Subgroups — #18003471

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924559	C₂	1	1	yes	maximal, characteristic
2	#14558015	C₂	1	2	yes	maximal, complemented
3	#14557636	C₂	1	2	yes	maximal, complemented
4	#14557631	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₂₉₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557615	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557533	C₂	1	2	yes	maximal, complemented
10	#14557487	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596966	C₂²	1	2	yes	—
13	#11924559	C₄	1	8	yes	complemented
14	C₂×#10596941	C₂²	1	2	yes	—
15	C₂×D₂₉₆	C₂²	1	1	yes	characteristic
16	C₂×#10596929	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
18	Q₈×Dic₃₇	C₂²	1	2	yes	—
19	C₄×D₁₄₈	—	2	2	no	—
20	C₄×D₁₄₈	C₂²	1	2	yes	—
21	#11924498	—	2	2	no	—
22	#11924498	C₂²	1	2	yes	—
23	#11924493	—	2	2	no	—
24	#11924493	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₃₇	—	2	2	no	—
26	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
27	#11924470	C₂²	1	1	yes	characteristic
28	#11924469	C₂²	1	2	yes	—
29	#11924460	C₂²	1	2	yes	—
30	#11924458	C₂²	1	2	yes	—
31	#11924456	C₂²	1	2	yes	—
32	#11924447	C₂²	1	1	yes	characteristic
33	#11924446	C₂²	1	1	yes	characteristic
34	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
37	#107×C₃₇	C₂²	1	1	yes	characteristic
38	#103×C₃₇	C₂²	1	2	yes	—
39	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
41	#10596966	—	2	4	no	—
42	#10596966	C₂×C₄	1	8	yes	complemented
43	C₂×D₁₄₈	—	2	2	no	—
44	C₂×D₁₄₈	C₂³	1	2	yes	—
45	C₂×C₄×D₃₇	—	2	2	no	—
46	C₂×C₄×D₃₇	—	2	2	no	—
47	C₂×C₄×D₃₇	—	2	2	no	—
48	C₂×C₄×D₃₇	—	2	2	no	—
49	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
50	Q₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₃₇	—	2	2	no	—
54	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
55	C₄²×C₃₇	—	2	2	no	—
56	C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	#10596941	C₂×C₄	1	8	yes	complemented
58	#10596938	—	2	2	no	—
59	#10596938	—	2	2	no	—
60	#10596938	D₄	1	1	yes	characteristic
61	#10596938	D₄	1	1	yes	characteristic
62	C₄⋊Dic₃₇	—	2	2	no	—
63	C₄⋊Dic₃₇	C₂³	1	2	yes	—
64	#10596936	—	2	2	no	—
65	#10596936	D₄	1	1	yes	characteristic
66	#10596936	D₄	1	1	yes	characteristic
67	C₄×Dic₃₇	—	2	2	no	—
68	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
69	C₂×#64436	C₂³	1	1	yes	characteristic
70	C₂×#64436	C₂³	1	1	yes	characteristic
71	D₂₉₆	C₂×C₄	1	4	yes	complemented
72	#10596929	—	2	2	no	—
73	C₈×D₃₇	C₂×C₄	1	4	yes	complemented
74	C₂²×D₃₇	—	2	2	no	—
75	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₃₇	—	2	2	no	—
77	Q₈×C₃₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₃₇	—	2	2	no	—
79	C₂×C₄×C₃₇	—	2	2	no	—
80	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₃₇	—	2	2	no	—
84	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
87	D₁₄₈	—	2	8	no	—
88	D₁₄₈	—	2	2	no	—
89	D₁₄₈	C₂²×C₄	1	4	yes	—
90	C₄×D₃₇	—	2	8	no	—
91	C₄×D₃₇	—	8	2	no	—
92	C₄×D₃₇	—	8	2	no	—
93	C₄×D₃₇	—	8	2	no	—
94	C₄×D₃₇	C₂²×C₄	1	2	yes	—
95	C₄×D₃₇	#41	1	2	yes	—
96	C₈×C₃₇	—	2	1	no	—
97	C₈×C₃₇	C₂²×C₄	1	2	yes	—
98	#64436	—	2	1	no	—
99	#64436	C₂²×C₄	1	2	yes	—
100	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₃₇	—	2	1	no	—
102	C₂×D₃₇	—	2	1	no	—
103	C₂×D₃₇	—	2	4	no	—
104	C₂×D₃₇	—	4	2	no	—
105	C₂×D₃₇	—	4	2	no	—
106	C₂×D₃₇	C₄×D₄	1	2	yes	—
107	C₄×C₃₇	—	2	4	no	—
108	C₄×C₃₇	—	2	1	no	—
109	C₄×C₃₇	—	2	1	no	—
110	C₄×C₃₇	—	4	2	no	—
111	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
113	Dic₃₇	—	2	1	no	—
114	Dic₃₇	—	2	1	no	—
115	Dic₃₇	—	4	2	no	—
116	Dic₃₇	C₄×D₄	1	2	yes	—
117	#2951	—	37	1	no	maximal, Sylow
118	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₃₇	#576	1	1	yes	characteristic
120	C₂×C₃₇	#578	1	1	yes	characteristic
121	D₃₇	—	2	2	no	—
122	D₃₇	—	4	4	no	—
123	C₂×#135	—	37	1	no	—
124	#520	—	37	2	no	—
125	#440	—	37	2	no	—
126	#440	—	37	2	no	—
127	C₄×Q₁₆	—	37	1	no	—
128	C₄×Q₁₆	D₃₇	1	1	yes	characteristic, complemented
