Subgroups — #18011407

Order 4736.

412 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558585	C₂	1	2	yes	maximal, complemented
2	#14558566	C₂	1	2	yes	maximal, complemented
3	#14558565	C₂	1	2	yes	maximal, complemented
4	#14558563	C₂	1	2	yes	maximal, complemented
5	#14558552	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558551	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10596953	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10596957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#14558001	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596953	C₂²	1	2	yes	—
13	C₂×#10596957	C₂²	1	2	yes	—
14	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
15	#11924603	C₂²	1	2	yes	—
16	#11924603	C₂²	1	2	yes	—
17	C₄⋊#64438	C₂²	1	2	yes	—
18	#11924593	C₂²	1	2	yes	—
19	#11924593	C₂²	1	2	yes	—
20	C₂²⋊#64438	C₂²	1	2	yes	—
21	#11924591	C₂²	1	2	yes	—
22	#11924591	C₂²	1	2	yes	—
23	C₄×#64438	C₂²	1	1	yes	characteristic
24	C₄×#64438	C₂²	1	1	yes	characteristic
25	C₄×#64438	C₂²	1	2	yes	—
26	C₄×#64438	C₂²	1	1	yes	characteristic
27	C₄×#64438	C₂²	1	1	yes	characteristic
28	C₂×#10596962	C₄	1	1	yes	characteristic
29	#11924501	C₄	1	1	yes	characteristic
30	#11924501	C₂²	1	1	yes	characteristic
31	#11924498	C₄	1	1	yes	characteristic
32	#11924498	C₂²	1	1	yes	characteristic
33	#11924496	C₄	1	1	yes	characteristic
34	#11924496	C₄	1	1	yes	characteristic
35	#11924496	C₂²	1	1	yes	characteristic
36	#11924496	C₂²	1	1	yes	characteristic
37	#11924493	C₄	1	1	yes	characteristic
38	#11924493	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
40	C₄²×D₃₇	C₂²	1	1	yes	characteristic
41	C₄²×D₃₇	C₂²	1	1	yes	characteristic
42	#117×C₃₇	C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
48	#10596962	—	2	4	no	—
49	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
56	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
57	#10596957	—	2	4	no	—
58	C₂×#64438	C₂³	1	2	yes	—
59	C₂×#64438	C₂³	1	2	yes	—
60	C₂×#64438	C₂³	1	2	yes	—
61	C₂×#64438	C₂³	1	2	yes	—
62	#10596953	—	2	4	no	—
63	D₃₇⋊C₈	—	2	4	no	—
64	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
70	#10596938	C₂×C₄	1	1	yes	characteristic
71	#10596938	C₂×C₄	1	1	yes	characteristic
72	#10596938	C₂×C₄	1	1	yes	characteristic
73	#10596938	C₂×C₄	1	1	yes	characteristic
74	#10596936	C₂×C₄	1	1	yes	characteristic
75	#10596936	C₂×C₄	1	1	yes	characteristic
76	#10596936	C₂×C₄	1	1	yes	characteristic
77	#10596936	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
84	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
85	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
86	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₃₇	—	2	2	no	—
97	C₂²⋊D₃₇	—	2	2	no	—
98	C₂×Dic₃₇	—	2	2	no	—
99	C₂×Dic₃₇	—	2	2	no	—
100	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
