Subgroups — #18008692

Order 4736.

992 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#14558261	C₂	1	3	yes	maximal, complemented
3	#14558256	C₂	1	3	yes	maximal, complemented
4	#14558218	C₂	1	6	yes	maximal, complemented
5	Q₈×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₂²⋊D₃₇	C₂	1	2	yes	maximal, complemented
7	#14558141	C₂	1	6	yes	maximal, complemented
8	#14558090	C₂	1	3	yes	maximal, complemented
9	#14558067	C₂	1	3	yes	maximal
10	#14558009	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₃₇	C₂²	1	2	yes	—
13	#11924536	C₂²	1	3	yes	—
14	#11924536	C₂²	1	3	yes	—
15	#11924534	C₂²	1	6	yes	—
16	#11924531	C₂²	1	12	yes	—
17	#11924530	C₂²	1	6	yes	—
18	#11924528	C₂²	1	3	yes	—
19	#11924528	C₂²	1	6	yes	—
20	#11924527	C₂²	1	6	yes	—
21	#11924526	C₂²	1	12	yes	—
22	#11924524	C₂²	1	3	yes	—
23	#11924519	C₂²	1	6	yes	—
24	#11924516	C₂²	1	6	yes	—
25	#11924515	C₂²	1	6	yes	—
26	#11924515	C₂²	1	3	yes	—
27	#11924512	C₂²	1	6	yes	—
28	#11924510	C₂²	1	1	yes	characteristic
29	Q₈×Dic₃₇	C₂²	1	2	yes	—
30	C₄×Dic₇₄	C₂²	1	6	yes	—
31	C₄×Dic₇₄	C₂²	1	3	yes	—
32	C₄×Dic₇₄	C₂²	1	6	yes	—
33	#11924501	C₂²	1	6	yes	—
34	C₄×D₁₄₈	C₂²	1	3	yes	—
35	C₄×C₂²⋊D₃₇	C₂²	1	6	yes	—
36	#11924495	C₂²	1	6	yes	—
37	#11924494	C₂²	1	6	yes	—
38	#11924493	C₂²	1	6	yes	—
39	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₃₇	C₂²	1	3	yes	—
41	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₃₇	C₂²	1	3	yes	—
44	C₂×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
45	C₂×C₄²×C₃₇	C₂²	1	3	yes	—
46	C₂×Q₈×C₃₇	—	2	4	no	—
47	C₂×Q₈×C₃₇	C₂³	1	3	yes	—
48	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
50	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
51	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₃₇	C₂³	1	2	yes	—
53	Q₈×D₃₇	—	2	8	no	—
54	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₃₇	C₂³	1	6	yes	—
56	C₂×Dic₇₄	C₂³	1	6	yes	—
57	C₂×Dic₇₄	C₂³	1	3	yes	—
58	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
59	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
60	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
61	C₄²×C₃₇	C₂³	1	3	yes	—
62	C₄²×C₃₇	C₂³	1	3	yes	—
63	C₄²×C₃₇	C₂³	1	3	yes	—
64	C₄²×C₃₇	C₂³	1	3	yes	—
65	C₂²⋊Dic₃₇	C₂³	1	6	yes	—
66	#10596938	C₂³	1	2	yes	—
67	#10596938	C₂³	1	6	yes	—
68	#10596938	C₂³	1	6	yes	—
69	#10596938	C₂³	1	6	yes	—
70	C₄⋊Dic₃₇	C₂³	1	6	yes	—
71	C₄⋊Dic₃₇	C₂³	1	6	yes	—
72	C₄⋊Dic₃₇	C₂³	1	6	yes	—
73	C₄⋊Dic₃₇	C₂³	1	3	yes	—
74	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
75	#10596936	C₂³	1	6	yes	—
76	#10596936	C₂³	1	6	yes	—
77	#10596936	C₂³	1	6	yes	—
78	#10596936	C₂³	1	12	yes	—
79	#10596936	C₂³	1	6	yes	—
80	C₄×Dic₃₇	C₂³	1	6	yes	—
