Subgroups — C₂×#14558158 #18007990

Order 4736.

1296 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558158	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924516	C₂	1	2	yes	maximal, complemented
3	C₂×#11924515	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11924514	C₂	1	2	yes	maximal, complemented
5	C₂×#11924514	C₂	1	2	yes	maximal, complemented
6	C₂×#11924513	C₂	1	2	yes	maximal, complemented
7	C₂×#11924512	C₂	1	2	yes	maximal, complemented
8	C₂²×#10596938	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊C₄×D₃₇	C₂	1	2	yes	maximal
10	C₂×C₂²⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₃₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₇₄	C₂²	1	2	yes	complemented
13	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
14	C₂³×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
17	#11924516	C₂²	1	16	yes	complemented
18	#11924515	C₂²	1	8	yes	complemented
19	#11924514	C₂²	1	16	yes	complemented
20	#11924514	C₂²	1	16	yes	complemented
21	#11924513	C₂²	1	16	yes	complemented
22	#11924512	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	complemented
24	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×#10596936	C₂²	1	2	yes	complemented
26	C₂×#10596936	C₂²	1	2	yes	complemented
27	C₂×#10596936	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₃₇	C₂²	1	2	yes	—
29	C₂×#10596938	C₂²	1	2	yes	—
30	C₂×#10596938	C₂²	1	2	yes	—
31	C₂×#10596938	C₂²	1	2	yes	—
32	C₂×#10596938	C₂²	1	2	yes	—
33	C₂×#10596938	C₂²	1	8	yes	—
34	C₂²⋊C₄×D₃₇	C₂²	1	16	yes	—
35	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₃₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
41	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
42	C₂³×D₃₇	—	4	2	no	—
43	C₂³×D₃₇	—	4	2	no	—
44	C₂³×D₃₇	Q₈	1	8	yes	complemented
45	C₂³×D₃₇	C₂³	1	2	yes	—
46	C₂³×D₃₇	C₂³	1	4	yes	—
47	C₂×Q₈×C₃₇	—	4	2	no	—
48	C₂×Q₈×C₃₇	C₂³	1	4	yes	complemented
49	C₂²×C₄×C₃₇	—	4	2	no	—
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	4	yes	—
55	C₂²×Dic₃₇	—	4	2	no	—
56	C₂²×Dic₃₇	C₂³	1	2	yes	—
57	C₂²×Dic₃₇	C₂³	1	2	yes	—
58	C₂²×Dic₃₇	C₂³	1	2	yes	—
59	C₂²×Dic₃₇	C₂³	1	4	yes	—
60	C₂×C₄×D₃₇	—	4	4	no	—
61	C₂×C₄×D₃₇	—	4	4	no	—
62	C₂×C₄×D₃₇	C₂³	1	8	yes	—
63	C₂×C₄×D₃₇	C₂³	1	8	yes	—
64	C₂×Dic₇₄	—	4	2	no	—
65	C₂×Dic₇₄	—	4	4	no	—
66	C₂×Dic₇₄	C₂³	1	8	yes	complemented
67	C₂×Dic₇₄	C₂³	1	4	yes	complemented
68	C₄⋊C₄×C₃₇	C₂³	1	4	yes	complemented
69	C₄⋊C₄×C₃₇	C₂³	1	8	yes	complemented
70	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	—
71	C₂²⋊Dic₃₇	C₂³	1	8	yes	—
72	#10596938	C₂³	1	8	yes	—
