Subgroups — #18008218

Order 4736.

1094 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558262	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558253	C₂	1	3	yes	maximal, complemented
3	#14558222	C₂	1	6	yes	maximal, complemented
4	Q₈×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558190	C₂	1	2	yes	maximal, complemented
6	#14558188	C₂	1	3	yes	maximal, complemented
7	#14558090	C₂	1	3	yes	maximal, complemented
8	#14558065	C₂	1	6	yes	maximal, complemented
9	C₄×#10596962	C₂	1	3	yes	maximal, complemented
10	C₄×Q₈×D₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	3	yes	—
13	C₂×Q₈×D₃₇	C₂²	1	2	yes	—
14	#11924536	C₂²	1	3	yes	—
15	#11924536	C₂²	1	3	yes	—
16	#11924529	C₂²	1	12	yes	—
17	#11924528	C₂²	1	6	yes	—
18	#11924527	C₂²	1	6	yes	—
19	#11924523	C₂²	1	6	yes	—
20	#11924518	C₂²	1	12	yes	—
21	#11924516	C₂²	1	6	yes	—
22	#11924515	C₂²	1	6	yes	—
23	#11924515	C₂²	1	6	yes	—
24	#11924507	C₂²	1	6	yes	—
25	Q₈×Dic₃₇	C₂²	1	2	yes	—
26	C₄×Dic₇₄	C₂²	1	6	yes	—
27	C₄×Dic₇₄	C₂²	1	3	yes	—
28	C₄×Dic₇₄	C₂²	1	3	yes	—
29	#11924501	C₂²	1	6	yes	—
30	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
31	C₄×D₁₄₈	C₂²	1	3	yes	—
32	C₄×C₂²⋊D₃₇	C₂²	1	6	yes	—
33	C₄×C₂²⋊D₃₇	C₂²	1	6	yes	—
34	#11924495	C₂²	1	3	yes	—
35	#11924495	C₂²	1	6	yes	—
36	#11924493	C₂²	1	6	yes	—
37	C₄⋊C₄×D₃₇	C₂²	1	6	yes	—
38	C₄²×D₃₇	C₂²	1	6	yes	—
39	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₃₇	C₂²	1	3	yes	—
43	C₄×D₄×C₃₇	C₂²	1	3	yes	—
44	#117×C₃₇	C₂²	1	3	yes	—
45	C₂×C₄²×C₃₇	C₂²	1	3	yes	—
46	#41×C₃₇	—	2	4	no	—
47	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₃₇	C₂³	1	3	yes	—
49	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
50	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
51	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₃₇	C₂³	1	6	yes	—
53	Q₈×D₃₇	—	2	8	no	—
54	#10596962	—	2	12	no	—
55	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₃₇	C₂³	1	2	yes	—
57	C₂×C₄×D₃₇	C₂³	1	6	yes	—
58	C₂×C₄×D₃₇	C₂³	1	6	yes	—
59	C₂×Dic₇₄	C₂³	1	6	yes	—
60	C₂×Dic₇₄	C₂³	1	3	yes	—
61	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
62	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
63	C₂²⋊C₄×C₃₇	C₂³	1	6	yes	—
64	C₄²×C₃₇	C₂³	1	3	yes	—
65	C₄²×C₃₇	C₂³	1	3	yes	—
66	C₄²×C₃₇	C₂³	1	1	yes	characteristic
67	C₄²×C₃₇	C₂³	1	3	yes	—
68	C₂²⋊Dic₃₇	C₂³	1	6	yes	—
69	C₂²⋊Dic₃₇	C₂³	1	12	yes	—
70	#10596938	C₂³	1	2	yes	—
71	#10596938	C₂³	1	6	yes	—
72	C₄⋊Dic₃₇	C₂³	1	6	yes	—
73	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₃₇	C₂³	1	3	yes	—
75	C₄⋊Dic₃₇	C₂³	1	6	yes	—
76	C₄⋊Dic₃₇	C₂³	1	6	yes	—
77	#10596936	C₂³	1	12	yes	—
78	#10596936	C₂³	1	6	yes	—
79	#10596936	C₂³	1	6	yes	—
80	C₄×Dic₃₇	C₂³	1	6	yes	—
81	C₄×Dic₃₇	C₂³	1	6	yes	—
82	C₄×Dic₃₇	C₂³	1	6	yes	—
83	C₂³×C₃₇	C₂⁴	1	3	yes	—
84	C₂²×D₃₇	C₂⁴	1	2	yes	—
85	Q₈×C₃₇	#41	1	4	yes	complemented