106	D₁₄₈	—	2	2	no	—
107	C₄×D₃₇	—	2	2	no	—
108	C₄×D₃₇	—	2	2	no	—
109	C₄×D₃₇	—	2	2	no	—
110	C₄×D₃₇	—	2	2	no	—
111	C₄×D₃₇	—	2	2	no	—
112	C₄×D₃₇	M₁₆	1	4	yes	complemented
113	Dic₇₄	—	2	2	no	—
114	#64438	—	2	2	no	—
115	#64438	—	2	2	no	—
116	#64438	—	2	2	no	—
117	#64438	#41	1	4	yes	complemented
118	C₂²×C₃₇	—	2	1	no	—
119	C₂²×C₃₇	—	2	1	no	—
120	C₂²×C₃₇	—	2	1	no	—
121	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₃₇	—	2	1	no	—
123	C₂×D₃₇	—	2	1	no	—
124	C₂×D₃₇	—	2	1	no	—
125	C₂×D₃₇	—	2	1	no	—
126	C₂×D₃₇	—	2	1	no	—
127	C₂×D₃₇	C₂×M₁₆	1	2	yes	—
128	C₄×C₃₇	—	2	1	no	—
129	C₄×C₃₇	—	2	1	no	—
130	C₄×C₃₇	—	2	1	no	—
131	C₄×C₃₇	—	2	1	no	—
132	C₄×C₃₇	—	2	1	no	—
133	C₄×C₃₇	C₂×M₁₆	1	2	yes	—
134	Dic₃₇	—	2	1	no	—
135	Dic₃₇	—	2	1	no	—
136	Dic₃₇	—	2	1	no	—
137	Dic₃₇	C₂×M₁₆	1	2	yes	—
138	Dic₃₇	#131	1	2	yes	—
139	Dic₃₇	C₂×#41	1	2	yes	—
140	#2965	—	37	1	no	maximal, Sylow
141	C₂×C₃₇	—	4	1	no	—
142	C₂×C₃₇	C₄×#41	1	1	yes	characteristic
143	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
144	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
145	D₃₇	—	2	2	no	—
146	D₃₇	—	4	1	no	—
147	C₄×#41	—	37	1	no	—
148	#446	—	37	2	no	—
149	#436	—	37	2	no	—
150	#435	—	37	2	no	—
151	#435	—	37	2	no	—
152	#431	—	37	1	no	—
153	#431	—	37	1	no	—
154	#431	—	37	1	no	—
155	C₄×M₁₆	—	37	1	no	—
156	C₄×M₁₆	—	37	1	no	—
157	C₂×C₄×C₈	—	37	1	no	—
158	C₃₇	#2965	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	37	1	no	—
160	C₂×M₁₆	—	37	2	no	—
161	C₂×M₁₆	—	37	2	no	—
162	C₂²×C₈	—	37	2	no	—
163	C₄×Q₈	—	37	1	no	—
164	C₄×Q₈	—	37	1	no	—
165	C₄×D₄	—	37	1	no	—
166	C₄×D₄	—	37	1	no	—
167	C₄×D₄	—	37	1	no	—
168	C₄×D₄	—	37	1	no	—
169	C₄×D₄	—	37	1	no	—
170	C₄×D₄	—	37	1	no	—
171	#117	—	37	1	no	—
172	#117	—	37	1	no	—
173	#117	#3866	1	1	yes	characteristic
174	C₂×C₄²	—	37	1	no	—
175	C₂×C₄²	—	37	1	no	—
176	C₂×C₄²	—	37	1	no	—
177	C₄⋊C₈	—	37	2	no	—
178	C₄⋊C₈	—	37	2	no	—
179	C₄⋊C₈	—	37	2	no	—
180	C₂²⋊C₈	—	37	2	no	—
181	C₂²⋊C₈	—	37	2	no	—
182	C₂²⋊C₈	—	37	2	no	—
183	#97	—	37	2	no	—
184	#97	—	37	2	no	—
185	C₄×C₈	—	37	1	no	—
186	C₄×C₈	—	37	1	no	—
187	C₄×C₈	—	37	2	no	—
188	C₄×C₈	—	37	1	no	—
189	C₄×C₈	—	37	1	no	—
190	#41	—	74	4	no	—
191	C₂×Q₈	—	37	1	no	—
192	C₂×D₄	—	37	1	no	—
193	C₂×D₄	—	37	1	no	—
194	C₂×D₄	—	37	1	no	—
195	C₂²×C₄	—	37	1	no	—
196	C₂²×C₄	—	37	1	no	—
197	C₂²×C₄	—	37	1	no	—
198	C₂²×C₄	—	37	1	no	—
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	37	1	no	—
201	C₂²×C₄	—	37	1	no	—
202	C₂²×C₄	—	37	1	no	—
203	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
204	M₁₆	—	74	4	no	—
205	M₁₆	—	74	4	no	—
206	C₂×C₈	—	37	2	no	—
207	C₂×C₈	—	37	2	no	—
208	C₂×C₈	—	37	2	no	—
209	C₂×C₈	—	37	2	no	—
210	C₂×C₈	—	74	4	no	—
211	C₄⋊C₄	—	37	1	no	—
212	C₄⋊C₄	—	37	1	no	—
213	C₄⋊C₄	—	37	1	no	—
214	C₄⋊C₄	—	37	1	no	—
215	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
