Subgroups — C₂×#14558163 #18007996

Order 4736.

1080 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558163	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924536	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924535	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11924535	C₂	1	2	yes	maximal, complemented
5	C₂×#11924527	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924526	C₂	1	2	yes	maximal, complemented
7	C₂×#11924516	C₂	1	2	yes	maximal
8	C₂×#11924514	C₂	1	2	yes	maximal
9	C₂×#11924494	C₂	1	1	yes	maximal, characteristic
10	C₂×#11924493	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₇₄	C₂²	1	2	yes	—
14	C₂²×Dic₇₄	C₂²	1	2	yes	—
15	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
17	#11924536	C₂²	1	8	yes	complemented
18	#11924535	C₂²	1	8	yes	complemented
19	#11924535	C₂²	1	16	yes	complemented
20	#11924527	C₂²	1	8	yes	complemented
21	#11924526	C₂²	1	16	yes	complemented
22	#11924516	C₂²	1	16	yes	—
23	#11924514	C₂²	1	16	yes	—
24	#11924494	C₂²	1	8	yes	—
25	#11924493	C₂²	1	8	yes	—
26	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
30	C₂×#10596936	C₂²	1	2	yes	—
31	C₂×#10596936	C₂²	1	1	yes	characteristic
32	C₂×#10596936	C₂²	1	1	yes	characteristic
33	C₂×#10596936	C₂²	1	2	yes	—
34	C₂×#10596936	C₂²	1	2	yes	—
35	C₄⋊C₄×D₃₇	C₂²	1	8	yes	—
36	C₂×#10596938	C₂²	1	1	yes	characteristic
37	C₂×#10596938	C₂²	1	1	yes	characteristic
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	#125×C₃₇	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
47	C₂³×D₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₃₇	C₂³	1	2	yes	—
56	C₂²×Dic₃₇	C₂³	1	2	yes	—
57	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₃₇	—	4	4	no	—
59	C₂×C₄×D₃₇	Q₈	1	8	yes	complemented
60	C₂×C₄×D₃₇	C₂³	1	8	yes	—
61	C₂×C₄×D₃₇	C₂³	1	4	yes	—
62	C₂×Dic₇₄	—	4	4	no	—
63	C₂×Dic₇₄	—	4	4	no	—
64	C₂×Dic₇₄	C₂³	1	8	yes	—
65	C₂×Dic₇₄	C₂³	1	8	yes	—
66	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
67	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
68	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
69	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
70	C₄²×C₃₇	C₂³	1	4	yes	—
71	#10596938	C₂³	1	4	yes	—
72	#10596938	C₂³	1	4	yes	—
73	#10596938	C₂³	1	8	yes	—
74	C₄⋊Dic₃₇	C₂³	1	4	yes	—
75	C₄⋊Dic₃₇	C₂³	1	4	yes	—
76	C₄⋊Dic₃₇	C₂³	1	8	yes	—
77	C₄⋊Dic₃₇	C₂³	1	8	yes	—
78	#10596936	C₂³	1	8	yes	—
79	#10596936	C₂³	1	4	yes	—
80	#10596936	C₂³	1	4	yes	—
81	#10596936	C₂³	1	8	yes	—
82	#10596936	C₂³	1	8	yes	—
83	C₄×Dic₃₇	C₂³	1	4	yes	—
84	C₄×Dic₃₇	C₂³	1	8	yes	—
85	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
86	C₂²×D₃₇	—	2	2	no	—
87	C₂²×D₃₇	—	2	2	no	—
88	C₂²×D₃₇	C₂×Q₈	1	4	yes	—
89	C₂²×D₃₇	C₂⁴	1	2	yes	—
90	C₂×C₄×C₃₇	—	2	2	no	—
91	C₂×C₄×C₃₇	—	2	4	no	—
92	C₂×C₄×C₃₇	—	2	4	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
95	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
