Subgroups — #18003190

Order 4736.

476 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924544	C₂	1	1	yes	maximal, characteristic
2	#14558014	C₂	1	1	yes	maximal, characteristic
3	#14558013	C₂	1	1	yes	maximal, characteristic
4	#14557575	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557574	C₂	1	2	yes	maximal, complemented
6	#14557572	C₂	1	2	yes	maximal, complemented
7	#14557570	C₂	1	2	yes	maximal, complemented
8	#14557569	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11924467	C₂	1	1	yes	maximal, characteristic, complemented
10	#106×D₃₇	C₂	1	2	yes	maximal, complemented
11	C₃₇×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924544	C₄	1	8	yes	complemented
14	C₂×#10596929	C₂²	1	2	yes	—
15	C₂²×#64436	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₃₇	C₂²	1	2	yes	—
17	C₄×Dic₇₄	—	2	1	no	—
18	#11924501	—	2	1	no	—
19	C₄×D₁₄₈	—	2	1	no	—
20	#11924498	—	2	1	no	—
21	C₄×C₂²⋊D₃₇	—	2	2	no	—
22	#11924496	—	2	2	no	—
23	#11924495	C₂²	1	1	yes	characteristic
24	#11924494	C₂²	1	2	yes	—
25	#11924494	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
28	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
29	#11924470	C₂²	1	2	yes	—
30	#11924469	C₂²	1	1	yes	characteristic
31	#11924469	C₂²	1	1	yes	characteristic
32	#11924468	C₂²	1	2	yes	—
33	#11924467	C₂²	1	1	yes	characteristic
34	#11924467	C₂²	1	2	yes	—
35	#11924467	C₂²	1	2	yes	—
36	#11924467	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
38	#117×C₃₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
40	#107×C₃₇	C₂²	1	2	yes	—
41	#106×C₃₇	C₂²	1	2	yes	—
42	C₂²×C₄×C₃₇	—	2	1	no	—
43	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₇	D₄	1	2	yes	—
45	C₂²×Dic₃₇	—	2	1	no	—
46	#10596962	C₂×C₄	1	4	yes	—
47	#10596962	Q₈	1	4	yes	—
48	C₂×D₁₄₈	D₄	1	1	yes	characteristic
49	C₂×C₄×D₃₇	—	2	2	no	—
50	C₂×C₄×D₃₇	—	2	2	no	—
51	C₂×C₄×D₃₇	C₂³	1	2	yes	—
52	C₂×Dic₇₄	D₄	1	1	yes	characteristic
53	M₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
54	C₂×C₈×C₃₇	C₂³	1	2	yes	—
55	C₄⋊C₄×C₃₇	—	2	1	no	—
56	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
58	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
59	C₂²⋊C₄×C₃₇	—	2	1	no	—
60	C₄²×C₃₇	—	2	1	no	—
61	C₂²⋊Dic₃₇	—	2	1	no	—
62	#10596938	—	2	2	no	—
63	#10596938	—	2	2	no	—
64	C₄⋊Dic₃₇	—	2	1	no	—
65	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₃₇	C₂³	1	2	yes	—
67	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
68	#10596936	—	2	2	no	—
69	#10596936	—	2	2	no	—
70	C₄×Dic₃₇	—	2	1	no	—
71	C₄×Dic₃₇	—	2	2	no	—
72	C₄×Dic₃₇	—	2	2	no	—
73	C₂×#64436	C₂×C₄	1	4	yes	complemented
74	C₂×#64436	C₂³	1	1	yes	characteristic
75	C₂×#64436	C₂³	1	1	yes	characteristic
76	#10596929	C₂×C₄	1	8	yes	complemented
77	C₈×D₃₇	C₂×C₄	1	8	yes	complemented
78	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×D₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₃₇	—	2	1	no	—
81	C₂×C₄×C₃₇	—	2	2	no	—
82	C₂×C₄×C₃₇	—	2	1	no	—
83	C₂×C₄×C₃₇	—	8	1	no	—
84	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₃₇	C₄⋊C₄	1	8	yes	—
89	C₂×Dic₃₇	—	2	1	no	—
90	C₂×Dic₃₇	—	2	1	no	—
91	C₂×Dic₃₇	—	2	2	no	—
92	C₂×Dic₃₇	—	8	1	no	—
93	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
94	D₁₄₈	C₄⋊C₄	1	4	yes	—
95	C₄×D₃₇	—	8	2	no	—
96	C₄×D₃₇	—	8	2	no	—
97	C₄×D₃₇	C₂²×C₄	1	4	yes	—
98	C₄×D₃₇	C₂×Q₈	1	4	yes	—
99	Dic₇₄	C₄⋊C₄	1	4	yes	—
100	C₈×C₃₇	C₂²×C₄	1	4	yes	—
101	#64436	C₂²×C₄	1	2	yes	—
102	#64436	C₂²×C₄	1	2	yes	—
103	C₂²×C₃₇	—	2	1	no	—
104	C₂²×C₃₇	—	2	1	no	—
105	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₃₇	—	2	2	no	—
108	C₂×D₃₇	—	2	2	no	—
109	C₂×D₃₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₄×C₃₇	—	4	1	no	—
111	C₄×C₃₇	—	4	2	no	—
112	C₄×C₃₇	—	4	1	no	—
113	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
116	Dic₃₇	—	4	1	no	—
117	Dic₃₇	—	4	2	no	—
118	Dic₃₇	—	4	1	no	—
119	Dic₃₇	C₂×C₄⋊C₄	1	4	yes	—
120	#2912	—	37	1	no	maximal, Sylow
121	C₂×C₃₇	—	2	2	no	—
122	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₃₇	#577	1	1	yes	characteristic
124	C₂×C₃₇	#577	1	1	yes	characteristic
125	D₃₇	—	2	4	no	—
126	C₂×#131	—	37	1	no	—
