Subgroups — C₂×#14558258 #18008165

Order 4736.

1048 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558258	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924534	C₂	1	2	yes	maximal
3	C₂×#11924527	C₂	1	2	yes	maximal
4	C₂×#11924526	C₂	1	4	yes	maximal
5	C₂×Q₈×Dic₃₇	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924501	C₂	1	2	yes	maximal
9	C₂×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
11	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
12	#11924534	C₂²	1	16	yes	—
13	#11924527	C₂²	1	16	yes	—
14	#11924526	C₂²	1	32	yes	—
15	Q₈×Dic₃₇	C₂²	1	16	yes	—
16	C₄×Dic₇₄	C₂²	1	8	yes	—
17	C₄×Dic₇₄	C₂²	1	8	yes	—
18	#11924501	C₂²	1	16	yes	—
19	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
22	C₂×#10596936	C₂²	1	4	yes	—
23	C₂×#10596936	C₂²	1	2	yes	—
24	C₂×#10596936	C₂²	1	2	yes	—
25	C₂×#10596936	C₂²	1	4	yes	—
26	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
28	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
30	C₄⋊Q₈×C₃₇	C₂²	1	8	yes	—
31	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
34	C₂×Q₈×C₃₇	—	2	8	no	—
35	C₂×Q₈×C₃₇	C₂³	1	8	yes	—
36	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
37	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃₇	C₂³	1	2	yes	—
41	C₂²×Dic₃₇	C₂³	1	2	yes	—
42	C₂²×Dic₃₇	C₂³	1	2	yes	—
43	C₂²×Dic₃₇	C₂³	1	2	yes	—
44	C₂×Dic₇₄	—	2	4	no	—
45	C₂×Dic₇₄	Q₈	1	8	yes	complemented
46	C₂×Dic₇₄	C₂³	1	4	yes	—
47	C₂×Dic₇₄	C₂³	1	4	yes	—
48	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
49	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
50	C₄²×C₃₇	C₂³	1	4	yes	—
51	C₄⋊Dic₃₇	C₂³	1	4	yes	—
52	C₄⋊Dic₃₇	C₂³	1	4	yes	—
53	C₄⋊Dic₃₇	C₂³	1	16	yes	—
54	#10596936	C₂³	1	16	yes	—
55	#10596936	C₂³	1	8	yes	—
56	#10596936	C₂³	1	8	yes	—
57	#10596936	C₂³	1	16	yes	—
58	C₄×Dic₃₇	C₂³	1	8	yes	—
59	C₄×Dic₃₇	C₂³	1	16	yes	—
60	C₄×Dic₃₇	C₂³	1	8	yes	—
61	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
62	Q₈×C₃₇	—	2	16	no	—
63	C₂×C₄×C₃₇	—	2	8	no	—
64	C₂×C₄×C₃₇	—	2	2	no	—
65	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₃₇	#41	1	4	yes	—
67	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
68	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
69	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
70	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
71	C₂×Dic₃₇	—	2	4	no	—
72	C₂×Dic₃₇	—	2	4	no	—
73	C₂×Dic₃₇	—	2	4	no	—
74	C₂×Dic₃₇	C₂×Q₈	1	8	yes	—
75	C₂×Dic₃₇	C₂⁴	1	4	yes	—
76	C₂×Dic₃₇	C₂⁴	1	4	yes	—
77	C₂×Dic₃₇	C₂⁴	1	4	yes	—
78	C₂×Dic₃₇	C₂⁴	1	4	yes	—
79	Dic₇₄	—	2	8	no	—
80	Dic₇₄	C₂×Q₈	1	16	yes	complemented
81	C₂²×C₃₇	C₂²×Q₈	1	2	yes	—
82	C₂²×C₃₇	C₂×#41	1	2	yes	—
83	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
84	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
85	C₄×C₃₇	—	2	8	no	—
86	C₄×C₃₇	—	2	2	no	—
87	C₄×C₃₇	C₂²×Q₈	1	4	yes	—
88	C₄×C₃₇	C₂×#41	1	4	yes	—
89	Dic₃₇	—	2	4	no	—
90	Dic₃₇	—	2	4	no	—
91	Dic₃₇	—	2	4	no	—
92	Dic₃₇	C₂²×Q₈	1	8	yes	—
93	C₂×#557	—	37	1	no	maximal, Sylow
94	C₂×C₃₇	#557	1	4	yes	complemented
95	C₂×C₃₇	C₂³×Q₈	1	1	yes	characteristic
96	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
97	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
98	#557	—	37	16	no	—
99	C₂×C₄⋊Q₈	—	37	2	no	—
100	C₂×C₄⋊Q₈	D₃₇	1	1	yes	characteristic
