Subgroups — #18003469

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924556	C₂	1	1	yes	maximal, characteristic
2	#14558013	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557633	C₂	1	2	yes	maximal, complemented
4	#14557631	C₂	1	2	yes	maximal, complemented
5	C₄×#10596942	C₂	1	2	yes	maximal, complemented
6	C₄×#10596939	C₂	1	2	yes	maximal, complemented
7	C₂×#11924469	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557534	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557532	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924556	C₄	1	8	yes	complemented
14	C₂×#10596942	C₂²	1	1	yes	characteristic
15	C₂×#10596940	C₂²	1	1	yes	characteristic
16	C₂×#10596941	C₂²	1	1	yes	characteristic
17	C₂×#10596939	C₂²	1	1	yes	characteristic
18	C₂×#10596934	C₂²	1	1	yes	characteristic
19	C₂²×#64436	C₂²	1	1	yes	characteristic
20	C₄×Dic₇₄	C₂²	1	2	yes	—
21	C₄×D₁₄₈	C₂²	1	2	yes	—
22	#11924496	—	2	2	no	—
23	#11924494	—	2	1	no	—
24	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₃₇	—	2	1	no	—
26	#11924469	C₂²	1	2	yes	—
27	#11924469	C₂²	1	2	yes	—
28	#11924460	C₂²	1	2	yes	—
29	#11924459	C₂²	1	2	yes	—
30	#11924457	C₂²	1	2	yes	—
31	#11924456	C₂²	1	2	yes	—
32	#11924447	C₂²	1	2	yes	—
33	C₄×#64436	C₂²	1	2	yes	—
34	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
36	C₄×D₄×C₃₇	—	2	2	no	—
37	C₄×D₄×C₃₇	C₂²	1	2	yes	—
38	#117×C₃₇	—	2	1	no	—
39	#117×C₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
42	#41×C₃₇	—	2	2	no	—
43	#41×C₃₇	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₇	—	2	1	no	—
46	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₇	—	2	1	no	—
48	C₂²×C₄×C₃₇	—	2	1	no	—
49	C₂²×C₄×C₃₇	—	2	2	no	—
50	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₃₇	—	2	1	no	—
52	C₂²×Dic₃₇	—	2	1	no	—
53	#10596962	—	2	2	no	—
54	#10596962	C₂×C₄	1	4	yes	complemented
55	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₃₇	—	2	1	no	—
57	C₂×C₄×D₃₇	—	2	2	no	—
58	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₃₇	—	2	1	no	—
60	C₄⋊C₄×C₃₇	—	2	2	no	—
61	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₃₇	—	2	2	no	—
65	C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
66	C₄²×C₃₇	—	2	2	no	—
67	C₄²×C₃₇	C₂³	1	2	yes	—
68	#10596942	C₂×C₄	1	4	yes	complemented
69	#10596941	C₂×C₄	1	4	yes	complemented
70	#10596940	C₂×C₄	1	4	yes	complemented
71	#10596939	C₂×C₄	1	4	yes	complemented
72	#10596938	—	2	2	no	—
73	C₄⋊Dic₃₇	—	2	1	no	—
74	C₄⋊Dic₃₇	C₂³	1	2	yes	—
75	#10596936	—	2	2	no	—
76	C₄×Dic₃₇	—	2	2	no	—
77	#10596934	—	2	2	no	—
78	C₂×#64436	C₂×C₄	1	4	yes	complemented
79	C₂×#64436	C₂³	1	2	yes	—
80	C₂×#64436	C₂³	1	1	yes	characteristic
81	C₂×#64436	C₂³	1	1	yes	characteristic
82	C₂³×C₃₇	—	2	1	no	—
83	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₃₇	—	2	1	no	—
85	Q₈×C₃₇	—	2	1	no	—
86	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
87	D₄×C₃₇	—	2	4	no	—
88	D₄×C₃₇	—	2	1	no	—
89	D₄×C₃₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₃₇	—	2	1	no	—
91	C₂×C₄×C₃₇	—	2	4	no	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	8	2	no	—
94	C₂×C₄×C₃₇	—	8	1	no	—
95	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₇	#41	1	2	yes	—
100	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₃₇	—	2	4	no	—
102	C₂×Dic₃₇	—	2	1	no	—
103	C₂×Dic₃₇	—	2	2	no	—
104	C₂×Dic₃₇	—	8	1	no	—
105	D₁₄₈	—	2	1	no	—
106	D₁₄₈	C₂²×C₄	1	2	yes	—
107	C₄×D₃₇	—	2	4	no	—
108	C₄×D₃₇	—	8	2	no	—
109	Dic₇₄	—	2	1	no	—
110	Dic₇₄	C₂²×C₄	1	2	yes	—
111	#64436	—	2	2	no	—
112	#64436	C₂²×C₄	1	2	yes	—
113	#64436	C₂²×C₄	1	2	yes	—
114	C₂²×C₃₇	—	2	1	no	—
115	C₂²×C₃₇	—	2	1	no	—
116	C₂²×C₃₇	—	2	2	no	—
117	C₂²×C₃₇	—	4	1	no	—
118	C₂²×C₃₇	—	4	1	no	—
119	C₂²×C₃₇	C₄×D₄	1	2	yes	—
120	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₃₇	—	2	2	no	—
122	C₂×D₃₇	—	4	1	no	—
123	C₂×D₃₇	—	4	1	no	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	—	2	2	no	—
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	—	4	2	no	—
128	C₄×C₃₇	C₄×D₄	1	2	yes	—
129	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	Dic₃₇	—	2	2	no	—
132	Dic₃₇	—	4	2	no	—
133	#2951	—	37	1	no	maximal, Sylow
134	C₂×C₃₇	—	2	2	no	—
135	C₂×C₃₇	—	4	2	no	—
136	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₃₇	#576	1	1	yes	characteristic
138	C₂×C₃₇	#578	1	1	yes	characteristic