81	C₄×Dic₃₇	C₂³	1	12	yes	—
82	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
83	C₂²×D₃₇	C₂⁴	1	2	yes	—
84	Q₈×C₃₇	—	2	6	no	—
85	Q₈×C₃₇	#41	1	4	yes	—
86	C₂×C₄×C₃₇	—	2	6	no	—
87	C₂×C₄×C₃₇	—	2	6	no	—
88	C₂×C₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
90	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
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	C₂²⋊D₃₇	—	2	4	no	—
96	C₂×Dic₃₇	C₂⁴	1	2	yes	—
97	C₂×Dic₃₇	C₂⁴	1	6	yes	—
98	C₂×Dic₃₇	C₂⁴	1	6	yes	—
99	D₁₄₈	—	2	2	no	—
100	C₄×D₃₇	—	2	12	no	—
101	Dic₇₄	—	2	12	no	—
102	Dic₇₄	—	2	6	no	—
103	C₂²×C₃₇	—	2	1	no	—
104	C₂²×C₃₇	—	2	1	no	—
105	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
106	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
107	C₂×D₃₇	—	2	2	no	—
108	C₂×D₃₇	—	2	2	no	—
109	C₂×D₃₇	—	2	2	no	—
110	C₄×C₃₇	—	2	3	no	—
111	C₄×C₃₇	—	2	3	no	—
112	C₄×C₃₇	—	2	3	no	—
113	C₄×C₃₇	—	2	1	no	—
114	C₄×C₃₇	C₂×#41	1	6	yes	—
115	C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
116	Dic₃₇	—	2	6	no	—
117	Dic₃₇	—	2	6	no	—
118	Dic₃₇	—	2	2	no	—
119	#3523	—	37	1	no	maximal, Sylow
120	C₂×C₃₇	—	2	2	no	—
121	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
122	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
124	D₃₇	—	4	2	no	—
125	#557	—	37	3	no	—
126	#557	—	37	1	no	—
127	#556	—	37	3	no	—
128	#552	—	37	6	no	—
129	D₄×Q₈	—	37	2	no	—
130	D₄×Q₈	—	37	1	no	—
131	#541	—	37	6	no	—
132	#533	—	37	3	no	—
133	#529	—	37	3	no	—
134	#519	—	37	2	no	—
135	C₂×C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
136	C₃₇	#3523	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	37	2	no	—
138	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
139	C₄⋊Q₈	—	37	6	no	—
140	C₄⋊Q₈	—	37	3	no	—
141	C₄⋊Q₈	—	37	3	no	—
142	#126	—	37	12	no	—
143	#126	—	37	6	no	—
144	#125	—	37	12	no	—
145	#125	—	37	6	no	—
146	#125	—	37	3	no	—
147	#125	—	37	6	no	—
148	#124	—	37	3	no	—
149	#123	—	37	6	no	—
150	C₂²⋊Q₈	—	37	6	no	—
151	C₂²⋊Q₈	—	37	3	no	—
152	C₂²⋊Q₈	—	37	6	no	—
153	C₂²⋊Q₈	—	37	6	no	—
154	#121	—	37	1	no	—
155	C₄×Q₈	—	37	2	no	—
156	C₄×Q₈	—	37	6	no	—
157	C₄×Q₈	—	37	6	no	—
158	C₄×Q₈	—	37	3	no	—
159	C₄×Q₈	—	37	6	no	—
160	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
161	C₄×Q₈	C₂×D₃₇	1	3	yes	—
162	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₃₇	1	3	yes	—
164	C₄×D₄	—	37	6	no	—
165	C₄×D₄	—	37	3	no	—
166	#117	—	37	6	no	—
167	#117	—	37	6	no	—
168	#117	—	37	6	no	—
169	C₂×C₄⋊C₄	C₂×D₃₇	1	3	yes	—
170	C₂×C₄²	C₂×D₃₇	1	3	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	37	3	no	—
173	C₂×Q₈	—	37	6	no	—
174	C₂×Q₈	—	74	8	no	—
175	C₂×Q₈	C₂²×D₃₇	1	3	yes	—
176	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
