Subgroups — C₃₇×#28827 #22650080

Order 9472.

678 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2552	C₂	1	2	yes	maximal
2	C₃₇×#2083	C₂	1	4	yes	maximal
3	C₃₇×#1947	C₂	1	2	yes	maximal
4	C₃₇×#1942	C₂	1	2	yes	maximal
5	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#1845	C₂	1	2	yes	maximal
7	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₃₇	—	2	2	no	—
10	C₂×#107×C₃₇	C₂²	1	2	yes	—
11	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₃₇	C₂²	1	4	yes	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₃₇×#398	—	2	4	no	—
19	C₃₇×#389	—	2	2	no	—
20	C₃₇×#389	C₂²	1	4	yes	—
21	C₃₇×#384	C₂²	1	2	yes	—
22	C₃₇×#384	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#340	C₂²	1	4	yes	—
26	C₃₇×#340	C₂²	1	2	yes	—
27	C₃₇×#336	C₂²	1	2	yes	—
28	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	#125×C₃₇	—	4	8	no	—
32	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₃₇	—	2	2	no	—
43	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
45	#107×C₃₇	—	2	2	no	—
46	#107×C₃₇	—	2	4	no	—
47	#107×C₃₇	—	2	2	no	—
48	#107×C₃₇	—	4	4	no	—
49	#107×C₃₇	—	4	2	no	—
50	C₄⋊C₈×C₃₇	Q₈	1	4	yes	—
51	C₄×C₈×C₃₇	—	2	4	no	—
52	C₄×C₈×C₃₇	Q₈	1	4	yes	—
53	#95×C₃₇	—	2	2	no	—
54	#95×C₃₇	—	2	4	no	—
55	#95×C₃₇	D₄	1	2	yes	—
56	C₂²×C₄×C₃₇	—	2	2	no	—
57	C₂²×C₄×C₃₇	—	2	2	no	—
58	C₂²×C₄×C₃₇	—	2	4	no	—
59	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₃₇	—	2	2	no	—
65	C₂×C₈×C₃₇	—	2	1	no	—
66	C₂×C₈×C₃₇	—	2	2	no	—
67	C₂×C₈×C₃₇	—	2	1	no	—
68	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₃₇	#41	1	2	yes	—
72	C₂×C₈×C₃₇	#41	1	2	yes	—
73	C₄⋊C₄×C₃₇	—	2	2	no	—
74	C₄⋊C₄×C₃₇	—	2	2	no	—
75	C₄⋊C₄×C₃₇	—	2	4	no	—
76	C₄⋊C₄×C₃₇	—	2	2	no	—
77	C₄⋊C₄×C₃₇	—	2	4	no	—
78	C₄⋊C₄×C₃₇	—	2	2	no	—
79	C₄⋊C₄×C₃₇	—	4	8	no	—
80	C₄⋊C₄×C₃₇	—	4	8	no	—
81	C₄⋊C₄×C₃₇	—	4	4	no	—
82	C₄⋊C₄×C₃₇	—	8	4	no	—
83	C₄⋊C₄×C₃₇	—	8	2	no	—
84	C₄⋊C₄×C₃₇	D₈	1	4	yes	—
85	C₄⋊C₄×C₃₇	Q₁₆	1	4	yes	—
86	C₄²×C₃₇	—	2	2	no	—
87	C₄²×C₃₇	—	2	2	no	—
88	C₄²×C₃₇	—	4	4	no	—
89	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
91	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	1	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	1	no	—
98	C₂×C₄×C₃₇	—	4	2	no	—
99	C₂×C₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	—	4	4	no	—
101	C₂×C₄×C₃₇	—	4	2	no	—
102	C₂×C₄×C₃₇	—	4	2	no	—
103	C₂×C₄×C₃₇	—	4	4	no	—
104	C₂×C₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	4	4	no	—
106	C₂×C₄×C₃₇	—	4	2	no	—
107	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₃₇	C₂×D₈	1	4	yes	—
110	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
113	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈×C₃₇	—	2	2	no	—
120	C₈×C₃₇	—	2	4	no	—
121	C₈×C₃₇	#125	1	4	yes	complemented
122	#28827	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₂²×C₃₇	#547	1	1	yes	characteristic
125	C₂²×C₃₇	#550	1	1	yes	characteristic
126	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
127	C₂²×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂²×C₃₇	#576	1	1	yes	characteristic
129	C₂²×C₃₇	#578	1	1	yes	characteristic
130	C₄×C₃₇	—	2	4	no	—
131	C₄×C₃₇	—	2	4	no	—
132	C₄×C₃₇	—	2	2	no	—
133	C₄×C₃₇	—	8	2	no	—
134	C₄×C₃₇	—	8	2	no	—
135	C₄×C₃₇	—	8	4	no	—
136	C₄×C₃₇	#500	1	4	yes	—
137	C₄×C₃₇	C₂×#125	1	1	yes	characteristic
138	C₄×C₃₇	#544	1	1	yes	characteristic
139	C₄×C₃₇	#554	1	1	yes	characteristic
140	C₄×C₃₇	#557	1	1	yes	characteristic
141	#2552	C₂×C₃₇	1	2	yes	—
142	#2083	C₂×C₃₇	1	4	yes	—
143	#1947	C₂×C₃₇	1	2	yes	—
144	#1942	C₂×C₃₇	1	2	yes	—
145	#1845	C₂×C₃₇	1	2	yes	—
146	#1845	C₂×C₃₇	1	1	yes	characteristic
147	C₄×#107	C₂×C₃₇	1	1	yes	characteristic
148	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
149	C₂×C₃₇	#2552	1	1	yes	characteristic
150	C₂×C₃₇	C₂×#500	1	1	yes	characteristic
151	C₂×C₃₇	#3164	1	1	yes	characteristic
152	C₂×C₃₇	#3334	1	1	yes	characteristic