96	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
97	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
98	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
99	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
100	C₂×Dic₃₇	—	2	2	no	—
101	C₂×Dic₃₇	—	2	4	no	—
102	C₂×Dic₃₇	—	2	4	no	—
103	C₂×Dic₃₇	—	2	2	no	—
104	C₂×Dic₃₇	C₂×Q₈	1	4	yes	—
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	—	4	8	no	—
111	C₄×D₃₇	C₂×Q₈	1	16	yes	complemented
112	Dic₇₄	—	4	8	no	—
113	Dic₇₄	—	4	8	no	—
114	C₂²×C₃₇	C₂²×Q₈	1	2	yes	—
115	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
117	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
118	C₂×D₃₇	—	2	8	no	—
119	C₂×D₃₇	—	2	2	no	—
120	C₂×D₃₇	—	2	2	no	—
121	C₂×D₃₇	C₂²×Q₈	1	4	yes	—
122	C₄×C₃₇	—	2	2	no	—
123	C₄×C₃₇	—	2	4	no	—
124	C₄×C₃₇	—	2	4	no	—
125	C₄×C₃₇	—	2	2	no	—
126	C₄×C₃₇	C₂²×Q₈	1	4	yes	—
127	Dic₃₇	—	2	2	no	—
128	Dic₃₇	—	2	4	no	—
129	Dic₃₇	—	2	4	no	—
130	Dic₃₇	—	2	2	no	—
131	Dic₃₇	C₂²×Q₈	1	4	yes	—
132	C₂×#544	—	37	1	no	maximal, Sylow
133	C₂×C₃₇	#544	1	4	yes	complemented
134	C₂×C₃₇	C₂³×Q₈	1	1	yes	characteristic
135	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
137	D₃₇	—	2	4	no	—
138	#544	—	37	16	no	—
139	C₂×C₄⋊Q₈	—	37	2	no	—
140	C₂×C₄⋊Q₈	—	37	1	no	—
141	C₂×C₄⋊Q₈	—	37	1	no	—
142	C₂×#125	—	37	2	no	—
143	C₂×#125	—	37	1	no	—
144	C₂×#125	D₃₇	1	1	yes	characteristic, complemented
145	C₂×C₂²⋊Q₈	—	37	2	no	—
146	C₂×C₂²⋊Q₈	—	37	2	no	—
147	C₂×#117	—	37	1	no	—
148	C₂×#117	—	37	1	no	—
149	C₂²×C₄⋊C₄	—	37	1	no	—
150	C₃₇	C₂×#544	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	37	2	no	—
152	C₂²×Q₈	—	37	2	no	—
153	C₂³×C₄	—	37	1	no	—
154	C₂³×C₄	—	37	2	no	—
155	C₄⋊Q₈	—	37	16	no	—
156	C₄⋊Q₈	—	37	8	no	—
157	C₄⋊Q₈	—	37	8	no	—
158	#125	—	37	16	no	—
159	#125	—	37	8	no	—
160	#125	C₂×D₃₇	1	8	yes	complemented
161	C₂²⋊Q₈	—	37	16	no	—
162	C₂²⋊Q₈	—	37	16	no	—
163	#117	—	37	8	no	—
164	#117	—	37	8	no	—
165	C₂×C₄⋊C₄	—	37	2	no	—
166	C₂×C₄⋊C₄	—	37	1	no	—
167	C₂×C₄⋊C₄	—	37	1	no	—
168	C₂×C₄⋊C₄	—	37	1	no	—
169	C₂×C₄⋊C₄	—	37	1	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	—	37	2	no	—
174	C₂×C₄⋊C₄	—	37	8	no	—
175	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
179	C₂×C₂²⋊C₄	—	37	1	no	—
180	C₂×C₂²⋊C₄	—	37	2	no	—
181	C₂×C₂²⋊C₄	—	37	1	no	—
182	C₂×C₄²	—	37	2	no	—
183	C₂×C₄²	—	37	1	no	—
184	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
185	C₂⁴	—	37	1	no	—
186	C₂×Q₈	—	37	8	no	—
187	C₂×Q₈	—	37	8	no	—
188	C₂×Q₈	—	148	4	no	—
189	C₂×Q₈	—	148	4	no	—
190	C₂²×C₄	—	37	1	no	—
191	C₂²×C₄	—	37	8	no	—
192	C₂²×C₄	—	37	4	no	—
193	C₂²×C₄	—	37	1	no	—
194	C₂²×C₄	—	37	1	no	—
195	C₂²×C₄	—	37	2	no	—
196	C₂²×C₄	—	37	2	no	—
197	C₂²×C₄	—	37	8	no	—
198	C₂²×C₄	—	148	4	no	—
199	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
200	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
203	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
204	C₄⋊C₄	—	37	8	no	—
