Subgroups — #18008681

Order 4736.

1104 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558268	C₂	1	3	yes	maximal, complemented
2	Q₈×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558243	C₂	1	3	yes	maximal, complemented
4	#14558205	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558194	C₂	1	2	yes	maximal, complemented
6	#14558184	C₂	1	6	yes	maximal, complemented
7	#14558155	C₂	1	6	yes	maximal, complemented
8	#14558083	C₂	1	3	yes	maximal, complemented
9	#14558067	C₂	1	3	yes	maximal, complemented
10	#14558015	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596966	C₂²	1	2	yes	—
13	C₂×#10596962	C₂²	1	6	yes	—
14	#11924536	C₂²	1	3	yes	—
15	#11924535	C₂²	1	6	yes	—
16	#11924533	C₂²	1	3	yes	—
17	#11924530	C₂²	1	6	yes	—
18	#11924528	C₂²	1	3	yes	—
19	#11924525	C₂²	1	6	yes	—
20	#11924524	C₂²	1	3	yes	—
21	#11924524	C₂²	1	6	yes	—
22	#11924521	C₂²	1	12	yes	—
23	#11924519	C₂²	1	6	yes	—
24	#11924515	C₂²	1	1	yes	characteristic
25	#11924514	C₂²	1	12	yes	—
26	#11924512	C₂²	1	6	yes	—
27	#11924511	C₂²	1	6	yes	—
28	#11924510	C₂²	1	3	yes	—
29	#11924510	C₂²	1	6	yes	—
30	Q₈×Dic₃₇	C₂²	1	2	yes	—
31	C₄×Dic₇₄	C₂²	1	3	yes	—
32	C₄×D₁₄₈	C₂²	1	6	yes	—
33	C₄×D₁₄₈	C₂²	1	3	yes	—
34	C₄×D₁₄₈	C₂²	1	6	yes	—
35	#11924498	C₂²	1	6	yes	—
36	C₄×C₂²⋊D₃₇	C₂²	1	6	yes	—
37	#11924493	C₂²	1	6	yes	—
38	C₄⋊C₄×D₃₇	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	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₃₇	C₂³	1	6	yes	—
53	#10596966	—	2	8	no	—
54	#10596962	—	4	12	no	—
55	C₂×D₁₄₈	C₂³	1	6	yes	—
56	C₂×D₁₄₈	C₂³	1	3	yes	—
57	C₂×D₁₄₈	C₂³	1	6	yes	—
58	C₂×C₄×D₃₇	C₂³	1	6	yes	—
59	C₂×C₄×D₃₇	C₂³	1	12	yes	—
60	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
61	C₂×Dic₇₄	C₂³	1	6	yes	—
62	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
63	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
64	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
65	C₄²×C₃₇	C₂³	1	3	yes	—
66	C₄²×C₃₇	C₂³	1	3	yes	—
67	C₄²×C₃₇	C₂³	1	3	yes	—
68	C₄²×C₃₇	C₂³	1	3	yes	—
69	C₂²⋊Dic₃₇	C₂³	1	2	yes	—
70	#10596938	C₂³	1	6	yes	—
71	#10596938	C₂³	1	6	yes	—
72	#10596938	C₂³	1	6	yes	—
73	#10596938	C₂³	1	12	yes	—
74	#10596938	C₂³	1	6	yes	—
75	C₄⋊Dic₃₇	C₂³	1	6	yes	—
76	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₃₇	C₂³	1	3	yes	—
78	#10596936	C₂³	1	6	yes	—
79	#10596936	C₂³	1	2	yes	—
80	#10596936	C₂³	1	6	yes	—
81	#10596936	C₂³	1	6	yes	—
82	C₄×Dic₃₇	C₂³	1	6	yes	—
83	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
84	C₂²×D₃₇	C₂⁴	1	6	yes	—
85	Q₈×C₃₇	—	2	6	no	—
86	Q₈×C₃₇	#41	1	4	yes	complemented
