Subgroups — C₂×#14558141 #18007960

Order 4736.

1004 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558141	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924534	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924531	C₂	1	2	yes	maximal
4	C₂×#11924530	C₂	1	2	yes	maximal
5	C₂×#11924528	C₂	1	2	yes	maximal, complemented
6	C₂×#11924527	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11924526	C₂	1	2	yes	maximal, complemented
8	C₂×#11924516	C₂	1	1	yes	maximal, characteristic
9	C₂×#11924512	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#11924493	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
15	#11924534	C₂²	1	8	yes	complemented
16	#11924531	C₂²	1	16	yes	—
17	#11924530	C₂²	1	16	yes	—
18	#11924528	C₂²	1	16	yes	complemented
19	#11924527	C₂²	1	8	yes	complemented
20	#11924526	C₂²	1	16	yes	complemented
21	#11924516	C₂²	1	8	yes	—
22	#11924512	C₂²	1	8	yes	—
23	C₄×Dic₇₄	C₂²	1	8	yes	complemented
24	#11924493	C₂²	1	8	yes	—
25	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
28	C₂×#10596936	C₂²	1	1	yes	characteristic
29	C₂×#10596936	C₂²	1	1	yes	characteristic
30	C₂×#10596936	C₂²	1	2	yes	—
31	C₂×#10596936	C₂²	1	2	yes	—
32	C₂×#10596936	C₂²	1	2	yes	—
33	C₂×#10596936	C₂²	1	2	yes	—
34	C₂×#10596936	C₂²	1	2	yes	—
35	C₂×#10596938	C₂²	1	1	yes	characteristic
36	C₂×#10596938	C₂²	1	1	yes	characteristic
37	C₂×#10596938	C₂²	1	2	yes	—
38	C₂×#10596938	C₂²	1	2	yes	—
39	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₃₇	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
45	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₃₇	C₂²	1	2	yes	—
47	C₂³×D₃₇	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₃₇	—	2	4	no	—
49	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
50	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
57	C₂²×Dic₃₇	C₂³	1	2	yes	—
58	C₂²×Dic₃₇	C₂³	1	2	yes	—
59	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₃₇	—	2	4	no	—
61	C₂×C₄×D₃₇	C₂³	1	4	yes	—
62	C₂×Dic₇₄	—	2	4	no	—
63	C₂×Dic₇₄	C₂³	1	4	yes	—
64	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
65	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
66	C₄²×C₃₇	C₂³	1	4	yes	—
67	C₄²×C₃₇	C₂³	1	8	yes	—
68	#10596938	C₂³	1	4	yes	—
69	#10596938	C₂³	1	4	yes	—
70	#10596938	C₂³	1	8	yes	—
71	#10596938	C₂³	1	8	yes	—
72	C₄⋊Dic₃₇	C₂³	1	4	yes	—
73	C₄⋊Dic₃₇	C₂³	1	8	yes	—
74	C₄⋊Dic₃₇	C₂³	1	8	yes	—
75	#10596936	C₂³	1	4	yes	—
76	#10596936	C₂³	1	4	yes	—
77	#10596936	C₂³	1	8	yes	—
78	#10596936	C₂³	1	8	yes	—
79	#10596936	C₂³	1	8	yes	—
80	#10596936	C₂³	1	8	yes	—
81	#10596936	C₂³	1	8	yes	—
82	C₄×Dic₃₇	C₂³	1	4	yes	—
83	C₄×Dic₃₇	C₂³	1	8	yes	—
84	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
85	C₂²×D₃₇	—	2	2	no	—
86	C₂²×D₃₇	—	2	2	no	—
87	C₂²×D₃₇	—	2	2	no	—
88	C₂²×D₃₇	C₂⁴	1	2	yes	—
89	Q₈×C₃₇	—	2	8	no	—
90	C₂×C₄×C₃₇	—	2	4	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	—	2	4	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	#41	1	4	yes	—
