Subgroups — C₃₇×#28695 #22649948

Order 9472.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#501	C₂	1	2	yes	maximal
2	C₃₇×#2862	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2085	C₂	1	4	yes	maximal
5	C₃₇×#1993	C₂	1	2	yes	maximal
6	C₃₇×#1949	C₂	1	4	yes	maximal
7	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
9	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#501	—	2	8	no	—
11	C₂×#107×C₃₇	C₂²	1	4	yes	—
12	C₂×#106×C₃₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×#97×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#391	—	2	2	no	—
16	C₃₇×#391	—	2	2	no	—
17	C₃₇×#389	C₂²	1	2	yes	—
18	C₃₇×#387	—	2	2	no	—
19	C₃₇×#382	C₂²	1	2	yes	—
20	C₃₇×#382	C₂²	1	2	yes	—
21	C₃₇×#382	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₄³×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#340	C₂²	1	8	yes	—
26	C₃₇×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₃₇	—	2	2	no	—
28	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
29	C₄⋊Q₈×C₃₇	—	2	4	no	—
30	C₄⋊Q₈×C₃₇	—	2	4	no	—
31	#125×C₃₇	—	2	2	no	—
32	#125×C₃₇	D₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
39	C₂×C₄²×C₃₇	—	2	2	no	—
40	C₂×C₄²×C₃₇	D₄	1	2	yes	—
41	C₂×C₄²×C₃₇	D₄	1	2	yes	—
42	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
45	#107×C₃₇	—	2	8	no	—
46	#106×C₃₇	—	2	8	no	—
47	C₄⋊C₈×C₃₇	Q₈	1	8	yes	complemented
48	#97×C₃₇	—	2	4	no	—
49	#95×C₃₇	—	2	2	no	—
50	#95×C₃₇	—	2	2	no	—
51	#95×C₃₇	—	2	2	no	—
52	#95×C₃₇	—	2	4	no	—
53	#95×C₃₇	—	2	2	no	—
54	#95×C₃₇	—	2	2	no	—
55	C₂×Q₈×C₃₇	—	4	4	no	—
56	C₂×Q₈×C₃₇	—	4	4	no	—
57	C₂×Q₈×C₃₇	—	4	4	no	—
58	C₂²×C₄×C₃₇	—	2	2	no	—
59	C₂²×C₄×C₃₇	—	2	2	no	—
60	C₂²×C₄×C₃₇	—	2	4	no	—
61	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₃₇	—	2	4	no	—
67	C₂×C₈×C₃₇	—	2	4	no	—
68	C₂×C₈×C₃₇	C₂×Q₈	1	8	yes	—
69	C₄⋊C₄×C₃₇	—	2	8	no	—
70	C₄⋊C₄×C₃₇	—	2	2	no	—
71	C₄⋊C₄×C₃₇	—	2	4	no	—
72	C₄⋊C₄×C₃₇	—	2	4	no	—
73	C₄⋊C₄×C₃₇	—	2	4	no	—
74	C₄⋊C₄×C₃₇	—	4	4	no	—
75	C₄⋊C₄×C₃₇	—	4	4	no	—
76	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
77	C₄²×C₃₇	—	2	2	no	—
78	C₄²×C₃₇	—	4	2	no	—
79	C₄²×C₃₇	—	4	2	no	—
80	C₄²×C₃₇	—	4	4	no	—
81	C₄²×C₃₇	C₂×D₄	1	2	yes	—
82	C₄²×C₃₇	C₂×Q₈	1	4	yes	—
83	C₄²×C₃₇	#41	1	2	yes	—
84	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₃₇	—	8	8	no	—
86	C₂×C₄×C₃₇	—	2	2	no	—
87	C₂×C₄×C₃₇	—	2	2	no	—
88	C₂×C₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	—	2	4	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	4	2	no	—
95	C₂×C₄×C₃₇	—	4	2	no	—
96	C₂×C₄×C₃₇	—	4	4	no	—
97	C₂×C₄×C₃₇	—	4	2	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	4	no	—
102	C₂×C₄×C₃₇	#121	1	4	yes	—
103	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₇	C₂²×Q₈	1	2	yes	—
107	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
109	C₈×C₃₇	—	2	8	no	—
110	#28695	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
112	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₃₇	#536	1	1	yes	characteristic
114	C₂²×C₃₇	#575	1	2	yes	—
115	C₂²×C₃₇	#575	1	2	yes	—