73	#10596938	C₂³	1	8	yes	—
74	#10596938	C₂³	1	8	yes	—
75	#10596938	C₂³	1	8	yes	—
76	C₄⋊Dic₃₇	C₂³	1	8	yes	complemented
77	C₄⋊Dic₃₇	C₂³	1	4	yes	complemented
78	#10596936	C₂³	1	8	yes	complemented
79	#10596936	C₂³	1	8	yes	complemented
80	#10596936	C₂³	1	8	yes	complemented
81	C₂³×C₃₇	—	2	2	no	—
82	C₂³×C₃₇	—	2	2	no	—
83	C₂³×C₃₇	C₂×Q₈	1	4	yes	—
84	C₂³×C₃₇	C₂⁴	1	2	yes	—
85	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
86	C₂²×D₃₇	—	2	4	no	—
87	C₂²×D₃₇	—	2	4	no	—
88	C₂²×D₃₇	—	4	16	no	—
89	C₂²×D₃₇	—	4	4	no	—
90	C₂²×D₃₇	—	4	4	no	—
91	C₂²×D₃₇	C₂×Q₈	1	8	yes	—
92	C₂²×D₃₇	C₂×Q₈	1	16	yes	complemented
93	C₂²×D₃₇	C₂⁴	1	4	yes	—
94	Q₈×C₃₇	—	4	4	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	4	4	no	—
100	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
101	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
102	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
103	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
104	C₂×Dic₃₇	—	2	4	no	—
105	C₂×Dic₃₇	—	2	4	no	—
106	C₂×Dic₃₇	—	2	4	no	—
107	C₂×Dic₃₇	—	4	4	no	—
108	C₂×Dic₃₇	C₂⁴	1	4	yes	—
109	C₂×Dic₃₇	C₂⁴	1	4	yes	—
110	C₂×Dic₃₇	C₂⁴	1	4	yes	—
111	C₄×D₃₇	—	4	8	no	—
112	C₄×D₃₇	—	4	8	no	—
113	Dic₇₄	—	4	4	no	—
114	Dic₇₄	—	4	8	no	—
115	C₂²×C₃₇	—	2	8	no	—
116	C₂²×C₃₇	—	2	2	no	—
117	C₂²×C₃₇	—	2	2	no	—
118	C₂²×C₃₇	C₂²×Q₈	1	4	yes	—
119	C₂²×C₃₇	C₂²×Q₈	1	2	yes	—
120	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
123	C₂×D₃₇	—	2	16	no	—
124	C₂×D₃₇	—	2	4	no	—
125	C₂×D₃₇	—	2	4	no	—
126	C₂×D₃₇	—	4	16	no	—
127	C₂×D₃₇	C₂²×Q₈	1	8	yes	—
128	C₄×C₃₇	—	2	4	no	—
129	C₄×C₃₇	—	2	2	no	—
130	C₄×C₃₇	—	2	4	no	—
131	C₄×C₃₇	—	2	2	no	—
132	Dic₃₇	—	2	4	no	—
133	Dic₃₇	—	2	4	no	—
134	Dic₃₇	—	2	4	no	—
135	C₂×#543	—	37	1	no	maximal, Sylow
136	C₂×C₃₇	—	2	4	no	—
137	C₂×C₃₇	#543	1	4	yes	complemented
138	C₂×C₃₇	C₂³×Q₈	1	1	yes	characteristic
139	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	D₃₇	—	2	8	no	—
142	#543	—	37	16	no	—
143	C₂×C₂²⋊Q₈	—	37	2	no	—
144	C₂×C₂²⋊Q₈	—	37	1	no	—
145	C₂×C₂²⋊Q₈	—	37	2	no	—
146	C₂×C₂²⋊Q₈	—	37	2	no	—
147	C₂×C₂²⋊Q₈	—	37	2	no	—
148	C₂×C₂²⋊Q₈	—	37	2	no	—
149	C₂×C₂²⋊Q₈	D₃₇	1	1	yes	characteristic, complemented
150	C₂²×C₂²⋊C₄	—	37	1	no	—
151	C₂²×C₂²⋊C₄	—	37	2	no	—
152	C₃₇	C₂×#543	1	1	yes	characteristic, complemented, Sylow
153	C₂⁵	—	37	1	no	—
154	C₂²×Q₈	—	37	1	no	—
155	C₂²×Q₈	—	37	2	no	—
156	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic, complemented
157	C₂³×C₄	—	37	1	no	—
158	C₂³×C₄	—	37	2	no	—
159	C₂³×C₄	—	37	2	no	—
160	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