95	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
96	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
97	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
98	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
99	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
100	C₂×Dic₃₇	—	2	4	no	—
101	C₂×Dic₃₇	—	2	2	no	—
102	C₂×Dic₃₇	—	2	2	no	—
103	C₂×Dic₃₇	—	2	2	no	—
104	C₂×Dic₃₇	—	2	4	no	—
105	C₂×Dic₃₇	C₂⁴	1	2	yes	—
106	C₂×Dic₃₇	C₂⁴	1	2	yes	—
107	C₂×Dic₃₇	C₂⁴	1	4	yes	—
108	C₂×Dic₃₇	C₂⁴	1	4	yes	—
109	C₂×Dic₃₇	C₂⁴	1	2	yes	—
110	C₄×D₃₇	—	2	8	no	—
111	Dic₇₄	—	2	8	no	—
112	C₂²×C₃₇	C₂×#41	1	2	yes	—
113	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
114	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
116	C₂×D₃₇	—	2	2	no	—
117	C₂×D₃₇	—	2	2	no	—
118	C₂×D₃₇	—	2	2	no	—
119	C₂×D₃₇	—	4	4	no	—
120	C₄×C₃₇	—	2	4	no	—
121	C₄×C₃₇	—	2	2	no	—
122	C₄×C₃₇	—	2	4	no	—
123	C₄×C₃₇	—	2	2	no	—
124	C₄×C₃₇	C₂×#41	1	4	yes	—
125	Dic₃₇	—	2	4	no	—
126	Dic₃₇	—	2	2	no	—
127	Dic₃₇	—	2	2	no	—
128	Dic₃₇	—	2	2	no	—
129	Dic₃₇	—	2	4	no	—
130	C₂×#541	—	37	1	no	maximal, Sylow
131	C₂×C₃₇	#541	1	4	yes	complemented
132	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
133	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
135	D₃₇	—	4	2	no	—
136	#541	—	37	16	no	—
137	C₂×C₄⋊Q₈	—	37	1	no	—
138	C₂×#126	—	37	2	no	—
139	C₂×#126	—	37	2	no	—
140	C₂×#125	—	37	2	no	—
141	C₂×#125	—	37	1	no	—
142	C₂×#125	—	37	2	no	—
143	C₂×C₂²⋊Q₈	—	37	1	no	—
144	C₂×C₂²⋊Q₈	—	37	1	no	—
145	C₂×C₄×Q₈	—	37	1	no	—
146	C₂×C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
147	C₂×#117	—	37	1	no	—
148	C₃₇	C₂×#541	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	37	1	no	—
150	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
151	C₂³×C₄	—	37	1	no	—
152	C₄⋊Q₈	—	37	8	no	—
153	#126	—	37	16	no	—
154	#126	—	37	16	no	—
155	#125	—	37	16	no	—
156	#125	—	37	8	no	—
157	#125	—	37	16	no	—
158	C₂²⋊Q₈	—	37	8	no	—
159	C₂²⋊Q₈	—	37	8	no	—
160	C₄×Q₈	—	37	8	no	—
161	C₄×Q₈	C₂×D₃₇	1	8	yes	complemented
162	#117	—	37	8	no	—
163	C₂×C₄⋊C₄	—	37	1	no	—
164	C₂×C₄⋊C₄	—	37	1	no	—
165	C₂×C₄⋊C₄	—	37	2	no	—
166	C₂×C₄⋊C₄	—	37	1	no	—
167	C₂×C₄⋊C₄	—	37	2	no	—
168	C₂×C₄⋊C₄	—	37	2	no	—
169	C₂×C₄⋊C₄	—	37	2	no	—
170	C₂×C₄⋊C₄	—	37	2	no	—
171	C₂×C₄⋊C₄	—	37	2	no	—
172	C₂×C₄⋊C₄	—	37	2	no	—
173	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
175	C₂×C₂²⋊C₄	—	37	1	no	—
176	C₂×C₂²⋊C₄	—	37	1	no	—
177	C₂×C₂²⋊C₄	—	37	2	no	—
178	C₂×C₂²⋊C₄	—	37	2	no	—
179	C₂×C₄²	—	37	1	no	—
180	C₂×C₄²	—	37	2	no	—
181	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×D₃₇	1	2	yes	—
183	C₂⁴	—	37	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	37	4	no	—
186	C₂×Q₈	—	74	4	no	—
187	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
188	C₂²×C₄	—	37	1	no	—
189	C₂²×C₄	—	37	1	no	—
190	C₂²×C₄	—	37	2	no	—
191	C₂²×C₄	—	37	2	no	—
192	C₂²×C₄	—	37	1	no	—
193	C₂²×C₄	—	37	4	no	—
194	C₂²×C₄	—	74	4	no	—