139	D₃₇	—	4	2	no	—
140	C₂×#135	—	37	1	no	—
141	#520	—	37	1	no	—
142	#520	D₃₇	1	1	yes	characteristic, complemented
143	#440	—	37	2	no	—
144	#440	—	37	2	no	—
145	C₄×Q₁₆	—	37	2	no	—
146	C₄×D₈	—	37	2	no	—
147	C₂×#107	—	37	1	no	—
148	#417	—	37	1	no	—
149	#417	—	37	1	no	—
150	#405	—	37	1	no	—
151	C₃₇	#2951	1	1	yes	characteristic, complemented, Sylow
152	C₂×#41	—	37	1	no	—
153	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
154	#135	—	37	8	no	—
155	C₂×Q₁₆	—	37	1	no	—
156	C₂×SD₁₆	—	37	1	no	—
157	C₂×SD₁₆	—	37	1	no	—
158	C₂×D₈	—	37	1	no	—
159	C₂×M₁₆	—	37	1	no	—
160	C₂²×C₈	—	37	1	no	—
161	C₄×Q₈	—	37	2	no	—
162	C₄×Q₈	C₂×D₃₇	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	37	2	no	—
165	C₄×D₄	—	74	2	no	—
166	C₄×D₄	C₂×D₃₇	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	74	1	no	—
169	#117	C₂×D₃₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	37	1	no	—
172	C₂×C₄⋊C₄	—	74	1	no	—
173	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
174	#107	—	37	2	no	—
175	#107	—	37	2	no	—
176	#103	—	37	2	no	—
177	#103	—	37	2	no	—
178	#102	—	37	2	no	—
179	#102	—	37	2	no	—
180	#97	—	37	2	no	—
181	C₄×C₈	—	37	2	no	—
182	#41	—	2	2	no	—
183	#41	—	37	4	no	—
184	#41	—	74	2	no	—
185	#41	C₄×D₃₇	1	4	yes	complemented
186	C₂×Q₈	—	37	1	no	—
187	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	37	1	no	—
190	C₂×D₄	—	74	1	no	—
191	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	74	1	no	—
196	C₂²×C₄	—	74	2	no	—
197	C₂²×C₄	—	74	1	no	—
198	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
199	Q₁₆	—	37	4	no	—
200	SD₁₆	—	37	4	no	—
201	SD₁₆	—	37	4	no	—
202	D₈	—	37	4	no	—
203	M₁₆	—	74	2	no	—
204	C₂×C₈	—	37	4	no	—
205	C₂×C₈	—	37	1	no	—
206	C₂×C₈	—	37	1	no	—
207	C₂×C₈	—	37	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	37	2	no	—
211	C₄⋊C₄	—	74	2	no	—
212	C₄⋊C₄	—	74	1	no	—
213	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	74	2	no	—
218	C₂²⋊C₄	C₂²⋊D₃₇	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	74	2	no	—
221	C₄²	C₂²×D₃₇	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	74	1	no	—
224	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	37	2	no	—
227	Q₈	—	74	1	no	—
228	Q₈	C₂×C₄×D₃₇	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	37	2	no	—
232	D₄	—	74	4	no	—
233	D₄	—	74	1	no	—
234	D₄	C₂×C₄×D₃₇	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	296	2	no	—
241	C₂×C₄	—	296	1	no	—
242	C₂×C₄	—	74	1	no	—
243	C₂×C₄	—	74	4	no	—
244	C₂×C₄	—	74	2	no	—
245	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
246	C₂×C₄	#10596962	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
251	C₈	—	37	2	no	—
252	C₈	—	37	2	no	—
253	C₈	—	74	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	74	2	no	—
260	C₂²	—	148	1	no	—
261	C₂²	—	148	1	no	—
262	C₂²	C₄×C₂²⋊D₃₇	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	74	2	no	—
269	C₄	—	148	2	no	—
270	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
271	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
272	C₄	C₂×#10596962	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	148	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₃₇	1	1	yes	characteristic
277	C₂	#14558336	1	1	yes	characteristic
278	C₂	#14558348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#520	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924556	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#520	D₃₇
hypocenter	C₃₇	#2951
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14558013	C₂	semidirect
3. #14557633	C₂	semidirect
4. #14557631	C₂	semidirect
5. C₄×#10596942	C₂	semidirect
6. C₄×#10596939	C₂	semidirect
7. C₂×#11924469	C₂	semidirect
8. #14557534	C₂	semidirect
9. #14557532	C₂	semidirect
10. #14557489	C₂	semidirect
11. C₃₇×#520	C₂	semidirect
13. #11924556	C₄	semidirect
43. #41×C₃₇	C₂×C₄	semidirect
54. #10596962	C₂×C₄	semidirect
68. #10596942	C₂×C₄	semidirect
69. #10596941	C₂×C₄	semidirect
70. #10596940	C₂×C₄	semidirect
71. #10596939	C₂×C₄	semidirect
78. C₂×#64436	C₂×C₄	semidirect
142. #520	D₃₇	semidirect
151. C₃₇	#2951	semidirect
185. #41	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×#11924556	2
#14558013	2
#14557633	2
#14557631	2
C₄×#10596942	2
C₄×#10596939	2
C₂×#11924469	2
#14557534	2
#14557532	2
#14557489	2
C₃₇×#520	2
#2951	37