161	C₂²⋊Q₈	—	37	16	no	—
162	C₂²⋊Q₈	—	37	8	no	—
163	C₂²⋊Q₈	—	37	16	no	—
164	C₂²⋊Q₈	—	37	16	no	—
165	C₂²⋊Q₈	—	37	16	no	—
166	C₂²⋊Q₈	—	37	16	no	—
167	C₂²⋊Q₈	C₂×D₃₇	1	8	yes	complemented
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	1	no	—
172	C₂×C₄⋊C₄	—	37	2	no	—
173	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic, complemented
174	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	complemented
175	C₂×C₂²⋊C₄	—	37	2	no	—
176	C₂×C₂²⋊C₄	—	37	2	no	—
177	C₂×C₂²⋊C₄	—	37	2	no	—
178	C₂×C₂²⋊C₄	—	37	2	no	—
179	C₂×C₂²⋊C₄	—	37	2	no	—
180	C₂×C₂²⋊C₄	—	37	8	no	—
181	C₂×C₂²⋊C₄	—	37	16	no	—
182	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
183	C₂⁴	—	37	2	no	—
184	C₂⁴	—	37	8	no	—
185	C₂⁴	—	37	4	no	—
186	C₂⁴	—	148	2	no	—
187	C₂⁴	—	148	2	no	—
188	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	37	4	no	—
191	C₂×Q₈	—	37	8	no	—
192	C₂×Q₈	—	148	2	no	—
193	C₂×Q₈	—	148	4	no	—
194	C₂×Q₈	C₂²×D₃₇	1	4	yes	complemented
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	37	2	no	—
197	C₂²×C₄	—	37	2	no	—
198	C₂²×C₄	—	37	2	no	—
199	C₂²×C₄	—	37	4	no	—
200	C₂²×C₄	—	37	8	no	—
201	C₂²×C₄	—	37	8	no	—
202	C₂²×C₄	—	148	4	no	—
203	C₂²×C₄	—	148	4	no	—
204	C₂²×C₄	—	148	2	no	—
205	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
208	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
210	C₄⋊C₄	—	37	8	no	—
211	C₄⋊C₄	—	37	8	no	—
212	C₄⋊C₄	—	37	8	no	—
213	C₄⋊C₄	—	37	4	no	—
214	C₄⋊C₄	—	37	8	no	—
215	C₄⋊C₄	C₂²×D₃₇	1	4	yes	complemented
216	C₄⋊C₄	C₂²×D₃₇	1	8	yes	complemented
217	C₂²⋊C₄	—	37	8	no	—
218	C₂²⋊C₄	—	37	8	no	—
219	C₂²⋊C₄	—	37	8	no	—
220	C₂²⋊C₄	—	37	8	no	—
221	C₂²⋊C₄	—	37	8	no	—
222	C₂²⋊C₄	C₂²×D₃₇	1	8	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	37	8	no	—
226	C₂³	—	37	4	no	—
227	C₂³	—	37	16	no	—
228	C₂³	—	74	4	no	—
229	C₂³	—	74	4	no	—
230	C₂³	—	148	4	no	—
231	C₂³	—	148	4	no	—
232	C₂³	—	148	16	no	—
233	C₂³	Q₈×D₃₇	1	4	yes	—
234	C₂³	C₂³×D₃₇	1	2	yes	—
235	C₂³	C₂³×D₃₇	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	148	4	no	—
238	Q₈	—	148	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	37	4	no	—
245	C₂×C₄	—	37	4	no	—
246	C₂×C₄	—	37	4	no	—
247	C₂×C₄	—	74	4	no	—
248	C₂×C₄	—	74	4	no	—
249	C₂×C₄	—	74	4	no	—
250	C₂×C₄	—	148	8	no	—
251	C₂×C₄	—	148	8	no	—
252	C₂×C₄	—	148	4	no	—
253	C₂×C₄	C₂³×D₃₇	1	2	yes	—
254	C₂×C₄	C₂³×D₃₇	1	2	yes	—
255	C₂×C₄	C₂³×D₃₇	1	4	yes	—
256	C₂×C₄	C₂³×D₃₇	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	37	8	no	—
261	C₂²	—	74	4	no	—
262	C₂²	—	74	4	no	—
263	C₂²	—	74	16	no	—
