Subgroups — #18007805

Order 4736.

1492 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924584	C₂	1	1	yes	maximal, characteristic
2	#14558015	C₂	1	4	yes	maximal, complemented
3	#14558013	C₂	1	6	yes	maximal, complemented
4	#14558009	C₂	1	4	yes	maximal, complemented
5	C₄×#10596966	C₂	1	4	yes	maximal, complemented
6	#14558001	C₂	1	6	yes	maximal, complemented
7	C₂×Q₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₃₇	C₂	1	4	yes	maximal, complemented
9	C₃₇×#519	C₂	1	1	yes	maximal, characteristic, complemented
10	#11924584	C₄	1	16	yes	complemented
11	C₂×#10596966	C₂²	1	4	yes	—
12	C₂×#10596962	C₂²	1	6	yes	—
13	C₂×Q₈×D₃₇	C₂²	1	4	yes	—
14	Q₈×Dic₃₇	C₂²	1	4	yes	—
15	Q₈×Dic₃₇	C₂²	1	4	yes	—
16	C₄×Dic₇₄	C₂²	1	12	yes	—
17	#11924501	C₂²	1	6	yes	—
18	#11924501	C₂²	1	6	yes	—
19	C₄×D₁₄₈	C₂²	1	12	yes	—
20	#11924498	C₂²	1	6	yes	—
21	#11924498	C₂²	1	6	yes	—
22	#11924496	C₂²	1	4	yes	—
23	#11924496	C₂²	1	12	yes	—
24	#11924494	C₂²	1	12	yes	—
25	#11924493	C₂²	1	12	yes	—
26	C₂×C₄⋊Dic₃₇	C₂²	1	3	yes	—
27	C₄⋊C₄×D₃₇	C₂²	1	12	yes	—
28	C₂×C₄×Dic₃₇	C₂²	1	3	yes	—
29	C₄²×D₃₇	C₂²	1	12	yes	—
30	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₃₇	C₂²	1	4	yes	—
32	C₄×Q₈×C₃₇	C₂²	1	4	yes	—
33	#117×C₃₇	C₂²	1	6	yes	—
34	C₂×Q₈×C₃₇	C₂×C₄	1	8	yes	complemented
35	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
37	C₂×C₂²⋊D₃₇	C₂³	1	4	yes	—
38	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃₇	C₂³	1	3	yes	—
40	#10596966	C₂×C₄	1	32	yes	complemented
41	Q₈×D₃₇	C₂×C₄	1	32	yes	complemented
42	#10596962	C₂×C₄	1	48	yes	complemented
43	C₂×D₁₄₈	C₂³	1	6	yes	—
44	C₂×C₄×D₃₇	C₂³	1	4	yes	—
45	C₂×C₄×D₃₇	C₂³	1	12	yes	—
46	C₂×C₄×D₃₇	C₂³	1	12	yes	—
47	C₂×Dic₇₄	C₂³	1	6	yes	—
48	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
49	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
50	C₂²⋊C₄×C₃₇	C₂³	1	4	yes	—
51	C₄²×C₃₇	C₂³	1	12	yes	—
52	#10596938	C₂³	1	4	yes	—
53	#10596938	C₂³	1	12	yes	—
54	C₄⋊Dic₃₇	C₂³	1	12	yes	—
55	#10596936	C₂³	1	12	yes	—
56	#10596936	C₂³	1	4	yes	—
57	C₄×Dic₃₇	C₂³	1	4	yes	—
58	C₄×Dic₃₇	C₂³	1	6	yes	—
59	C₄×Dic₃₇	C₂³	1	12	yes	—
60	C₄×Dic₃₇	C₂³	1	6	yes	—
61	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
62	C₂²×D₃₇	C₂⁴	1	4	yes	—
63	Q₈×C₃₇	C₂²×C₄	1	16	yes	—
64	C₂×C₄×C₃₇	C₂²×C₄	1	12	yes	—
65	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