86	D₄×C₃₇	—	2	6	no	—
87	C₂×C₄×C₃₇	—	2	6	no	—
88	C₂×C₄×C₃₇	—	2	6	no	—
89	C₂×C₄×C₃₇	—	2	6	no	—
90	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
91	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
93	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
94	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₃₇	—	2	12	no	—
97	C₂×Dic₃₇	C₂⁴	1	2	yes	—
98	C₂×Dic₃₇	C₂⁴	1	6	yes	—
99	C₂×Dic₃₇	C₂⁴	1	6	yes	—
100	D₁₄₈	#41	1	4	yes	complemented
101	C₄×D₃₇	—	2	12	no	—
102	C₄×D₃₇	—	2	12	no	—
103	C₄×D₃₇	—	2	4	no	—
104	Dic₇₄	—	2	6	no	—
105	Dic₇₄	—	2	12	no	—
106	C₂²×C₃₇	—	2	3	no	—
107	C₂²×C₃₇	—	2	3	no	—
108	C₂²×C₃₇	—	2	3	no	—
109	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
110	C₂×D₃₇	—	2	2	no	—
111	C₂×D₃₇	—	2	2	no	—
112	C₂×D₃₇	C₂×#41	1	4	yes	—
113	C₄×C₃₇	—	2	3	no	—
114	C₄×C₃₇	—	2	1	no	—
115	C₄×C₃₇	—	2	3	no	—
116	C₄×C₃₇	C₂×#41	1	2	yes	—
117	C₄×C₃₇	C₂×#41	1	6	yes	—
118	C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
119	Dic₃₇	—	2	6	no	—
120	Dic₃₇	—	2	6	no	—
121	Dic₃₇	—	2	2	no	—
122	#3482	—	37	1	no	maximal, Sylow
123	C₂×C₃₇	—	4	3	no	—
124	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
125	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
126	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
127	D₃₇	—	2	4	no	—
128	#557	—	37	1	no	—
129	#556	—	37	3	no	—
130	#552	—	37	6	no	—
131	D₄×Q₈	—	37	1	no	—
132	#548	—	37	2	no	—
133	#547	—	37	3	no	—
134	#533	—	37	3	no	—
135	#529	—	37	6	no	—
136	C₄×#41	—	37	3	no	—
137	C₄×#41	D₃₇	1	1	yes	characteristic, complemented
138	C₂×C₄×Q₈	—	37	2	no	—
139	C₃₇	#3482	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	37	3	no	—
141	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
142	C₂²×Q₈	—	37	2	no	—
143	C₄⋊Q₈	—	37	3	no	—
144	C₄⋊Q₈	—	37	3	no	—
145	#126	—	37	12	no	—
146	#125	—	37	6	no	—
147	#125	—	37	6	no	—
148	#124	—	37	6	no	—
149	#123	—	37	12	no	—
150	C₂²⋊Q₈	—	37	6	no	—
151	C₂²⋊Q₈	—	37	6	no	—
152	C₂²⋊Q₈	—	37	6	no	—
153	#121	—	37	6	no	—
154	C₄×Q₈	—	37	2	no	—
155	C₄×Q₈	—	37	6	no	—
156	C₄×Q₈	—	37	6	no	—
157	C₄×Q₈	—	37	3	no	—
158	C₄×Q₈	—	37	3	no	—
159	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
161	C₄×D₄	—	37	6	no	—
162	C₄×D₄	—	37	6	no	—
163	C₄×D₄	—	37	1	no	—
164	C₄×D₄	—	37	3	no	—
165	C₄×D₄	C₂×D₃₇	1	3	yes	—
166	C₄×D₄	C₂×D₃₇	1	3	yes	—
167	#117	—	37	3	no	—
168	#117	—	37	6	no	—
169	#117	—	37	6	no	—
170	#117	C₂×D₃₇	1	3	yes	—
171	C₂×C₄⋊C₄	—	37	6	no	—
172	C₂×C₄²	—	37	6	no	—
173	C₂×C₄²	C₂×D₃₇	1	3	yes	—
174	#41	—	2	4	no	—
175	#41	—	74	12	no	—
176	C₂×Q₈	—	37	6	no	—
177	C₂×Q₈	—	37	3	no	—
178	C₂×Q₈	—	74	8	no	—
179	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
180	C₂×D₄	—	37	6	no	—
181	C₂×D₄	—	37	1	no	—
182	C₂×D₄	C₂²×D₃₇	1	3	yes	—
183	C₂²×C₄	—	37	6	no	—
184	C₂²×C₄	—	37	6	no	—
185	C₂²×C₄	—	37	2	no	—
186	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