129	C₄×D₈	—	37	1	no	—
130	C₄×D₈	—	37	1	no	—
131	C₂×#107	—	37	1	no	—
132	#417	—	37	2	no	—
133	#405	—	37	1	no	—
134	C₃₇	#2951	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	37	2	no	—
136	#135	—	37	8	no	—
137	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
138	C₂×SD₁₆	—	37	2	no	—
139	C₂×D₈	—	37	1	no	—
140	C₂×M₁₆	—	37	1	no	—
141	C₂²×C₈	—	37	1	no	—
142	C₄×Q₈	—	37	2	no	—
143	C₄×Q₈	C₂×D₃₇	1	2	yes	—
144	C₄×D₄	—	37	2	no	—
145	C₄×D₄	—	37	2	no	—
146	C₄×D₄	—	74	2	no	—
147	C₄×D₄	—	74	2	no	—
148	#117	—	37	1	no	—
149	#117	—	74	2	no	—
150	C₂×C₄⋊C₄	—	37	2	no	—
151	C₂×C₄⋊C₄	—	74	2	no	—
152	#107	—	37	1	no	—
153	#107	—	37	2	no	—
154	#107	C₂×D₃₇	1	1	yes	characteristic
155	#103	—	37	2	no	—
156	#103	C₂×D₃₇	1	2	yes	—
157	#102	—	37	2	no	—
158	#102	—	37	2	no	—
159	#97	—	37	1	no	—
160	#97	—	37	1	no	—
161	C₄×C₈	—	37	1	no	—
162	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
163	#41	—	37	8	no	—
164	#41	—	74	4	no	—
165	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
166	C₂×D₄	—	37	2	no	—
167	C₂×D₄	—	74	2	no	—
168	C₂²×C₄	—	37	1	no	—
169	C₂²×C₄	—	74	2	no	—
170	C₂²×C₄	—	74	2	no	—
171	C₂²×C₄	—	74	2	no	—
172	C₂²×C₄	—	74	2	no	—
173	Q₁₆	C₄×D₃₇	1	4	yes	complemented
174	SD₁₆	—	37	8	no	—
175	D₈	—	37	4	no	—
176	M₁₆	—	74	2	no	—
177	C₂×C₈	—	37	1	no	—
178	C₂×C₈	—	37	4	no	—
179	C₂×C₈	—	37	1	no	—
180	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	37	1	no	—
184	C₄⋊C₄	—	37	1	no	—
185	C₄⋊C₄	—	37	2	no	—
186	C₄⋊C₄	—	74	2	no	—
187	C₄⋊C₄	—	74	2	no	—
188	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
189	C₂²⋊C₄	—	37	1	no	—
190	C₂²⋊C₄	—	37	1	no	—
191	C₂²⋊C₄	—	74	2	no	—
192	C₂²⋊C₄	—	74	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	37	1	no	—
195	C₄²	—	74	2	no	—
196	C₄²	C₂²×D₃₇	1	1	yes	characteristic
197	C₂³	—	37	1	no	—
198	C₂³	—	74	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	C₂×C₄×D₃₇	1	4	yes	—
201	D₄	—	37	4	no	—
202	D₄	—	74	8	no	—
203	D₄	—	74	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	37	1	no	—
207	C₂×C₄	—	37	2	no	—
208	C₂×C₄	—	37	2	no	—
209	C₂×C₄	—	37	1	no	—
210	C₂×C₄	—	37	1	no	—
211	C₂×C₄	—	296	2	no	—
212	C₂×C₄	—	296	2	no	—
213	C₂×C₄	—	296	2	no	—
214	C₂×C₄	—	74	8	no	—
215	C₂×C₄	—	74	2	no	—
216	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
217	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	37	2	no	—
221	C₈	—	74	1	no	—
222	C₈	C₂×C₄×D₃₇	1	2	yes	—
223	C₂²	—	37	2	no	—
224	C₂²	—	74	1	no	—
225	C₂²	—	74	1	no	—
226	C₂²	—	74	4	no	—
227	C₂²	—	148	2	no	—
228	C₂²	—	148	2	no	—
229	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
230	C₄	—	2	4	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	37	2	no	—
235	C₄	—	74	1	no	—
236	C₄	—	74	1	no	—
237	C₄	—	148	2	no	—
238	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
239	C₄	#41×D₃₇	1	1	yes	characteristic
240	C₂	—	74	2	no	—
241	C₂	—	148	4	no	—
242	C₂	C₄×D₄×D₃₇	1	1	yes	characteristic
243	C₂	#14558335	1	1	yes	characteristic
244	C₂	#14558346	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×Q₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924559	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×Q₁₆	D₃₇
hypocenter	C₃₇	#2951
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #14558015	C₂	semidirect
3. #14557636	C₂	semidirect
4. #14557631	C₂	semidirect
5. Q₁₆×Dic₃₇	C₂	semidirect
6. C₄×D₂₉₆	C₂	semidirect
7. #14557615	C₂	semidirect
8. #107×D₃₇	C₂	semidirect
9. #14557533	C₂	semidirect
10. #14557487	C₂	semidirect
11. C₄×Q₁₆×C₃₇	C₂	semidirect
13. #11924559	C₄	semidirect
42. #10596966	C₂×C₄	semidirect
50. Q₁₆×C₃₇	C₂×C₄	semidirect
57. #10596941	C₂×C₄	semidirect
71. D₂₉₆	C₂×C₄	semidirect
73. C₈×D₃₇	C₂×C₄	semidirect
128. C₄×Q₁₆	D₃₇	semidirect
134. C₃₇	#2951	semidirect
173. Q₁₆	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924559	2
#14558015	2
#14557636	2
#14557631	2
Q₁₆×Dic₃₇	2
C₄×D₂₉₆	2
#14557615	2
#107×D₃₇	2
#14557533	2
#14557487	2
C₄×Q₁₆×C₃₇	2
#2951	37