Subgroups — #18003336

Order 4736.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924542	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558079	C₂	1	1	yes	maximal, characteristic
4	#14557608	C₂	1	2	yes	maximal, complemented
5	#14557607	C₂	1	2	yes	maximal, complemented
6	#14557601	C₂	1	2	yes	maximal, complemented
7	#14557596	C₂	1	2	yes	maximal, complemented
8	C₂×#11924465	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557501	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924543	—	2	4	no	—
14	#11924542	—	2	4	no	—
15	C₂×#10596934	C₂²	1	2	yes	—
16	C₂×#10596934	C₂²	1	1	yes	characteristic
17	C₂×#10596929	C₂²	1	2	yes	—
18	C₂²×#64436	C₂²	1	1	yes	characteristic
19	C₂²×#64436	C₂²	1	2	yes	—
20	C₂×C₈×D₃₇	C₂²	1	2	yes	—
21	#11924534	C₄	1	1	yes	characteristic
22	#11924532	C₄	1	1	yes	characteristic
23	#11924525	C₄	1	1	yes	characteristic
24	#11924523	C₄	1	1	yes	characteristic
25	#11924508	C₄	1	2	yes	—
26	#11924508	C₄	1	2	yes	—
27	C₄×C₂²⋊D₃₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₃₇	C₂²	1	2	yes	—
29	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	#11924465	C₂²	1	1	yes	characteristic
31	#11924465	C₂²	1	1	yes	characteristic
32	#11924465	C₂²	1	1	yes	characteristic
33	#11924465	C₂²	1	1	yes	characteristic
34	#11924448	C₂²	1	2	yes	—
35	#11924448	C₂²	1	2	yes	—
36	#11924446	C₂²	1	2	yes	—
37	C₈×Dic₃₇	C₂²	1	2	yes	—
38	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
40	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
42	C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
43	#41×C₃₇	—	2	2	no	—
44	#41×C₃₇	—	2	2	no	—
45	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₃₇	C₂×C₄	1	2	yes	—
47	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
53	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
55	#10596962	—	2	2	no	—
56	#10596962	—	2	2	no	—
57	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₃₇	C₂³	1	2	yes	—
59	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₃₇	—	2	2	no	—
61	C₂×C₈×C₃₇	—	2	2	no	—
62	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₃₇	C₂³	1	2	yes	—
65	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
67	#10596938	C₂×C₄	1	2	yes	—
68	#10596938	C₂×C₄	1	2	yes	—
69	#10596936	C₂×C₄	1	2	yes	—
70	#10596936	C₂×C₄	1	2	yes	—
71	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
74	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
75	#10596934	—	2	4	no	—
76	#10596934	D₄	1	4	yes	complemented
77	C₂×#64436	—	2	2	no	—
78	C₂×#64436	—	2	4	no	—
79	C₂×#64436	C₂³	1	1	yes	characteristic
80	C₂×#64436	C₂³	1	1	yes	characteristic
81	C₂×#64436	C₂³	1	2	yes	—
82	#10596929	—	2	4	no	—
83	C₈×D₃₇	—	2	4	no	—
84	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
85	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
87	Q₈×C₃₇	—	2	2	no	—
88	D₄×C₃₇	—	2	4	no	—
89	D₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	—	2	2	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₇	#41	1	2	yes	—
96	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₃₇	—	2	4	no	—
98	C₂²⋊D₃₇	—	2	4	no	—
99	C₂²⋊D₃₇	—	4	2	no	—
100	C₂×Dic₃₇	—	2	2	no	—
101	C₂×Dic₃₇	—	4	1	no	—
102	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
106	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
107	D₁₄₈	—	2	2	no	—
108	C₄×D₃₇	—	2	2	no	—
109	C₄×D₃₇	—	2	2	no	—
110	Dic₇₄	—	2	2	no	—
111	C₈×C₃₇	—	2	1	no	—
112	C₈×C₃₇	—	2	1	no	—
113	C₈×C₃₇	—	2	2	no	—
114	#64436	—	2	1	no	—
115	#64436	—	2	1	no	—
116	#64436	C₂×D₄	1	4	yes	—
117	C₂²×C₃₇	—	2	1	no	—
118	C₂²×C₃₇	—	2	1	no	—
119	C₂²×C₃₇	—	2	2	no	—
120	C₂²×C₃₇	—	2	2	no	—
121	C₂²×C₃₇	—	2	2	no	—
122	C₂²×C₃₇	C₄×D₄	1	2	yes	—
123	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₃₇	—	2	2	no	—
125	C₂×D₃₇	—	2	2	no	—
126	C₂×D₃₇	—	2	2	no	—
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	C₄×D₄	1	2	yes	—
129	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	Dic₃₇	—	2	2	no	—
132	Dic₃₇	—	2	1	no	—
133	Dic₃₇	—	2	1	no	—
134	Dic₃₇	—	2	1	no	—
135	Dic₃₇	#131	1	2	yes	complemented
136	#2935	—	37	1	no	maximal, Sylow
137	C₂×C₃₇	—	2	2	no	—
138	C₂×C₃₇	—	4	2	no	—
139	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
141	C₂×C₃₇	#568	1	1	yes	characteristic
142	D₃₇	—	4	2	no	—
143	C₂×#131	—	37	1	no	—
