Subgroups — C₂×#14558211 #18008095

Order 4736.

1328 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	#14558211	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924529	C₂	1	2	yes	maximal, complemented
3	C₂×#11924523	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11924518	C₂	1	2	yes	maximal, complemented
5	C₂×#11924514	C₂	1	2	yes	maximal, complemented
6	C₂×#11924504	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₂²⋊D₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×#11924494	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×D₃₇	C₂	1	2	yes	maximal, complemented
10	C₂×#123×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₃₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₂²⋊D₃₇	C₂²	1	2	yes	—
14	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
16	#11924529	C₂²	1	16	yes	complemented
17	#11924523	C₂²	1	8	yes	complemented
18	#11924518	C₂²	1	16	yes	complemented
19	#11924514	C₂²	1	16	yes	complemented
20	#11924504	C₂²	1	8	yes	—
21	C₄×C₂²⋊D₃₇	C₂²	1	16	yes	complemented
22	#11924494	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	complemented
24	C₂×#10596936	C₂²	1	2	yes	complemented
25	C₂×#10596936	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊Dic₃₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₃₇	C₂²	1	2	yes	complemented
29	C₂×#10596938	C₂²	1	2	yes	—
30	C₂×#10596938	C₂²	1	2	yes	—
31	C₂×#10596938	C₂²	1	2	yes	—
32	C₂²⋊C₄×D₃₇	C₂²	1	16	yes	complemented
33	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×Dic₃₇	C₂²	1	2	yes	complemented
35	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic, complemented
37	#123×C₃₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
42	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
43	C₂³×D₃₇	—	2	8	no	—
44	C₂³×D₃₇	—	4	2	no	—
45	C₂³×D₃₇	C₂³	1	4	yes	—
46	C₂³×D₃₇	C₂³	1	2	yes	—
47	C₂×D₄×C₃₇	—	4	2	no	—
48	C₂×D₄×C₃₇	C₂³	1	4	yes	—
49	C₂²×C₄×C₃₇	—	4	2	no	—
50	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₇	C₂³	1	4	yes	complemented
54	C₂×C₂²⋊D₃₇	—	2	8	no	—
55	C₂×C₂²⋊D₃₇	C₂³	1	8	yes	—
56	C₂²×Dic₃₇	C₂³	1	2	yes	—
57	C₂²×Dic₃₇	C₂³	1	2	yes	—
58	C₂²×Dic₃₇	C₂³	1	2	yes	—
59	C₂×C₄×D₃₇	—	2	8	no	—
60	C₂×C₄×D₃₇	—	4	4	no	—
61	C₂×C₄×D₃₇	C₂³	1	8	yes	—
62	C₂×C₄×D₃₇	C₂³	1	8	yes	—
63	C₂×Dic₇₄	—	4	2	no	—
64	C₂×Dic₇₄	C₂³	1	4	yes	complemented
65	C₄⋊C₄×C₃₇	C₂³	1	8	yes	complemented
66	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	—
67	C₂²⋊C₄×C₃₇	C₂³	1	4	yes	complemented
68	C₂²⋊Dic₃₇	C₂³	1	8	yes	—
69	C₂²⋊Dic₃₇	C₂³	1	4	yes	—
70	C₂²⋊Dic₃₇	C₂³	1	8	yes	complemented
71	#10596938	C₂³	1	8	yes	—
72	#10596938	C₂³	1	8	yes	—
73	#10596938	C₂³	1	8	yes	—
74	C₄⋊Dic₃₇	C₂³	1	8	yes	complemented
75	#10596936	C₂³	1	8	yes	complemented
76	#10596936	C₂³	1	8	yes	complemented
77	C₄×Dic₃₇	C₂³	1	4	yes	complemented
78	C₄×Dic₃₇	C₂³	1	8	yes	complemented
79	C₂³×C₃₇	—	2	4	no	—
80	C₂³×C₃₇	—	2	4	no	—
81	C₂³×C₃₇	—	2	2	no	—
82	C₂³×C₃₇	—	2	2	no	—
83	C₂³×C₃₇	C₂⁴	1	2	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	—	2	16	no	—
89	C₂²×D₃₇	—	4	4	no	—
90	C₂²×D₃₇	—	4	16	no	—
91	C₂²×D₃₇	#41	1	8	yes	—
92	C₂²×D₃₇	C₂⁴	1	4	yes	—
93	D₄×C₃₇	—	4	4	no	—
94	C₂×C₄×C₃₇	—	2	4	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	2	4	no	—
97	C₂×C₄×C₃₇	—	4	4	no	—
98	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