195	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
198	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
199	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
200	C₄⋊C₄	—	37	4	no	—
201	C₄⋊C₄	—	37	4	no	—
202	C₄⋊C₄	—	37	8	no	—
203	C₄⋊C₄	—	37	8	no	—
204	C₄⋊C₄	—	37	8	no	—
205	C₄⋊C₄	—	37	8	no	—
206	C₄⋊C₄	—	37	8	no	—
207	C₄⋊C₄	—	37	4	no	—
208	C₄⋊C₄	—	37	8	no	—
209	C₄⋊C₄	—	37	8	no	—
210	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
211	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
212	C₂²⋊C₄	—	37	4	no	—
213	C₂²⋊C₄	—	37	4	no	—
214	C₂²⋊C₄	—	37	8	no	—
215	C₂²⋊C₄	—	37	8	no	—
216	C₄²	—	37	4	no	—
217	C₄²	—	37	8	no	—
218	C₄²	C₂²×D₃₇	1	4	yes	—
219	C₄²	C₂²×D₃₇	1	8	yes	—
220	C₂³	—	37	2	no	—
221	C₂³	—	74	2	no	—
222	C₂³	—	74	2	no	—
223	C₂³	—	74	2	no	—
224	C₂³	C₂³×D₃₇	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	74	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	37	2	no	—
232	C₂×C₄	—	37	2	no	—
233	C₂×C₄	—	37	4	no	—
234	C₂×C₄	—	37	4	no	—
235	C₂×C₄	—	37	2	no	—
236	C₂×C₄	—	74	2	no	—
237	C₂×C₄	—	74	4	no	—
238	C₂×C₄	—	74	2	no	—
239	C₂×C₄	—	74	2	no	—
240	C₂×C₄	—	74	4	no	—
241	C₂×C₄	—	74	8	no	—
242	C₂×C₄	#10596962	1	4	yes	—
243	C₂×C₄	C₂³×D₃₇	1	2	yes	—
244	C₂×C₄	C₂³×D₃₇	1	2	yes	—
245	C₂×C₄	C₂³×D₃₇	1	4	yes	—
246	C₂×C₄	C₂³×D₃₇	1	4	yes	—
247	C₂×C₄	C₂³×D₃₇	1	2	yes	—
248	C₂²	—	74	2	no	—
249	C₂²	—	74	2	no	—
250	C₂²	—	74	2	no	—
251	C₂²	—	148	4	no	—
252	C₂²	C₂×#10596962	1	2	yes	—
253	C₂²	#11924583	1	2	yes	—
254	C₂²	#11924584	1	2	yes	—
255	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	74	2	no	—
261	C₄	—	74	4	no	—
262	C₄	—	74	2	no	—
263	C₄	—	74	2	no	—
264	C₄	—	74	4	no	—
265	C₄	C₂×#10596962	1	4	yes	—
266	C₂	—	148	2	no	—
267	C₂	#14558141	1	4	yes	direct product factor
268	C₂	C₂²×#10596962	1	1	yes	characteristic
269	C₂	C₂×#11924583	1	1	yes	characteristic
270	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	C₂³×C₃₇	C₂⁴
prime radical	C₂³×D₃₇	C₂³
hypercenter	C₂×C₄×Q₈	D₃₇
hypocenter	C₃₇	C₂×#541
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. #14558141	C₂	direct (N × H)
2. C₂×#11924534	C₂	semidirect
5. C₂×#11924528	C₂	semidirect
6. C₂×#11924527	C₂	semidirect
7. C₂×#11924526	C₂	semidirect
10. C₂×C₄×Dic₇₄	C₂	semidirect
12. C₂×C₄×Q₈×C₃₇	C₂	semidirect
15. #11924534	C₂²	semidirect
18. #11924528	C₂²	semidirect
19. #11924527	C₂²	semidirect
20. #11924526	C₂²	semidirect
23. C₄×Dic₇₄	C₂²	semidirect
42. C₄×Q₈×C₃₇	C₂²	semidirect
131. C₂×C₃₇	#541	semidirect
146. C₂×C₄×Q₈	D₃₇	semidirect
148. C₃₇	C₂×#541	semidirect
161. C₄×Q₈	C₂×D₃₇	semidirect
267. C₂	#14558141	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558141	2
C₂×#11924534	2
C₂×#11924531	2
C₂×#11924530	2
C₂×#11924528	2
C₂×#11924527	2
C₂×#11924526	2
C₂×#11924516	2
C₂×#11924512	2
C₂×C₄×Dic₇₄	2
C₂×#11924493	2
C₂×C₄×Q₈×C₃₇	2
C₂×#541	37

Subgroups — C2×#14558141 #18007960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558141 #18007960