66	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
67	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
68	C₂²⋊D₃₇	C₂²×C₄	1	16	yes	—
69	C₂×Dic₃₇	—	2	2	no	—
70	C₂×Dic₃₇	—	4	3	no	—
71	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₃₇	C₂⁴	1	6	yes	—
73	C₂×Dic₃₇	C₂⁴	1	4	yes	—
74	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
75	D₁₄₈	C₂²×C₄	1	24	yes	—
76	C₄×D₃₇	—	2	8	no	—
77	C₄×D₃₇	—	4	12	no	—
78	C₄×D₃₇	C₂²×C₄	1	48	yes	—
79	Dic₇₄	C₂²×C₄	1	24	yes	—
80	C₂²×C₃₇	—	2	1	no	—
81	C₂²×C₃₇	—	2	1	no	—
82	C₂²×C₃₇	C₂³×C₄	1	2	yes	—
83	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
84	C₂×D₃₇	—	2	4	no	—
85	C₂×D₃₇	—	2	4	no	—
86	C₂×D₃₇	C₂³×C₄	1	8	yes	—
87	C₄×C₃₇	—	2	4	no	—
88	C₄×C₃₇	—	2	4	no	—
89	C₄×C₃₇	C₂³×C₄	1	12	yes	—
90	Dic₃₇	—	2	1	no	—
91	Dic₃₇	—	2	6	no	—
92	Dic₃₇	C₂³×C₄	1	8	yes	—
93	Dic₃₇	2₋¹⁺⁴	1	2	yes	complemented
94	C₄×2₋¹⁺⁴	—	37	1	no	maximal, Sylow
95	C₂×C₃₇	—	2	2	no	—
96	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
97	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
98	C₂×C₃₇	#585	1	1	yes	characteristic
99	D₃₇	—	2	8	no	—
100	C₂×2₋¹⁺⁴	—	37	1	no	—
101	#520	—	37	4	no	—
102	#520	—	37	6	no	—
103	#519	—	37	4	no	—
104	#519	D₃₇	1	1	yes	characteristic, complemented
105	C₄×#41	—	37	4	no	—
106	C₄×#41	—	37	6	no	—
107	C₂×C₄×Q₈	—	37	1	no	—
108	C₂×C₄×Q₈	—	37	4	no	—
109	C₃₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
110	2₋¹⁺⁴	—	37	16	no	—
111	C₂×#41	—	37	4	no	—
112	C₂×#41	—	37	6	no	—
113	C₂²×Q₈	—	37	4	no	—
114	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
115	C₄×Q₈	—	37	4	no	—
116	C₄×Q₈	—	37	4	no	—
117	C₄×Q₈	—	37	6	no	—
118	C₄×Q₈	—	37	6	no	—
119	C₄×Q₈	—	37	12	no	—
120	C₄×Q₈	C₂×D₃₇	1	4	yes	—
121	C₄×Q₈	C₂×D₃₇	1	4	yes	—
122	C₄×D₄	—	37	6	no	—
123	C₄×D₄	—	37	6	no	—
124	C₄×D₄	—	37	4	no	—
125	C₄×D₄	—	37	12	no	—
126	C₄×D₄	—	37	12	no	—
127	#117	—	37	12	no	—
128	#117	—	37	12	no	—
129	#117	C₂×D₃₇	1	6	yes	—
130	C₂×C₄⋊C₄	—	37	3	no	—
131	C₂×C₄⋊C₄	—	37	12	no	—
132	C₂×C₄²	—	37	3	no	—
133	C₂×C₄²	—	37	12	no	—
134	#41	—	37	32	no	—
135	#41	—	37	48	no	—
136	C₂×Q₈	—	37	32	no	—
137	C₂×Q₈	—	37	6	no	—
138	C₂×Q₈	C₄×D₃₇	1	8	yes	complemented
139	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
140	C₂×D₄	—	37	4	no	—
141	C₂×D₄	—	37	6	no	—
142	C₂²×C₄	—	37	12	no	—
143	C₂²×C₄	—	37	1	no	—