153	C₂×C₃₇	#3338	1	1	yes	characteristic
154	C₂×C₃₇	#3391	1	1	yes	characteristic
155	C₂×C₃₇	#3395	1	1	yes	characteristic
156	C₂×#125	—	2	2	no	—
157	C₂×#107	C₂²×C₃₇	1	2	yes	—
158	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
160	C₂×#107	C₂²×C₃₇	1	2	yes	—
161	C₂×#107	C₂²×C₃₇	1	4	yes	—
162	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
165	#398	—	2	4	no	—
166	#389	—	2	2	no	—
167	#389	C₂²×C₃₇	1	4	yes	—
168	#384	C₂²×C₃₇	1	4	yes	—
169	#384	C₂²×C₃₇	1	2	yes	—
170	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
171	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
172	#340	C₂²×C₃₇	1	2	yes	—
173	#340	C₂²×C₃₇	1	4	yes	—
174	#336	C₂²×C₃₇	1	2	yes	—
175	C₃₇	#28827	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
178	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
179	#125	—	4	8	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
186	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
188	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
193	#107	—	2	2	no	—
194	#107	—	2	4	no	—
195	#107	—	2	2	no	—
196	#107	—	4	2	no	—
197	#107	—	4	4	no	—
198	C₄⋊C₈	Q₈×C₃₇	1	4	yes	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	Q₈×C₃₇	1	4	yes	—
201	#95	—	2	2	no	—
202	#95	—	2	4	no	—
203	#95	D₄×C₃₇	1	2	yes	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
211	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
217	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
218	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
219	C₂×C₈	#41×C₃₇	1	2	yes	—
220	C₂×C₈	#41×C₃₇	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	8	no	—
228	C₄⋊C₄	—	4	8	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	8	4	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₄⋊C₄	D₈×C₃₇	1	4	yes	—
233	C₄⋊C₄	Q₁₆×C₃₇	1	4	yes	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
238	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
239	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
256	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
257	C₂×C₄	C₂×D₈×C₃₇	1	4	yes	—
258	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
259	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
260	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
261	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
264	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
265	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
266	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	4	no	—
269	C₈	#125×C₃₇	1	4	yes	complemented
270	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
271	C₂²	C₃₇×#547	1	1	yes	characteristic
272	C₂²	C₃₇×#550	1	1	yes	characteristic
273	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
274	C₂²	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
275	C₂²	C₃₇×#576	1	1	yes	characteristic
276	C₂²	C₃₇×#578	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₃₇×#500	1	4	yes	—
284	C₄	C₂×#125×C₃₇	1	1	yes	characteristic
285	C₄	C₃₇×#544	1	1	yes	characteristic
286	C₄	C₃₇×#554	1	1	yes	characteristic
287	C₄	C₃₇×#557	1	1	yes	characteristic
288	C₂	C₃₇×#2552	1	1	yes	characteristic
289	C₂	C₂×C₃₇×#500	1	1	yes	characteristic
290	C₂	C₃₇×#3164	1	1	yes	characteristic
291	C₂	C₃₇×#3334	1	1	yes	characteristic
292	C₂	C₃₇×#3338	1	1	yes	characteristic
293	C₂	C₃₇×#3391	1	1	yes	characteristic
294	C₂	C₃₇×#3395	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄×C₄⋊C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
121. C₈×C₃₇	#125	semidirect
122. #28827	C₃₇	direct (N × H)
175. C₃₇	#28827	direct (N × H)
269. C₈	#125×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#2552	2
C₃₇×#2083	2
C₃₇×#1947	2
C₃₇×#1942	2
C₃₇×#1845	2
C₄×#107×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
#28827	37

Subgroups — C37×#28827 #22650080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28827 #22650080