Subgroups — #18003467

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	#14558016	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558014	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557636	C₂	1	2	yes	maximal, complemented
5	#14557630	C₂	1	2	yes	maximal, complemented
6	#14557625	C₂	1	2	yes	maximal, complemented
7	#14557613	C₂	1	2	yes	maximal, complemented
8	C₂×#11924469	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557535	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#417	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	C₂²×#64436	C₂²	1	1	yes	characteristic
19	Q₈×Dic₃₇	C₂²	1	2	yes	—
20	#11924501	C₂²	1	2	yes	—
21	D₄×Dic₃₇	—	2	2	no	—
22	D₄×Dic₃₇	C₂²	1	2	yes	—
23	#11924498	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₃₇	—	2	2	no	—
25	#11924495	—	2	1	no	—
26	#11924495	C₂²	1	1	yes	characteristic
27	#11924494	—	2	1	no	—
28	C₂×C₄⋊Dic₃₇	—	2	1	no	—
29	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₃₇	—	2	1	no	—
31	#11924469	C₂²	1	2	yes	—
32	#11924469	C₂²	1	2	yes	—
33	#11924461	C₂²	1	2	yes	—
34	#11924458	C₂²	1	2	yes	—
35	#11924446	C₂²	1	2	yes	—
36	C₈×Dic₃₇	C₂²	1	2	yes	—
37	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
40	#103×C₃₇	C₂²	1	2	yes	—
41	#102×C₃₇	C₂²	1	2	yes	—
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	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₃₇	—	2	1	no	—
51	C₂²×Dic₃₇	—	2	1	no	—
52	C₂²×Dic₃₇	—	2	2	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	M₁₆×C₃₇	—	2	2	no	—
60	C₂×C₈×C₃₇	C₂³	1	2	yes	—
61	C₄⋊C₄×C₃₇	—	2	1	no	—
62	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
63	C₂²⋊Dic₃₇	—	2	2	no	—
64	C₂²⋊Dic₃₇	D₄	1	2	yes	—
65	#10596942	C₂×C₄	1	4	yes	complemented
66	#10596941	C₂×C₄	1	4	yes	complemented
67	#10596940	C₂×C₄	1	4	yes	complemented
68	#10596939	C₂×C₄	1	4	yes	complemented
69	#10596938	—	2	2	no	—
70	C₄⋊Dic₃₇	—	2	1	no	—
71	C₄⋊Dic₃₇	—	2	2	no	—
72	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
73	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
74	C₄⋊Dic₃₇	C₂³	1	2	yes	—
75	#10596936	—	2	2	no	—
76	C₄×Dic₃₇	—	2	2	no	—
77	C₄×Dic₃₇	—	2	2	no	—
78	C₄×Dic₃₇	C₂³	1	2	yes	—
79	C₂×#64436	C₂×C₄	1	4	yes	complemented
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	1	no	—
94	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₇	#41	1	2	yes	—
97	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₃₇	—	2	4	no	—
99	C₂×Dic₃₇	—	2	1	no	—
100	C₂×Dic₃₇	—	2	2	no	—
101	C₂×Dic₃₇	—	8	2	no	—
102	C₂×Dic₃₇	—	8	1	no	—
103	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
104	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
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	C₈×C₃₇	—	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	—	4	2	no	—
126	C₄×C₃₇	C₄×D₄	1	2	yes	—
127	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
129	Dic₃₇	—	2	2	no	—
130	Dic₃₇	—	2	2	no	—
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	—	37	1	no	—
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	D₃₇	1	1	yes	characteristic, complemented
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	C₂×D₃₇	1	1	yes	characteristic
160	C₂²×C₈	—	37	1	no	—
161	C₄×Q₈	—	37	2	no	—
162	C₄×Q₈	—	37	2	no	—
163	C₄×D₄	—	37	2	no	—
164	C₄×D₄	—	37	2	no	—
165	C₄×D₄	—	74	2	no	—
166	C₄×D₄	—	74	2	no	—
167	#117	—	37	1	no	—
168	#117	—	74	1	no	—
169	#117	—	74	1	no	—
170	C₂×C₄⋊C₄	—	37	1	no	—
171	C₂×C₄⋊C₄	—	74	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	C₂×D₃₇	1	2	yes	—
178	#102	—	37	2	no	—
179	#102	C₂×D₃₇	1	2	yes	—
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	—	74	2	no	—
195	C₂²×C₄	—	74	2	no	—
196	C₂²×C₄	—	74	1	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	—	2	2	no	—
204	C₂×C₈	—	37	4	no	—
205	C₂×C₈	—	37	1	no	—
206	C₂×C₈	—	37	1	no	—
207	C₂×C₈	C₂²×D₃₇	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	37	1	no	—
210	C₄⋊C₄	—	37	1	no	—
211	C₄⋊C₄	—	37	2	no	—
212	C₄⋊C₄	—	74	2	no	—
213	C₄⋊C₄	—	74	2	no	—
214	C₄⋊C₄	—	74	1	no	—
215	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
216	C₂²⋊C₄	—	37	2	no	—
217	C₂²⋊C₄	—	74	2	no	—
218	C₂²⋊C₄	—	74	2	no	—
219	C₄²	—	37	2	no	—
220	C₄²	—	74	2	no	—
221	C₄²	—	74	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	74	1	no	—
224	C₂³	D₄×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	1	no	—
239	C₂×C₄	—	37	2	no	—
240	C₂×C₄	—	37	2	no	—
241	C₂×C₄	—	296	2	no	—
242	C₂×C₄	—	296	2	no	—
243	C₂×C₄	—	296	1	no	—
244	C₂×C₄	—	74	1	no	—
245	C₂×C₄	—	74	4	no	—
246	C₂×C₄	—	74	2	no	—
247	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
248	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
249	C₂×C₄	#10596964	1	2	yes	—
250	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	37	2	no	—
253	C₈	—	37	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	#11924496	1	2	yes	—
263	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	74	2	no	—
267	C₄	—	74	2	no	—
268	C₄	—	74	2	no	—
269	C₄	—	148	2	no	—
270	C₄	#11924496	1	2	yes	—
271	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
272	C₄	C₂×#10596964	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	148	2	no	—
276	C₂	C₂×#11924496	1	1	yes	characteristic
277	C₂	#14558337	1	1	yes	characteristic
278	C₂	#14558349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#417	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924556	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#417	D₃₇
hypocenter	C₃₇	#2951
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #14558016	C₂	semidirect
3. #14558014	C₂	semidirect
4. #14557636	C₂	semidirect
5. #14557630	C₂	semidirect
6. #14557625	C₂	semidirect
7. #14557613	C₂	semidirect
8. C₂×#11924469	C₂	semidirect
9. #14557535	C₂	semidirect
10. #14557491	C₂	semidirect
11. C₃₇×#417	C₂	semidirect
13. #11924556	C₄	semidirect
43. #41×C₃₇	C₂×C₄	semidirect
54. #10596962	C₂×C₄	semidirect
65. #10596942	C₂×C₄	semidirect
66. #10596941	C₂×C₄	semidirect
67. #10596940	C₂×C₄	semidirect
68. #10596939	C₂×C₄	semidirect
79. C₂×#64436	C₂×C₄	semidirect
149. #417	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
#14558016	2
#14558014	2
#14557636	2
#14557630	2
#14557625	2
#14557613	2
C₂×#11924469	2
#14557535	2
#14557491	2
C₃₇×#417	2
#2951	37