144	C₂×#131	—	37	1	no	—
145	#532	—	37	1	no	—
146	#436	—	37	2	no	—
147	#435	—	37	2	no	—
148	#435	—	37	2	no	—
149	C₈×D₄	—	37	2	no	—
150	C₂×C₄⋊C₈	—	37	1	no	—
151	#408	—	37	1	no	—
152	#408	D₃₇	1	1	yes	characteristic, complemented
153	C₄×M₁₆	—	37	1	no	—
154	C₃₇	#2935	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	37	1	no	—
156	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
157	#131	—	74	4	no	—
158	#131	—	74	4	no	—
159	C₂×M₁₆	—	37	1	no	—
160	C₂×M₁₆	—	37	2	no	—
161	C₂×M₁₆	—	37	2	no	—
162	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
163	C₂²×C₈	—	37	1	no	—
164	C₂²×C₈	—	37	2	no	—
165	C₂²×C₈	—	37	2	no	—
166	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	37	1	no	—
168	#127	—	37	1	no	—
169	#124	—	37	1	no	—
170	#124	—	37	1	no	—
171	#121	—	37	2	no	—
172	#121	—	37	2	no	—
173	C₄×D₄	—	37	2	no	—
174	C₄×D₄	—	37	2	no	—
175	C₂×C₄²	—	37	1	no	—
176	C₄⋊C₈	—	37	1	no	—
177	C₄⋊C₈	—	37	1	no	—
178	C₄⋊C₈	—	37	1	no	—
179	C₄⋊C₈	—	37	1	no	—
180	C₂²⋊C₈	—	37	2	no	—
181	C₂²⋊C₈	—	37	2	no	—
182	C₂²⋊C₈	C₂×D₃₇	1	2	yes	—
183	C₂²⋊C₈	C₂×D₃₇	1	2	yes	—
184	#97	—	37	2	no	—
185	C₄×C₈	—	37	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	74	2	no	—
189	#41	—	74	2	no	—
190	C₂×Q₈	—	37	1	no	—
191	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
192	C₂×D₄	—	37	2	no	—
193	C₂×D₄	—	37	2	no	—
194	C₂×D₄	—	37	2	no	—
195	C₂×D₄	—	37	1	no	—
196	C₂×D₄	C₄×D₃₇	1	2	yes	—
197	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
198	C₂²×C₄	—	37	1	no	—
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	37	2	no	—
201	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	37	4	no	—
205	M₁₆	—	74	4	no	—
206	M₁₆	—	74	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	37	2	no	—
209	C₂×C₈	—	37	1	no	—
210	C₂×C₈	—	37	1	no	—
211	C₂×C₈	—	74	4	no	—
212	C₂×C₈	—	74	4	no	—
213	C₂×C₈	—	74	2	no	—
214	C₂×C₈	C₂²×D₃₇	1	2	yes	—
215	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
217	C₄⋊C₄	—	37	2	no	—
218	C₄⋊C₄	—	37	2	no	—
219	C₂²⋊C₄	—	37	2	no	—
220	C₂²⋊C₄	—	37	2	no	—
221	C₂²⋊C₄	—	37	2	no	—
222	C₂²⋊C₄	—	37	2	no	—
223	C₄²	—	37	1	no	—
224	C₄²	—	37	1	no	—
225	C₄²	—	37	1	no	—
226	C₄²	—	37	1	no	—
227	C₂³	—	37	2	no	—
228	C₂³	C₂×C₄×D₃₇	1	2	yes	—
229	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	74	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	74	4	no	—
235	D₄	—	74	4	no	—
236	D₄	—	74	2	no	—
237	D₄	—	148	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	37	1	no	—
241	C₂×C₄	—	37	1	no	—
242	C₂×C₄	—	37	1	no	—
243	C₂×C₄	—	37	1	no	—
244	C₂×C₄	—	37	2	no	—
245	C₂×C₄	—	74	2	no	—
246	C₂×C₄	—	74	2	no	—
247	C₂×C₄	—	74	2	no	—
248	C₂×C₄	—	148	1	no	—
249	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
251	C₂×C₄	D₄×D₃₇	1	2	yes	—
252	C₂×C₄	#10596964	1	2	yes	—
253	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	37	4	no	—
258	C₈	—	74	1	no	—
259	C₈	—	74	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	74	2	no	—
266	C₂²	—	74	2	no	—
267	C₂²	—	74	2	no	—
268	C₂²	#11924496	1	2	yes	—
269	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	37	2	no	—
272	C₄	—	74	2	no	—
273	C₄	—	74	1	no	—
274	C₄	—	74	1	no	—
275	C₄	—	74	1	no	—
276	C₄	#11924496	1	2	yes	—
277	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
278	C₄	C₂×#10596964	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	148	2	no	—
282	C₂	C₂×#11924496	1	1	yes	characteristic
283	C₂	#131×D₃₇	1	1	yes	characteristic
284	C₂	#14558298	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#408	C₂
prime radical	#14558079	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#408	D₃₇
hypocenter	C₃₇	#2935
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#11924543	C₂	semidirect
2. C₂×#11924542	C₂	semidirect
4. #14557608	C₂	semidirect
5. #14557607	C₂	semidirect
6. #14557601	C₂	semidirect
7. #14557596	C₂	semidirect
8. C₂×#11924465	C₂	semidirect
9. #14557501	C₂	semidirect
10. M₁₆×Dic₃₇	C₂	semidirect
11. C₃₇×#408	C₂	semidirect
76. #10596934	D₄	semidirect
135. Dic₃₇	#131	semidirect
152. #408	D₃₇	semidirect
154. C₃₇	#2935	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924543	2
C₂×#11924542	2
#14558079	2
#14557608	2
#14557607	2
#14557601	2
#14557596	2
C₂×#11924465	2
#14557501	2
M₁₆×Dic₃₇	2
C₃₇×#408	2
#2935	37