Subgroups — C₂×#14558088 #18007885

Order 4736.

1168 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	#14558088	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924536	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924527	C₂	1	2	yes	maximal, complemented
4	C₂×#11924516	C₂	1	2	yes	maximal
5	C₂×#11924512	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₃₇	C₂	1	2	yes	maximal, complemented
7	C₂×#11924501	C₂	1	2	yes	maximal, complemented
8	C₂×#11924494	C₂	1	2	yes	maximal
9	C₂×C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₇₄	C₂²	1	2	yes	—
12	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
15	#11924536	C₂²	1	8	yes	complemented
16	#11924527	C₂²	1	16	yes	complemented
17	#11924516	C₂²	1	16	yes	—
18	#11924512	C₂²	1	16	yes	—
19	Q₈×Dic₃₇	C₂²	1	16	yes	complemented
20	#11924501	C₂²	1	16	yes	complemented
21	#11924494	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
25	C₂×#10596936	C₂²	1	4	yes	—
26	C₂×#10596938	C₂²	1	4	yes	—
27	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
31	C₄²×D₃₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₃₇	C₂²	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
37	C₂³×D₃₇	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₃₇	—	2	8	no	—
39	C₂×Q₈×C₃₇	C₂³	1	8	yes	—
40	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₃₇	C₂³	1	4	yes	—
47	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₃₇	—	2	4	no	—
49	C₂×C₄×D₃₇	—	2	4	no	—
50	C₂×C₄×D₃₇	Q₈	1	8	yes	complemented
51	C₂×C₄×D₃₇	C₂³	1	4	yes	—
52	C₂×C₄×D₃₇	C₂³	1	4	yes	—
53	C₂×C₄×D₃₇	C₂³	1	4	yes	—
54	C₂×Dic₇₄	—	2	8	no	—
55	C₂×Dic₇₄	C₂³	1	8	yes	—
56	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
57	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
58	C₄²×C₃₇	C₂³	1	4	yes	—
59	#10596938	C₂³	1	16	yes	—
60	C₄⋊Dic₃₇	C₂³	1	8	yes	—
61	C₄⋊Dic₃₇	C₂³	1	8	yes	—
62	C₄⋊Dic₃₇	C₂³	1	16	yes	—
63	#10596936	C₂³	1	16	yes	—
64	C₄×Dic₃₇	C₂³	1	4	yes	—
65	C₄×Dic₃₇	C₂³	1	4	yes	—
66	C₄×Dic₃₇	C₂³	1	4	yes	—
67	C₄×Dic₃₇	C₂³	1	16	yes	—
68	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
69	C₂²×D₃₇	—	2	2	no	—
70	C₂²×D₃₇	—	2	2	no	—
71	C₂²×D₃₇	C₂×Q₈	1	4	yes	—
72	C₂²×D₃₇	C₂⁴	1	2	yes	—
73	Q₈×C₃₇	—	2	16	no	—
74	C₂×C₄×C₃₇	—	2	8	no	—
75	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₃₇	#41	1	4	yes	—
77	C₂×C₄×C₃₇	#41	1	4	yes	—
78	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
79	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
80	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
81	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
82	C₂×Dic₃₇	—	2	8	no	—
83	C₂×Dic₃₇	—	2	2	no	—