177	C₂×D₄	—	37	2	no	—
178	C₂×D₄	—	37	1	no	—
179	C₂²×C₄	—	37	6	no	—
180	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
182	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
183	C₄⋊C₄	—	37	6	no	—
184	C₄⋊C₄	—	37	6	no	—
185	C₄⋊C₄	—	37	6	no	—
186	C₄⋊C₄	—	37	12	no	—
187	C₄⋊C₄	—	37	6	no	—
188	C₄⋊C₄	—	37	6	no	—
189	C₄⋊C₄	—	37	6	no	—
190	C₄⋊C₄	—	37	6	no	—
191	C₄⋊C₄	—	37	3	no	—
192	C₄⋊C₄	—	37	1	no	—
193	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
194	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
195	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
196	C₂²⋊C₄	—	37	6	no	—
197	C₂²⋊C₄	—	37	2	no	—
198	C₂²⋊C₄	—	37	6	no	—
199	C₂²⋊C₄	—	37	6	no	—
200	C₂²⋊C₄	—	37	6	no	—
201	C₄²	—	37	12	no	—
202	C₄²	—	37	6	no	—
203	C₄²	C₂²×D₃₇	1	3	yes	—
204	C₄²	C₂²×D₃₇	1	3	yes	—
205	C₄²	C₂²×D₃₇	1	3	yes	—
206	C₄²	C₂²×D₃₇	1	3	yes	—
207	C₂³	—	37	2	no	—
208	C₂³	C₂³×D₃₇	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	—	74	12	no	—
211	Q₈	—	74	6	no	—
212	Q₈	#10596962	1	4	yes	—
213	D₄	—	74	4	no	—
214	D₄	—	74	2	no	—
215	C₂×C₄	—	2	6	no	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	37	6	no	—
219	C₂×C₄	—	37	2	no	—
220	C₂×C₄	—	37	6	no	—
221	C₂×C₄	—	74	12	no	—
222	C₂×C₄	C₂³×D₃₇	1	3	yes	—
223	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
224	C₂×C₄	C₂³×D₃₇	1	3	yes	—
225	C₂×C₄	C₂³×D₃₇	1	3	yes	—
226	C₂×C₄	C₂³×D₃₇	1	3	yes	—
227	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	74	2	no	—
231	C₂²	—	74	2	no	—
232	C₂²	—	74	2	no	—
233	C₂²	#11924583	1	2	yes	—
234	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	—	2	3	no	—
238	C₄	—	2	1	no	—
239	C₄	—	74	6	no	—
240	C₄	—	74	2	no	—
241	C₄	—	74	6	no	—
242	C₄	C₂×#10596962	1	6	yes	—
243	C₄	#11924584	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	148	2	no	—
246	C₂	C₂²×#10596962	1	1	yes	characteristic
247	C₂	C₂×#11924583	1	1	yes	characteristic
248	C₂	C₂×#11924584	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Fitting	C₂×C₄×Q₈×C₃₇	C₂
prime radical	#11924510	C₂²
hypercenter	C₂×C₄×Q₈	D₃₇
hypocenter	C₃₇	#3523
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14558262	C₂	semidirect
2. #14558261	C₂	semidirect
3. #14558256	C₂	semidirect
4. #14558218	C₂	semidirect
5. Q₈×D₁₄₈	C₂	semidirect
6. Q₈×C₂²⋊D₃₇	C₂	semidirect
7. #14558141	C₂	semidirect
8. #14558090	C₂	semidirect
10. #14558009	C₂	semidirect
11. C₂×C₄×Q₈×C₃₇	C₂	semidirect
135. C₂×C₄×Q₈	D₃₇	semidirect
136. C₃₇	#3523	semidirect

Maximal subgroups

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

subgroup	index
#14558262	2
#14558261	2
#14558256	2
#14558218	2
Q₈×D₁₄₈	2
Q₈×C₂²⋊D₃₇	2
#14558141	2
#14558090	2
#14558067	2
#14558009	2
C₂×C₄×Q₈×C₃₇	2
#3523	37