205	C₄⋊C₄	—	37	4	no	—
206	C₄⋊C₄	—	37	4	no	—
207	C₄⋊C₄	—	37	8	no	—
208	C₄⋊C₄	—	37	8	no	—
209	C₄⋊C₄	—	37	4	no	—
210	C₄⋊C₄	—	37	4	no	—
211	C₄⋊C₄	—	37	8	no	—
212	C₄⋊C₄	—	37	8	no	—
213	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
214	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
215	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
216	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
217	C₂²⋊C₄	—	37	4	no	—
218	C₂²⋊C₄	—	37	8	no	—
219	C₂²⋊C₄	—	37	4	no	—
220	C₄²	—	37	8	no	—
221	C₄²	—	37	4	no	—
222	C₄²	C₂²×D₃₇	1	4	yes	—
223	C₂³	—	37	4	no	—
224	C₂³	—	37	2	no	—
225	C₂³	—	74	2	no	—
226	C₂³	—	74	2	no	—
227	C₂³	C₂³×D₃₇	1	1	yes	characteristic
228	Q₈	—	148	8	no	—
229	Q₈	—	148	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	37	4	no	—
235	C₂×C₄	—	37	2	no	—
236	C₂×C₄	—	37	16	no	—
237	C₂×C₄	—	37	2	no	—
238	C₂×C₄	—	37	2	no	—
239	C₂×C₄	—	37	4	no	—
240	C₂×C₄	—	37	4	no	—
241	C₂×C₄	—	74	4	no	—
242	C₂×C₄	—	74	2	no	—
243	C₂×C₄	—	74	4	no	—
244	C₂×C₄	—	74	2	no	—
245	C₂×C₄	—	148	8	no	—
246	C₂×C₄	Q₈×D₃₇	1	4	yes	—
247	C₂×C₄	C₂³×D₃₇	1	4	yes	—
248	C₂×C₄	C₂³×D₃₇	1	2	yes	—
249	C₂×C₄	C₂³×D₃₇	1	2	yes	—
250	C₂×C₄	C₂³×D₃₇	1	4	yes	—
251	C₂×C₄	C₂³×D₃₇	1	2	yes	—
252	C₂²	—	37	4	no	—
253	C₂²	—	74	2	no	—
254	C₂²	—	74	2	no	—
255	C₂²	—	74	8	no	—
256	C₂²	C₂×Q₈×D₃₇	1	2	yes	—
257	C₂²	#11924582	1	2	yes	—
258	C₂²	#11924583	1	2	yes	—
259	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	37	4	no	—
265	C₄	—	74	4	no	—
266	C₄	—	74	2	no	—
267	C₄	—	74	4	no	—
268	C₄	—	74	2	no	—
269	C₄	C₂×Q₈×D₃₇	1	4	yes	—
270	C₂	—	74	4	no	—
271	C₂	#14558163	1	4	yes	direct product factor
272	C₂	C₂²×Q₈×D₃₇	1	1	yes	characteristic
273	C₂	C₂×#11924582	1	1	yes	characteristic
274	C₂	C₂×#11924583	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#125×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂³×D₃₇	C₂³
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	C₂×#544
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #14558163	C₂	direct (N × H)
2. C₂×#11924536	C₂	semidirect
3. C₂×#11924535	C₂	semidirect
4. C₂×#11924535	C₂	semidirect
5. C₂×#11924527	C₂	semidirect
6. C₂×#11924526	C₂	semidirect
12. C₂×#125×C₃₇	C₂	semidirect
17. #11924536	C₂²	semidirect
18. #11924535	C₂²	semidirect
19. #11924535	C₂²	semidirect
20. #11924527	C₂²	semidirect
21. #11924526	C₂²	semidirect
41. #125×C₃₇	C₂²	semidirect
59. C₂×C₄×D₃₇	Q₈	semidirect
111. C₄×D₃₇	C₂×Q₈	semidirect
133. C₂×C₃₇	#544	semidirect
144. C₂×#125	D₃₇	semidirect
150. C₃₇	C₂×#544	semidirect
160. #125	C₂×D₃₇	semidirect
271. C₂	#14558163	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558163	2
C₂×#11924536	2
C₂×#11924535	2
C₂×#11924527	2
C₂×#11924526	2
C₂×#11924516	2
C₂×#11924514	2
C₂×#11924494	2
C₂×#11924493	2
C₂×C₄⋊C₄×D₃₇	2
C₂×#125×C₃₇	2
C₂×#544	37

Subgroups — C2×#14558163 #18007996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558163 #18007996