99	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
100	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
101	C₂²⋊D₃₇	—	2	16	no	—
102	C₂×Dic₃₇	—	2	4	no	—
103	C₂×Dic₃₇	—	2	4	no	—
104	C₂×Dic₃₇	—	2	4	no	—
105	C₂×Dic₃₇	C₂⁴	1	4	yes	—
106	C₂×Dic₃₇	C₂⁴	1	4	yes	—
107	C₂×Dic₃₇	C₂⁴	1	4	yes	—
108	C₄×D₃₇	—	2	16	no	—
109	C₄×D₃₇	—	4	8	no	—
110	Dic₇₄	—	4	4	no	—
111	C₂²×C₃₇	—	2	4	no	—
112	C₂²×C₃₇	—	2	4	no	—
113	C₂²×C₃₇	—	2	2	no	—
114	C₂²×C₃₇	—	2	2	no	—
115	C₂²×C₃₇	—	4	4	no	—
116	C₂²×C₃₇	—	4	4	no	—
117	C₂²×C₃₇	C₂×#41	1	4	yes	—
118	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
120	C₂×D₃₇	—	2	16	no	—
121	C₂×D₃₇	—	2	4	no	—
122	C₂×D₃₇	—	2	4	no	—
123	C₂×D₃₇	—	4	16	no	—
124	C₂×D₃₇	C₂×#41	1	8	yes	—
125	C₄×C₃₇	—	2	4	no	—
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	—	2	4	no	—
128	Dic₃₇	—	2	4	no	—
129	Dic₃₇	—	2	4	no	—
130	Dic₃₇	—	2	4	no	—
131	C₂×#551	—	37	1	no	maximal, Sylow
132	C₂×C₃₇	—	4	2	no	—
133	C₂×C₃₇	—	4	2	no	—
134	C₂×C₃₇	#551	1	4	yes	complemented
135	C₂×C₃₇	C₂²×#41	1	2	yes	—
136	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
137	D₃₇	—	2	8	no	—
138	#551	—	37	16	no	—
139	C₂×#126	—	37	2	no	—
140	C₂×#124	—	37	1	no	—
141	C₂×#123	—	37	2	no	—
142	C₂×#123	D₃₇	1	1	yes	characteristic, complemented
143	C₂×C₂²⋊Q₈	—	37	2	no	—
144	C₂×#120	—	37	1	no	—
145	C₂×C₄×D₄	—	37	2	no	—
146	C₂×#117	—	37	2	no	—
147	C₂²×C₂²⋊C₄	—	37	2	no	—
148	C₃₇	C₂×#551	1	1	yes	characteristic, complemented, Sylow
149	C₂⁵	—	37	1	no	—
150	C₂²×Q₈	—	37	1	no	—
151	C₂²×D₄	—	37	2	no	—
152	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
153	C₂³×C₄	—	37	2	no	—
154	C₂³×C₄	—	37	2	no	—
155	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic, complemented
156	#126	—	37	16	no	—
157	#124	—	37	8	no	—
158	#123	—	37	16	no	—
159	#123	C₂×D₃₇	1	8	yes	complemented
160	C₂²⋊Q₈	—	37	16	no	—
161	#120	—	37	8	no	—
162	C₄×D₄	—	37	16	no	—
163	#117	—	37	16	no	—
164	C₂×C₄⋊C₄	—	37	2	no	—
165	C₂×C₄⋊C₄	—	37	2	no	—
166	C₂×C₄⋊C₄	—	37	2	no	—
167	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	complemented
168	C₂×C₂²⋊C₄	—	37	2	no	—
169	C₂×C₂²⋊C₄	—	37	2	no	—
170	C₂×C₂²⋊C₄	—	37	1	no	—
171	C₂×C₂²⋊C₄	—	37	2	no	—
172	C₂×C₂²⋊C₄	—	37	2	no	—
173	C₂×C₂²⋊C₄	—	37	2	no	—
174	C₂×C₂²⋊C₄	—	37	16	no	—
175	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×D₃₇	1	1	yes	characteristic, complemented
177	C₂×C₄²	—	37	1	no	—
178	C₂×C₄²	—	37	2	no	—
179	C₂⁴	—	37	2	no	—
180	C₂⁴	—	37	4	no	—
181	C₂⁴	—	74	8	no	—
182	C₂⁴	—	148	2	no	—
183	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
184	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
185	C₂×Q₈	—	37	4	no	—
186	C₂×Q₈	—	148	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	37	8	no	—
189	C₂×D₄	—	74	8	no	—
190	C₂×D₄	C₂²×D₃₇	1	4	yes	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	37	2	no	—
193	C₂²×C₄	—	37	2	no	—
194	C₂²×C₄	—	37	2	no	—
195	C₂²×C₄	—	37	8	no	—
196	C₂²×C₄	—	37	8	no	—
197	C₂²×C₄	—	74	8	no	—
198	C₂²×C₄	—	148	4	no	—
199	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
201	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₃₇	1	4	yes	complemented
203	C₄⋊C₄	—	37	8	no	—
204	C₄⋊C₄	—	37	8	no	—
205	C₄⋊C₄	—	37	8	no	—
206	C₄⋊C₄	C₂²×D₃₇	1	8	yes	complemented
207	C₂²⋊C₄	—	37	8	no	—
208	C₂²⋊C₄	—	37	4	no	—
209	C₂²⋊C₄	—	37	8	no	—
210	C₂²⋊C₄	—	37	8	no	—
211	C₂²⋊C₄	—	37	8	no	—