144	C₂²×C₄	—	37	3	no	—
145	C₂²×C₄	—	37	4	no	—
146	C₂²×C₄	—	37	12	no	—
147	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
148	C₄⋊C₄	—	37	12	no	—
149	C₄⋊C₄	—	37	4	no	—
150	C₄⋊C₄	—	37	12	no	—
151	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
152	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
153	C₂²⋊C₄	—	37	4	no	—
154	C₂²⋊C₄	—	37	12	no	—
155	C₂²⋊C₄	C₂²×D₃₇	1	4	yes	—
156	C₄²	—	37	4	no	—
157	C₄²	—	37	6	no	—
158	C₄²	—	37	12	no	—
159	C₄²	—	37	6	no	—
160	C₄²	C₂²×D₃₇	1	12	yes	—
161	C₂³	—	37	4	no	—
162	C₂³	C₂³×D₃₇	1	1	yes	characteristic
163	Q₈	—	37	24	no	—
164	Q₈	C₂×C₄×D₃₇	1	16	yes	—
165	D₄	—	37	16	no	—
166	D₄	—	37	24	no	—
167	C₂×C₄	—	37	1	no	—
168	C₂×C₄	—	37	4	no	—
169	C₂×C₄	—	37	48	no	—
170	C₂×C₄	—	37	1	no	—
171	C₂×C₄	—	37	6	no	—
172	C₂×C₄	—	74	8	no	—
173	C₂×C₄	—	74	2	no	—
174	C₂×C₄	—	148	12	no	—
175	C₂×C₄	—	148	3	no	—
176	C₂×C₄	C₂×C₄×D₃₇	1	12	yes	—
177	C₂×C₄	C₂³×D₃₇	1	4	yes	—
178	C₂×C₄	C₂³×D₃₇	1	4	yes	—
179	C₂×C₄	C₂³×D₃₇	1	6	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	37	8	no	—
183	C₂²	—	74	4	no	—
184	C₂²	—	74	4	no	—
185	C₂²	C₂²×C₄×D₃₇	1	2	yes	—
186	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	37	8	no	—
190	C₄	—	37	2	no	—
191	C₄	—	74	1	no	—
192	C₄	—	74	6	no	—
193	C₄	C₂²×C₄×D₃₇	1	12	yes	—
194	C₂	—	2	2	no	—
195	C₂	—	74	8	no	—
196	C₂	C₂³×C₄×D₃₇	1	1	yes	characteristic
197	C₂	2₋¹⁺⁴×D₃₇	1	1	yes	characteristic
198	C₂	#14558370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₇
commutator	C₂×C₃₇	C₂⁴×C₄
Fitting	C₃₇×#519	C₂
socle, cosocle	C₂²×C₃₇	C₂⁵
prime radical	C₂×#11924584	C₂
hypercenter	#519	D₃₇
hypocenter	C₃₇	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
2. #14558015	C₂	semidirect
3. #14558013	C₂	semidirect
4. #14558009	C₂	semidirect
5. C₄×#10596966	C₂	semidirect
6. #14558001	C₂	semidirect
7. C₂×Q₈×Dic₃₇	C₂	semidirect
8. C₄×Q₈×D₃₇	C₂	semidirect
9. C₃₇×#519	C₂	semidirect
10. #11924584	C₄	semidirect
34. C₂×Q₈×C₃₇	C₂×C₄	semidirect
40. #10596966	C₂×C₄	semidirect
41. Q₈×D₃₇	C₂×C₄	semidirect
42. #10596962	C₂×C₄	semidirect
93. Dic₃₇	2₋¹⁺⁴	semidirect
104. #519	D₃₇	semidirect
109. C₃₇	C₄×2₋¹⁺⁴	semidirect
138. C₂×Q₈	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924584	2
#14558015	2
#14558013	2
#14558009	2
C₄×#10596966	2
#14558001	2
C₂×Q₈×Dic₃₇	2
C₄×Q₈×D₃₇	2
C₃₇×#519	2
C₄×2₋¹⁺⁴	37