84	C₂×Dic₃₇	—	2	2	no	—
85	C₂×Dic₃₇	C₂×Q₈	1	4	yes	—
86	C₂×Dic₃₇	C₂⁴	1	2	yes	—
87	C₂×Dic₃₇	C₂⁴	1	2	yes	—
88	C₂×Dic₃₇	C₂⁴	1	8	yes	—
89	C₂×Dic₃₇	C₂⁴	1	2	yes	—
90	C₄×D₃₇	—	2	8	no	—
91	C₄×D₃₇	—	2	8	no	—
92	C₄×D₃₇	C₂×Q₈	1	16	yes	complemented
93	Dic₇₄	—	2	16	no	—
94	C₂²×C₃₇	C₂²×Q₈	1	2	yes	—
95	C₂²×C₃₇	C₂×#41	1	2	yes	—
96	C₂²×C₃₇	C₂×#41	1	2	yes	—
97	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
98	C₂×D₃₇	—	2	8	no	—
99	C₂×D₃₇	—	2	2	no	—
100	C₂×D₃₇	—	2	2	no	—
101	C₂×D₃₇	C₂²×Q₈	1	4	yes	—
102	C₄×C₃₇	—	2	8	no	—
103	C₄×C₃₇	C₂²×Q₈	1	4	yes	—
104	C₄×C₃₇	C₂×#41	1	4	yes	—
105	C₄×C₃₇	C₂×#41	1	4	yes	—
106	Dic₃₇	—	2	8	no	—
107	Dic₃₇	—	2	2	no	—
108	Dic₃₇	—	2	2	no	—
109	Dic₃₇	C₂²×Q₈	1	4	yes	—
110	C₂×#533	—	37	1	no	maximal, Sylow
111	C₂×C₃₇	#533	1	4	yes	complemented
112	C₂×C₃₇	C₂³×Q₈	1	1	yes	characteristic
113	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
114	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
115	D₃₇	—	2	4	no	—
116	#533	—	37	16	no	—
117	C₂×C₄⋊Q₈	—	37	1	no	—
118	C₂×C₄⋊Q₈	D₃₇	1	1	yes	characteristic, complemented
119	C₂×#125	—	37	2	no	—
120	C₂×C₂²⋊Q₈	—	37	2	no	—
121	C₂×C₂²⋊Q₈	—	37	2	no	—
122	C₂×C₄×Q₈	—	37	2	no	—
123	C₂×C₄×Q₈	—	37	2	no	—
124	C₂×#117	—	37	2	no	—
125	C₂²×C₄²	—	37	1	no	—
126	C₃₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	37	2	no	—
128	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
129	C₂³×C₄	—	37	1	no	—
130	C₂³×C₄	—	37	1	no	—
131	C₂³×C₄	—	37	1	no	—
132	C₄⋊Q₈	—	37	8	no	—
133	C₄⋊Q₈	C₂×D₃₇	1	8	yes	complemented
134	#125	—	37	16	no	—
135	C₂²⋊Q₈	—	37	16	no	—
136	C₂²⋊Q₈	—	37	16	no	—
137	C₄×Q₈	—	37	16	no	—
138	C₄×Q₈	—	37	16	no	—
139	#117	—	37	16	no	—
140	C₂×C₄⋊C₄	—	37	2	no	—
141	C₂×C₄⋊C₄	—	37	4	no	—
142	C₂×C₄⋊C₄	—	37	4	no	—
143	C₂×C₄⋊C₄	—	37	2	no	—
144	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
145	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
146	C₂×C₂²⋊C₄	—	37	4	no	—
147	C₂×C₄²	—	37	4	no	—
148	C₂×C₄²	—	37	1	no	—
149	C₂×C₄²	—	37	1	no	—
150	C₂×C₄²	—	37	1	no	—
151	C₂×C₄²	—	37	8	no	—
152	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
153	C₂⁴	—	37	1	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	37	8	no	—
156	C₂×Q₈	—	74	8	no	—
157	C₂×Q₈	C₂²×D₃₇	1	8	yes	—
158	C₂²×C₄	—	37	1	no	—
159	C₂²×C₄	—	37	8	no	—
160	C₂²×C₄	—	37	4	no	—
161	C₂²×C₄	—	37	1	no	—
162	C₂²×C₄	—	37	1	no	—
163	C₂²×C₄	—	37	4	no	—
164	C₂²×C₄	—	37	4	no	—
165	C₂²×C₄	—	37	4	no	—
166	C₂²×C₄	—	74	4	no	—
167	C₂²×C₄	—	74	4	no	—
168	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
169	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
172	C₄⋊C₄	—	37	16	no	—
173	C₄⋊C₄	—	37	8	no	—