212	C₂²⋊C₄	—	37	8	no	—
213	C₂²⋊C₄	C₂²×D₃₇	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₃₇	1	4	yes	complemented
215	C₄²	—	37	4	no	—
216	C₄²	—	37	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	37	8	no	—
222	C₂³	—	37	4	no	—
223	C₂³	—	74	16	no	—
224	C₂³	—	74	4	no	—
225	C₂³	—	74	4	no	—
226	C₂³	—	148	4	no	—
227	C₂³	—	148	16	no	—
228	C₂³	C₂³×D₃₇	1	2	yes	—
229	C₂³	C₂³×D₃₇	1	2	yes	—
230	C₂³	C₂³×D₃₇	1	1	yes	characteristic
231	Q₈	—	148	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	74	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	37	4	no	—
239	C₂×C₄	—	37	4	no	—
240	C₂×C₄	—	37	4	no	—
241	C₂×C₄	—	74	16	no	—
242	C₂×C₄	—	74	4	no	—
243	C₂×C₄	—	74	4	no	—
244	C₂×C₄	—	74	4	no	—
245	C₂×C₄	—	148	8	no	—
246	C₂×C₄	C₂³×D₃₇	1	2	yes	—
247	C₂×C₄	C₂³×D₃₇	1	4	yes	—
248	C₂×C₄	C₂³×D₃₇	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	37	8	no	—
256	C₂²	—	74	16	no	—
257	C₂²	—	74	4	no	—
258	C₂²	—	74	4	no	—
259	C₂²	—	148	16	no	—
260	C₂²	#41×D₃₇	1	4	yes	—
261	C₂²	#11924581	1	2	yes	—
262	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	74	4	no	—
267	C₄	—	74	4	no	—
268	C₄	—	74	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	74	8	no	—
272	C₂	#14558211	1	4	yes	direct product factor
273	C₂	C₂×#41×D₃₇	1	2	yes	—
274	C₂	C₂×#11924581	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#123×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂×#11924504	C₂
hypercenter	C₂×#123	D₃₇
hypocenter	C₃₇	C₂×#551
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
1. #14558211	C₂	direct (N × H)
2. C₂×#11924529	C₂	semidirect
3. C₂×#11924523	C₂	semidirect
4. C₂×#11924518	C₂	semidirect
5. C₂×#11924514	C₂	semidirect
7. C₂×C₄×C₂²⋊D₃₇	C₂	semidirect
8. C₂×#11924494	C₂	semidirect
9. C₂×C₂²⋊C₄×D₃₇	C₂	semidirect
10. C₂×#123×C₃₇	C₂	semidirect
12. C₂²×Dic₇₄	C₂²	semidirect
16. #11924529	C₂²	semidirect
17. #11924523	C₂²	semidirect
18. #11924518	C₂²	semidirect
19. #11924514	C₂²	semidirect
21. C₄×C₂²⋊D₃₇	C₂²	semidirect
22. #11924494	C₂²	semidirect
23. C₂×C₄⋊Dic₃₇	C₂²	semidirect
24. C₂×#10596936	C₂²	semidirect
25. C₂×#10596936	C₂²	semidirect
28. C₂×C₂²⋊Dic₃₇	C₂²	semidirect
32. C₂²⋊C₄×D₃₇	C₂²	semidirect
33. C₂×C₄×Dic₃₇	C₂²	semidirect
34. C₂×C₄×Dic₃₇	C₂²	semidirect
36. C₂³×C₄×C₃₇	C₂²	semidirect
37. #123×C₃₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₃₇	C₂²	semidirect
53. C₂²×C₄×C₃₇	C₂³	semidirect
64. C₂×Dic₇₄	C₂³	semidirect
65. C₄⋊C₄×C₃₇	C₂³	semidirect
67. C₂²⋊C₄×C₃₇	C₂³	semidirect
70. C₂²⋊Dic₃₇	C₂³	semidirect
74. C₄⋊Dic₃₇	C₂³	semidirect
75. #10596936	C₂³	semidirect
76. #10596936	C₂³	semidirect
77. C₄×Dic₃₇	C₂³	semidirect
78. C₄×Dic₃₇	C₂³	semidirect
134. C₂×C₃₇	#551	semidirect
142. C₂×#123	D₃₇	semidirect
148. C₃₇	C₂×#551	semidirect
155. C₂³×C₄	C₂×D₃₇	semidirect
159. #123	C₂×D₃₇	semidirect
167. C₂×C₄⋊C₄	C₂×D₃₇	semidirect
176. C₂×C₂²⋊C₄	C₂×D₃₇	semidirect
202. C₂²×C₄	C₂²×D₃₇	semidirect
206. C₄⋊C₄	C₂²×D₃₇	semidirect
214. C₂²⋊C₄	C₂²×D₃₇	semidirect
272. C₂	#14558211	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558211	2
C₂×#11924529	2
C₂×#11924523	2
C₂×#11924518	2
C₂×#11924514	2
C₂×#11924504	2
C₂×C₄×C₂²⋊D₃₇	2
C₂×#11924494	2
C₂×C₂²⋊C₄×D₃₇	2
C₂×#123×C₃₇	2
C₂×#551	37

Subgroups — C2×#14558211 #18008095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558211 #18008095