₄₈	2
C₂×C₄⋊#64436	2
#14557501	2
C₄×M₁₆×C₃₇	2
#2935	37