174	C₄⋊C₄	—	37	16	no	—
175	C₄⋊C₄	—	37	8	no	—
176	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
177	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
178	C₂²⋊C₄	—	37	16	no	—
179	C₄²	—	37	4	no	—
180	C₄²	—	37	4	no	—
181	C₄²	—	37	4	no	—
182	C₄²	—	37	16	no	—
183	C₄²	C₂²×D₃₇	1	4	yes	—
184	C₂³	—	37	4	no	—
185	C₂³	—	37	2	no	—
186	C₂³	—	74	2	no	—
187	C₂³	—	74	2	no	—
188	C₂³	C₂³×D₃₇	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	74	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	37	4	no	—
193	C₂×C₄	—	37	2	no	—
194	C₂×C₄	—	37	16	no	—
195	C₂×C₄	—	37	2	no	—
196	C₂×C₄	—	37	2	no	—
197	C₂×C₄	—	37	8	no	—
198	C₂×C₄	—	74	2	no	—
199	C₂×C₄	—	74	2	no	—
200	C₂×C₄	—	74	8	no	—
201	C₂×C₄	—	74	8	no	—
202	C₂×C₄	—	74	8	no	—
203	C₂×C₄	#10596964	1	4	yes	—
204	C₂×C₄	Q₈×D₃₇	1	4	yes	—
205	C₂×C₄	#10596966	1	4	yes	—
206	C₂×C₄	C₂³×D₃₇	1	8	yes	—
207	C₂×C₄	C₂³×D₃₇	1	2	yes	—
208	C₂×C₄	C₂³×D₃₇	1	2	yes	—
209	C₂×C₄	C₂³×D₃₇	1	2	yes	—
210	C₂²	—	37	4	no	—
211	C₂²	—	74	8	no	—
212	C₂²	—	74	2	no	—
213	C₂²	—	74	2	no	—
214	C₂²	C₂×Q₈×D₃₇	1	2	yes	—
215	C₂²	C₂×#10596964	1	2	yes	—
216	C₂²	C₂×#10596966	1	2	yes	—
217	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
218	C₄	—	2	8	no	—
219	C₄	—	37	4	no	—
220	C₄	—	74	2	no	—
221	C₄	—	74	2	no	—
222	C₄	—	74	8	no	—
223	C₄	C₂×Q₈×D₃₇	1	4	yes	—
224	C₄	C₂×#10596964	1	4	yes	—
225	C₄	C₂×#10596966	1	4	yes	—
226	C₂	—	74	4	no	—
227	C₂	#14558088	1	4	yes	direct product factor
228	C₂	C₂²×Q₈×D₃₇	1	1	yes	characteristic
229	C₂	C₂²×#10596964	1	1	yes	characteristic
230	C₂	C₂²×#10596966	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	C₂³×C₃₇	C₂⁴
prime radical	C₂³×D₃₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₇
hypocenter	C₃₇	C₂×#533
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. #14558088	C₂	direct (N × H)
2. C₂×#11924536	C₂	semidirect
3. C₂×#11924527	C₂	semidirect
6. C₂×Q₈×Dic₃₇	C₂	semidirect
7. C₂×#11924501	C₂	semidirect
10. C₂×C₄⋊Q₈×C₃₇	C₂	semidirect
15. #11924536	C₂²	semidirect
16. #11924527	C₂²	semidirect
19. Q₈×Dic₃₇	C₂²	semidirect
20. #11924501	C₂²	semidirect
33. C₄⋊Q₈×C₃₇	C₂²	semidirect
50. C₂×C₄×D₃₇	Q₈	semidirect
92. C₄×D₃₇	C₂×Q₈	semidirect
111. C₂×C₃₇	#533	semidirect
118. C₂×C₄⋊Q₈	D₃₇	semidirect
126. C₃₇	C₂×#533	semidirect
133. C₄⋊Q₈	C₂×D₃₇	semidirect
227. C₂	#14558088	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558088	2
C₂×#11924536	2
C₂×#11924527	2
C₂×#11924516	2
C₂×#11924512	2
C₂×Q₈×Dic₃₇	2
C₂×#11924501	2
C₂×#11924494	2
C₂×C₄²×D₃₇	2
C₂×C₄⋊Q₈×C₃₇	2
C₂×#533	37

Subgroups — C2×#14558088 #18007885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558088 #18007885