127	#520	—	37	1	no	—
128	#520	—	37	1	no	—
129	#428	—	37	1	no	—
130	#428	—	37	2	no	—
131	#428	—	37	2	no	—
132	#428	D₃₇	1	1	yes	characteristic, complemented
133	#427	—	37	1	no	—
134	#427	—	37	2	no	—
135	C₂×#106	—	37	1	no	—
136	C₂×#106	—	37	2	no	—
137	C₃₇	#2912	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	37	1	no	—
139	#131	—	37	8	no	—
140	C₂×M₁₆	—	37	2	no	—
141	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
142	C₂²×C₈	—	37	1	no	—
143	C₂²×C₈	—	37	2	no	—
144	C₄×Q₈	—	74	1	no	—
145	C₄×Q₈	—	74	1	no	—
146	C₄×D₄	—	74	1	no	—
147	C₄×D₄	—	74	2	no	—
148	C₄×D₄	—	74	1	no	—
149	C₄×D₄	—	74	2	no	—
150	#117	—	37	1	no	—
151	#117	—	37	2	no	—
152	#117	—	37	2	no	—
153	#117	C₂×D₃₇	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	37	1	no	—
155	C₂×C₄⋊C₄	—	37	2	no	—
156	C₂×C₄⋊C₄	—	37	2	no	—
157	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
158	#107	—	37	2	no	—
159	#107	—	37	1	no	—
160	#107	—	37	1	no	—
161	#107	C₂×D₃₇	1	2	yes	—
162	#106	—	37	2	no	—
163	#106	—	37	2	no	—
164	#106	—	37	1	no	—
165	#106	—	37	2	no	—
166	#106	—	37	1	no	—
167	#106	C₂×D₃₇	1	2	yes	—
168	#41	—	37	4	no	—
169	#41	—	37	4	no	—
170	C₂×Q₈	—	37	1	no	—
171	C₂×D₄	—	37	2	no	—
172	C₂×D₄	—	37	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	37	2	no	—
175	C₂²×C₄	—	74	2	no	—
176	C₂²×C₄	—	74	2	no	—
177	C₂²×C₄	—	74	1	no	—
178	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
179	M₁₆	—	37	8	no	—
180	M₁₆	C₄×D₃₇	1	4	yes	complemented
181	C₂×C₈	—	37	4	no	—
182	C₂×C₈	—	37	1	no	—
183	C₂×C₈	—	37	1	no	—
184	C₂×C₈	—	37	8	no	—
185	C₂×C₈	C₂²×D₃₇	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	37	1	no	—
188	C₄⋊C₄	—	37	2	no	—
189	C₄⋊C₄	—	37	1	no	—
190	C₄⋊C₄	—	74	2	no	—
191	C₄⋊C₄	—	74	2	no	—
192	C₄⋊C₄	—	74	1	no	—
193	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
194	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	74	2	no	—
198	C₂²⋊C₄	—	74	2	no	—
199	C₂²⋊C₄	—	74	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	74	2	no	—
202	C₄²	—	74	2	no	—
203	C₄²	—	74	1	no	—
204	C₂³	—	37	2	no	—
205	C₂³	D₄×D₃₇	1	1	yes	characteristic
206	Q₈	—	37	4	no	—
207	D₄	—	37	8	no	—
208	D₄	—	37	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	37	2	no	—
214	C₂×C₄	—	37	4	no	—
215	C₂×C₄	—	37	4	no	—
216	C₂×C₄	—	296	2	no	—
217	C₂×C₄	—	296	2	no	—
218	C₂×C₄	—	296	1	no	—
219	C₂×C₄	—	74	1	no	—
220	C₂×C₄	—	74	2	no	—
221	C₂×C₄	—	74	1	no	—
222	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
223	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
224	C₂×C₄	Q₈×D₃₇	1	2	yes	—
225	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
226	C₈	—	37	2	no	—
227	C₈	—	37	2	no	—
228	C₈	C₂×C₄×D₃₇	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	37	4	no	—
232	C₂²	—	74	2	no	—
233	C₂²	—	74	2	no	—
234	C₂²	C₄⋊C₄×D₃₇	1	2	yes	—
235	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	37	4	no	—
240	C₄	—	148	2	no	—
241	C₄	—	148	1	no	—
242	C₄	—	148	1	no	—
243	C₄	C₄⋊C₄×D₃₇	1	2	yes	—
244	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
245	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	74	4	no	—
248	C₂	C₂×C₄⋊C₄×D₃₇	1	1	yes	characteristic
249	C₂	#14558342	1	1	yes	characteristic
250	C₂	#14558343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#428	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#10596962	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#428	D₃₇
hypocenter	C₃₇	#2912
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #14557575	C₂	semidirect
5. #14557574	C₂	semidirect
6. #14557572	C₂	semidirect
7. #14557570	C₂	semidirect
8. #14557569	C₂	semidirect
9. C₂×#11924467	C₂	semidirect
10. #106×D₃₇	C₂	semidirect
11. C₃₇×#428	C₂	semidirect
13. #11924544	C₄	semidirect
53. M₁₆×C₃₇	C₂×C₄	semidirect
73. C₂×#64436	C₂×C₄	semidirect
76. #10596929	C₂×C₄	semidirect
77. C₈×D₃₇	C₂×C₄	semidirect
132. #428	D₃₇	semidirect
137. C₃₇	#2912	semidirect
180. M₁₆	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×#11924544	2
#14558014	2
#14558013	2
#14557575	2
#14557574	2
#14557572	2
#14557570	2
#14557569	2
C₂×#11924467	2
#106×D₃₇	2
C₃₇×#428	2
#2912	37