264	C₂²	—	148	16	no	—
265	C₂²	C₂×Q₈×D₃₇	1	4	yes	—
266	C₂²	C₂×Q₈×D₃₇	1	2	yes	—
267	C₂²	#11924581	1	2	yes	—
268	C₂²	#11924582	1	2	yes	—
269	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	74	4	no	—
275	C₄	—	74	4	no	—
276	C₄	—	74	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	74	8	no	—
279	C₂	#14558158	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₃₇	1	1	yes	characteristic
281	C₂	C₂×#11924581	1	1	yes	characteristic
282	C₂	C₂×#11924582	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×C₂²⋊Q₈×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂⁴×D₃₇	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₃₇
hypocenter	C₃₇	C₂×#543
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14558158	C₂	direct (N × H)
2. C₂×#11924516	C₂	semidirect
3. C₂×#11924515	C₂	semidirect
4. C₂×#11924514	C₂	semidirect
5. C₂×#11924514	C₂	semidirect
6. C₂×#11924513	C₂	semidirect
7. C₂×#11924512	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₃₇	C₂	semidirect
12. C₂²×Dic₇₄	C₂²	semidirect
13. C₂²×Dic₇₄	C₂²	semidirect
17. #11924516	C₂²	semidirect
18. #11924515	C₂²	semidirect
19. #11924514	C₂²	semidirect
20. #11924514	C₂²	semidirect
21. #11924513	C₂²	semidirect
22. #11924512	C₂²	semidirect
23. C₂×C₄⋊Dic₃₇	C₂²	semidirect
24. C₂×C₄⋊Dic₃₇	C₂²	semidirect
25. C₂×#10596936	C₂²	semidirect
26. C₂×#10596936	C₂²	semidirect
27. C₂×#10596936	C₂²	semidirect
35. C₂²×Q₈×C₃₇	C₂²	semidirect
37. C₂²⋊Q₈×C₃₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
39. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
44. C₂³×D₃₇	Q₈	semidirect
48. C₂×Q₈×C₃₇	C₂³	semidirect
66. C₂×Dic₇₄	C₂³	semidirect
67. C₂×Dic₇₄	C₂³	semidirect
68. C₄⋊C₄×C₃₇	C₂³	semidirect
69. C₄⋊C₄×C₃₇	C₂³	semidirect
76. C₄⋊Dic₃₇	C₂³	semidirect
77. C₄⋊Dic₃₇	C₂³	semidirect
78. #10596936	C₂³	semidirect
79. #10596936	C₂³	semidirect
80. #10596936	C₂³	semidirect
92. C₂²×D₃₇	C₂×Q₈	semidirect
137. C₂×C₃₇	#543	semidirect
149. C₂×C₂²⋊Q₈	D₃₇	semidirect
152. C₃₇	C₂×#543	semidirect
156. C₂²×Q₈	C₂×D₃₇	semidirect
167. C₂²⋊Q₈	C₂×D₃₇	semidirect
173. C₂×C₄⋊C₄	C₂×D₃₇	semidirect
174. C₂×C₄⋊C₄	C₂×D₃₇	semidirect
194. C₂×Q₈	C₂²×D₃₇	semidirect
215. C₄⋊C₄	C₂²×D₃₇	semidirect
216. C₄⋊C₄	C₂²×D₃₇	semidirect
279. C₂	#14558158	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558158	2
C₂×#11924516	2
C₂×#11924515	2
C₂×#11924514	2
C₂×#11924513	2
C₂×#11924512	2
C₂²×#10596938	2
C₂×C₂²⋊C₄×D₃₇	2
C₂×C₂²⋊Q₈×C₃₇	2
C₂×#543	37

Subgroups — C2×#14558158 #18007990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558158 #18007990