101	C₂×#125	—	37	4	no	—
102	C₂×#125	—	37	2	no	—
103	C₂×C₄×Q₈	—	37	2	no	—
104	C₂×C₄×Q₈	—	37	2	no	—
105	C₂×C₄×Q₈	—	37	1	no	—
106	C₂×C₄×Q₈	—	37	1	no	—
107	C₃₇	C₂×#557	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	37	1	no	—
109	C₂²×Q₈	—	37	1	no	—
110	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
111	C₄⋊Q₈	—	37	16	no	—
112	C₄⋊Q₈	C₂×D₃₇	1	8	yes	—
113	#125	—	37	32	no	—
114	#125	—	37	16	no	—
115	C₄×Q₈	—	37	16	no	—
116	C₄×Q₈	—	37	16	no	—
117	C₄×Q₈	—	37	8	no	—
118	C₄×Q₈	—	37	8	no	—
119	C₂×C₄⋊C₄	—	37	1	no	—
120	C₂×C₄⋊C₄	—	37	2	no	—
121	C₂×C₄⋊C₄	—	37	2	no	—
122	C₂×C₄⋊C₄	—	37	4	no	—
123	C₂×C₄⋊C₄	—	37	4	no	—
124	C₂×C₄⋊C₄	—	37	1	no	—
125	C₂×C₄⋊C₄	—	37	4	no	—
126	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
127	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
128	C₂×C₄²	—	37	4	no	—
129	C₂×C₄²	—	37	2	no	—
130	C₂×C₄²	—	37	2	no	—
131	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	37	8	no	—
134	C₂×Q₈	—	37	4	no	—
135	C₂×Q₈	—	37	4	no	—
136	C₂×Q₈	—	74	4	no	—
137	C₂×Q₈	C₂²×D₃₇	1	8	yes	—
138	C₂²×C₄	—	37	2	no	—
139	C₂²×C₄	—	37	2	no	—
140	C₂²×C₄	—	37	2	no	—
141	C₂²×C₄	—	37	2	no	—
142	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
143	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
146	C₄⋊C₄	—	37	8	no	—
147	C₄⋊C₄	—	37	8	no	—
148	C₄⋊C₄	—	37	16	no	—
149	C₄⋊C₄	—	37	16	no	—
150	C₄⋊C₄	—	37	4	no	—
151	C₄⋊C₄	—	37	16	no	—
152	C₄⋊C₄	—	37	4	no	—
153	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
154	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
155	C₄²	—	37	16	no	—
156	C₄²	—	37	8	no	—
157	C₄²	—	37	8	no	—
158	C₄²	C₂²×D₃₇	1	4	yes	—
159	C₂³	C₂³×D₃₇	1	1	yes	characteristic
160	Q₈	—	2	16	no	—
161	Q₈	—	37	16	no	—
162	Q₈	—	74	8	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	37	8	no	—
166	C₂×C₄	—	37	4	no	—
167	C₂×C₄	—	37	4	no	—
168	C₂×C₄	—	37	4	no	—
169	C₂×C₄	—	37	4	no	—
170	C₂×C₄	—	74	4	no	—
171	C₂×C₄	—	74	4	no	—
172	C₂×C₄	—	74	4	no	—
173	C₂×C₄	#10596964	1	4	yes	—
174	C₂×C₄	Q₈×D₃₇	1	4	yes	—
175	C₂×C₄	C₂³×D₃₇	1	8	yes	—
176	C₂×C₄	C₂³×D₃₇	1	2	yes	—
177	C₂×C₄	C₂³×D₃₇	1	2	yes	—
178	C₂×C₄	C₂³×D₃₇	1	2	yes	—
179	C₂²	C₂×Q₈×D₃₇	1	2	yes	—
180	C₂²	C₂×#10596964	1	2	yes	—
181	C₂²	#11924584	1	2	yes	—
182	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
183	C₄	—	2	8	no	—
184	C₄	—	2	2	no	—
185	C₄	—	37	8	no	—
186	C₄	—	74	4	no	—
187	C₄	—	74	4	no	—
188	C₄	—	74	4	no	—
189	C₄	C₂×Q₈×D₃₇	1	4	yes	—
190	C₄	C₂×#10596964	1	4	yes	—
191	C₂	#14558258	1	4	yes	direct product factor
192	C₂	C₂²×Q₈×D₃₇	1	1	yes	characteristic
193	C₂	C₂²×#10596964	1	1	yes	characteristic
194	C₂	C₂×#11924584	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×C₄⋊Q₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₇
hypocenter	C₃₇	C₂×#557
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14558258	C₂	direct (N × H)
45. C₂×Dic₇₄	Q₈	semidirect
80. Dic₇₄	C₂×Q₈	semidirect
94. C₂×C₃₇	#557	semidirect
107. C₃₇	C₂×#557	semidirect
191. C₂	#14558258	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558258	2
C₂×#11924534	2
C₂×#11924527	2
C₂×#11924526	2
C₂×Q₈×Dic₃₇	2
C₂×C₄×Dic₇₄	2
C₂×#11924501	2
C₂×C₄⋊Q₈×C₃₇	2
C₂×#557	37

Subgroups — C2×#14558258 #18008165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558258 #18008165