87	C₂×C₄×C₃₇	—	2	6	no	—
88	C₂×C₄×C₃₇	—	2	6	no	—
89	C₂×C₄×C₃₇	—	2	2	no	—
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	3	yes	—
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	2	yes	—
100	D₁₄₈	—	2	12	no	—
101	D₁₄₈	—	2	6	no	—
102	D₁₄₈	—	4	6	no	—
103	C₄×D₃₇	—	2	12	no	—
104	C₄×D₃₇	—	4	12	no	—
105	Dic₇₄	—	2	2	no	—
106	Dic₇₄	—	4	6	no	—
107	C₂²×C₃₇	—	2	1	no	—
108	C₂²×C₃₇	—	2	1	no	—
109	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
111	C₂×D₃₇	—	2	6	no	—
112	C₂×D₃₇	—	2	6	no	—
113	C₂×D₃₇	—	2	6	no	—
114	C₄×C₃₇	—	2	3	no	—
115	C₄×C₃₇	—	2	3	no	—
116	C₄×C₃₇	—	2	3	no	—
117	C₄×C₃₇	—	2	1	no	—
118	C₄×C₃₇	C₂×#41	1	6	yes	—
119	C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
120	Dic₃₇	—	2	2	no	—
121	Dic₃₇	—	2	6	no	—
122	Dic₃₇	—	2	2	no	—
123	#3522	—	37	1	no	maximal, Sylow
124	C₂×C₃₇	—	2	2	no	—
125	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
126	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
128	D₃₇	—	4	6	no	—
129	#559	—	37	3	no	—
130	Q₈²	—	37	1	no	—
131	#554	—	37	3	no	—
132	#550	—	37	1	no	—
133	#548	—	37	2	no	—
134	#547	—	37	6	no	—
135	#542	—	37	6	no	—
136	#532	—	37	3	no	—
137	#529	—	37	3	no	—
138	#520	—	37	2	no	—
139	C₂×C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
140	C₃₇	#3522	1	1	yes	characteristic, complemented, Sylow
141	C₂×#41	—	37	2	no	—
142	C₂×#41	—	37	6	no	—
143	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
144	C₄⋊Q₈	—	37	6	no	—
145	C₄⋊Q₈	—	37	3	no	—
146	#127	—	37	3	no	—
147	#126	—	37	6	no	—
148	#125	—	37	3	no	—
149	#124	—	37	6	no	—
150	#124	—	37	3	no	—
151	#124	—	37	6	no	—
152	#123	—	37	6	no	—
153	#123	—	37	12	no	—
154	C₂²⋊Q₈	—	37	6	no	—
155	C₂²⋊Q₈	—	37	1	no	—
156	C₂²⋊Q₈	—	37	12	no	—
157	#121	—	37	3	no	—
158	#121	—	37	6	no	—
159	#121	—	37	6	no	—
160	C₄×Q₈	—	37	2	no	—
161	C₄×Q₈	—	37	3	no	—
162	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₃₇	1	3	yes	—
164	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
165	C₄×Q₈	C₂×D₃₇	1	3	yes	—
166	C₄×D₄	—	37	6	no	—
167	C₄×D₄	—	37	6	no	—
168	C₄×D₄	—	37	6	no	—
169	C₄×D₄	—	37	3	no	—
170	C₄×D₄	—	37	6	no	—
171	#117	—	37	6	no	—
172	C₂×C₄⋊C₄	—	37	6	no	—
173	C₂×C₄⋊C₄	C₂×D₃₇	1	3	yes	—
174	C₂×C₄²	C₂×D₃₇	1	3	yes	—
175	#41	—	74	8	no	—
176	#41	—	148	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	37	1	no	—
179	C₂×Q₈	—	37	6	no	—
180	C₂×Q₈	C₂²×D₃₇	1	3	yes	—
181	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
182	C₂×D₄	—	37	6	no	—
183	C₂×D₄	—	37	3	no	—
184	C₂×D₄	—	37	6	no	—
185	C₂×D₄	—	37	6	no	—
186	C₂²×C₄	—	37	6	no	—
187	C₂²×C₄	—	37	12	no	—