187	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
188	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
189	C₄⋊C₄	—	37	12	no	—
190	C₄⋊C₄	—	37	6	no	—
191	C₄⋊C₄	—	37	6	no	—
192	C₄⋊C₄	—	37	6	no	—
193	C₄⋊C₄	—	37	1	no	—
194	C₄⋊C₄	—	37	3	no	—
195	C₄⋊C₄	—	37	6	no	—
196	C₄⋊C₄	—	37	6	no	—
197	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
198	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
199	C₂²⋊C₄	—	37	6	no	—
200	C₂²⋊C₄	—	37	12	no	—
201	C₂²⋊C₄	—	37	2	no	—
202	C₂²⋊C₄	—	37	6	no	—
203	C₂²⋊C₄	C₂²×D₃₇	1	6	yes	—
204	C₄²	—	37	6	no	—
205	C₄²	—	37	6	no	—
206	C₄²	—	37	6	no	—
207	C₄²	C₂²×D₃₇	1	3	yes	—
208	C₄²	C₂²×D₃₇	1	3	yes	—
209	C₄²	C₂²×D₃₇	1	3	yes	—
210	C₄²	C₂²×D₃₇	1	1	yes	characteristic
211	C₂³	—	37	2	no	—
212	C₂³	C₂³×D₃₇	1	3	yes	—
213	Q₈	—	74	6	no	—
214	Q₈	—	74	12	no	—
215	Q₈	#10596962	1	4	yes	complemented
216	D₄	—	2	6	no	—
217	D₄	—	37	4	no	—
218	D₄	—	74	12	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	6	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	37	2	no	—
223	C₂×C₄	—	37	6	no	—
224	C₂×C₄	—	37	6	no	—
225	C₂×C₄	—	74	12	no	—
226	C₂×C₄	—	74	12	no	—
227	C₂×C₄	—	74	4	no	—
228	C₂×C₄	C₂³×D₃₇	1	3	yes	—
229	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
230	C₂×C₄	C₂³×D₃₇	1	3	yes	—
231	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
232	C₂×C₄	C₂³×D₃₇	1	3	yes	—
233	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
234	C₂²	—	2	3	no	—
235	C₂²	—	2	3	no	—
236	C₂²	—	2	3	no	—
237	C₂²	—	37	4	no	—
238	C₂²	—	74	2	no	—
239	C₂²	—	74	2	no	—
240	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
241	C₄	—	2	3	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	3	no	—
244	C₄	—	74	6	no	—
245	C₄	—	74	6	no	—
246	C₄	—	74	2	no	—
247	C₄	#41×D₃₇	1	2	yes	—
248	C₄	C₂×#10596962	1	6	yes	—
249	C₄	#11924583	1	2	yes	—
250	C₂	—	4	3	no	—
251	C₂	—	74	4	no	—
252	C₂	C₂×#41×D₃₇	1	1	yes	characteristic
253	C₂	C₂²×#10596962	1	1	yes	characteristic
254	C₂	C₂×#11924583	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₄×#41×C₃₇	C₂
hypercenter	C₄×#41	D₃₇
hypocenter	C₃₇	#3482
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14558262	C₂	semidirect
2. #14558253	C₂	semidirect
3. #14558222	C₂	semidirect
4. Q₈×D₁₄₈	C₂	semidirect
5. #14558190	C₂	semidirect
6. #14558188	C₂	semidirect
7. #14558090	C₂	semidirect
8. #14558065	C₂	semidirect
9. C₄×#10596962	C₂	semidirect
10. C₄×Q₈×D₃₇	C₂	semidirect
11. C₄×#41×C₃₇	C₂	semidirect
85. Q₈×C₃₇	#41	semidirect
100. D₁₄₈	#41	semidirect
137. C₄×#41	D₃₇	semidirect
139. C₃₇	#3482	semidirect
215. Q₈	#10596962	semidirect

Maximal subgroups

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

subgroup	index
#14558262	2
#14558253	2
#14558222	2
Q₈×D₁₄₈	2
#14558190	2
#14558188	2
#14558090	2
#14558065	2
C₄×#10596962	2
C₄×Q₈×D₃₇	2
C₄×#41×C₃₇	2
#3482	37