216	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
217	C₂²⋊C₄	—	37	1	no	—
218	C₂²⋊C₄	—	37	1	no	—
219	C₂²⋊C₄	—	37	1	no	—
220	C₂²⋊C₄	—	37	1	no	—
221	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
222	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
223	C₄²	—	37	1	no	—
224	C₄²	—	37	1	no	—
225	C₄²	—	37	1	no	—
226	C₄²	—	37	1	no	—
227	C₄²	—	37	1	no	—
228	C₄²	—	37	1	no	—
229	C₄²	—	37	1	no	—
230	C₄²	—	37	1	no	—
231	C₄²	C₂×#3866	1	1	yes	characteristic
232	C₄²	C₂×#3866	1	1	yes	characteristic
233	C₂³	—	37	1	no	—
234	C₂³	—	37	1	no	—
235	C₂³	C₂²×#3866	1	1	yes	characteristic
236	Q₈	—	74	2	no	—
237	D₄	—	74	2	no	—
238	D₄	—	74	2	no	—
239	D₄	—	74	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	37	1	no	—
242	C₂×C₄	—	37	1	no	—
243	C₂×C₄	—	37	1	no	—
244	C₂×C₄	—	37	4	no	—
245	C₂×C₄	—	37	1	no	—
246	C₂×C₄	—	37	1	no	—
247	C₂×C₄	—	37	1	no	—
248	C₂×C₄	—	74	2	no	—
249	C₂×C₄	—	74	2	no	—
250	C₂×C₄	—	74	2	no	—
251	C₂×C₄	—	74	2	no	—
252	C₂×C₄	—	74	2	no	—
253	C₂×C₄	—	74	2	no	—
254	C₂×C₄	—	74	2	no	—
255	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
256	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
257	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
258	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
259	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
260	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
261	C₈	—	37	4	no	—
262	C₈	—	74	2	no	—
263	C₈	—	74	2	no	—
264	C₈	—	74	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	37	2	no	—
269	C₂²	—	74	1	no	—
270	C₂²	—	74	1	no	—
271	C₂²	—	74	1	no	—
272	C₂²	—	74	1	no	—
273	C₂²	—	74	1	no	—
274	C₂²	C₂³×#3866	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	37	2	no	—
281	C₄	—	37	2	no	—
282	C₄	—	37	2	no	—
283	C₄	—	74	1	no	—
284	C₄	—	74	1	no	—
285	C₄	—	74	1	no	—
286	C₄	D₃₇⋊M₁₆	1	2	yes	—
287	C₂	—	4	1	no	—
288	C₂	—	74	2	no	—
289	C₂	—	148	1	no	—
290	C₂	#41×#3866	1	1	yes	characteristic
291	C₂	C₂×D₃₇⋊M₁₆	1	1	yes	characteristic
292	C₂	D₃₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#3866
Fitting	#117×C₃₇	C₄
prime radical	C₂×#10596962	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	#117	#3866
hypocenter	C₃₇	#2965
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558566	C₂	semidirect
3. #14558565	C₂	semidirect
4. #14558563	C₂	semidirect
5. #14558552	C₂	semidirect
6. #14558551	C₂	semidirect
7. #14558550	C₂	semidirect
8. C₄×#10596953	C₂	semidirect
9. C₄×#10596957	C₂	semidirect
10. C₄×D₃₇⋊C₈	C₂	semidirect
112. C₄×D₃₇	M₁₆	semidirect
117. #64438	#41	semidirect
158. C₃₇	#2965	semidirect

Maximal subgroups

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

subgroup	index
#14558585	2
#14558566	2
#14558565	2
#14558563	2
#14558552	2
#14558551	2
#14558550	2
C₄×#10596953	2
C₄×#10596957	2
C₄×D₃₇⋊C₈	2
#14558001	2
#2965	37