188	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
190	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
191	C₄⋊C₄	—	37	6	no	—
192	C₄⋊C₄	—	37	2	no	—
193	C₄⋊C₄	—	37	6	no	—
194	C₄⋊C₄	—	37	6	no	—
195	C₄⋊C₄	—	37	6	no	—
196	C₄⋊C₄	—	37	1	no	—
197	C₄⋊C₄	—	37	3	no	—
198	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
199	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
200	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
201	C₂²⋊C₄	—	37	2	no	—
202	C₂²⋊C₄	—	37	6	no	—
203	C₂²⋊C₄	—	37	6	no	—
204	C₂²⋊C₄	—	37	6	no	—
205	C₂²⋊C₄	—	37	12	no	—
206	C₂²⋊C₄	—	37	6	no	—
207	C₄²	—	37	6	no	—
208	C₄²	C₂²×D₃₇	1	3	yes	—
209	C₄²	C₂²×D₃₇	1	3	yes	—
210	C₄²	C₂²×D₃₇	1	3	yes	—
211	C₄²	C₂²×D₃₇	1	3	yes	—
212	C₂³	—	37	6	no	—
213	C₂³	C₂³×D₃₇	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	74	2	no	—
216	Q₈	—	148	6	no	—
217	Q₈	#10596962	1	4	yes	complemented
218	D₄	—	74	12	no	—
219	D₄	—	74	12	no	—
220	D₄	—	74	6	no	—
221	D₄	—	148	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	37	2	no	—
226	C₂×C₄	—	37	6	no	—
227	C₂×C₄	—	37	2	no	—
228	C₂×C₄	—	74	12	no	—
229	C₂×C₄	—	148	12	no	—
230	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₃₇	1	3	yes	—
232	C₂×C₄	C₂³×D₃₇	1	3	yes	—
233	C₂×C₄	C₂³×D₃₇	1	3	yes	—
234	C₂×C₄	C₂³×D₃₇	1	3	yes	—
235	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	74	6	no	—
239	C₂²	—	74	6	no	—
240	C₂²	—	74	6	no	—
241	C₂²	#11924582	1	2	yes	—
242	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	1	no	—
247	C₄	—	74	2	no	—
248	C₄	—	74	6	no	—
249	C₄	—	74	2	no	—
250	C₄	C₂×#10596962	1	6	yes	—
251	C₄	#11924584	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	148	6	no	—
254	C₂	C₂²×#10596962	1	1	yes	characteristic
255	C₂	C₂×#11924582	1	1	yes	characteristic
256	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
hypercenter	C₂×C₄×Q₈	D₃₇
hypocenter	C₃₇	#3522
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. #14558268	C₂	semidirect
2. Q₈×Dic₇₄	C₂	semidirect
3. #14558243	C₂	semidirect
4. #14558205	C₂	semidirect
5. #14558194	C₂	semidirect
6. #14558184	C₂	semidirect
7. #14558155	C₂	semidirect
8. #14558083	C₂	semidirect
9. #14558067	C₂	semidirect
10. #14558015	C₂	semidirect
11. C₂×C₄×Q₈×C₃₇	C₂	semidirect
86. Q₈×C₃₇	#41	semidirect
139. C₂×C₄×Q₈	D₃₇	semidirect
140. C₃₇	#3522	semidirect
217. Q₈	#10596962	semidirect

Maximal subgroups

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

subgroup	index
#14558268	2
Q₈×Dic₇₄	2
#14558243	2
#14558205	2
#14558194	2
#14558184	2
#14558155	2
#14558083	2
#14558067	2
#14558015	2
C₂×C₄×Q₈×C₃₇	2
#3522	37