116	C₄×C₃₇	—	2	2	no	—
117	C₄×C₃₇	—	2	2	no	—
118	C₄×C₃₇	—	2	4	no	—
119	C₄×C₃₇	—	4	2	no	—
120	C₄×C₃₇	—	4	4	no	—
121	C₄×C₃₇	—	8	2	no	—
122	C₄×C₃₇	—	8	4	no	—
123	C₄×C₃₇	D₄×Q₈	1	2	yes	—
124	C₄×C₃₇	#557	1	2	yes	—
125	C₂×#501	C₂×C₃₇	1	2	yes	—
126	#2862	C₂×C₃₇	1	1	yes	characteristic
127	C₄×#125	C₂×C₃₇	1	1	yes	characteristic
128	#2085	C₂×C₃₇	1	4	yes	—
129	#1993	C₂×C₃₇	1	2	yes	—
130	#1949	C₂×C₃₇	1	4	yes	—
131	#1767	C₂×C₃₇	1	1	yes	characteristic
132	C₂×C₃₇	#2546	1	1	yes	characteristic
133	C₂×C₃₇	#3040	1	2	yes	—
134	C₂×C₃₇	#3162	1	1	yes	characteristic
135	C₂×C₃₇	#3166	1	1	yes	characteristic
136	C₂×C₃₇	#3240	1	2	yes	—
137	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
138	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
139	#501	—	2	8	no	—
140	C₂×#107	C₂²×C₃₇	1	4	yes	—
141	C₂×#106	C₂²×C₃₇	1	4	yes	—
142	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
143	C₂×#97	C₂²×C₃₇	1	2	yes	—
144	#391	—	2	2	no	—
145	#391	—	2	2	no	—
146	#389	C₂²×C₃₇	1	2	yes	—
147	#387	—	2	2	no	—
148	#382	C₂²×C₃₇	1	1	yes	characteristic
149	#382	C₂²×C₃₇	1	2	yes	—
150	#382	C₂²×C₃₇	1	2	yes	—
151	C₄×C₄⋊C₄	—	2	2	no	—
152	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
153	C₄³	C₂²×C₃₇	1	1	yes	characteristic
154	#340	C₂²×C₃₇	1	8	yes	—
155	#336	C₂²×C₃₇	1	2	yes	—
156	C₃₇	#28695	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₈	—	2	2	no	—
158	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
159	C₄⋊Q₈	—	2	4	no	—
160	C₄⋊Q₈	—	2	4	no	—
161	#125	—	2	2	no	—
162	#125	D₄×C₃₇	1	4	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
167	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	D₄×C₃₇	1	2	yes	—
171	C₂×C₄²	D₄×C₃₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
175	#107	—	2	8	no	—
176	#106	—	2	8	no	—
177	C₄⋊C₈	Q₈×C₃₇	1	8	yes	complemented
178	#97	—	2	4	no	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	#95	—	2	4	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂×Q₈×C₃₇	1	8	yes	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
207	C₄²	—	2	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×D₄×C₃₇	1	2	yes	—
212	C₄²	C₂×Q₈×C₃₇	1	4	yes	—
213	C₄²	#41×C₃₇	1	2	yes	—
214	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
215	Q₈	—	8	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	#121×C₃₇	1	4	yes	—
233	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
234	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×Q₈×C₃₇	1	2	yes	—
237	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
238	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
239	C₈	—	2	8	no	—
240	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
242	C₂²	C₃₇×#536	1	1	yes	characteristic
243	C₂²	C₃₇×#575	1	2	yes	—
244	C₂²	C₃₇×#575	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	4	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	4	no	—
252	C₄	D₄×Q₈×C₃₇	1	2	yes	—
253	C₄	C₃₇×#557	1	2	yes	—
254	C₂	C₃₇×#2546	1	1	yes	characteristic
255	C₂	C₃₇×#3040	1	2	yes	—
256	C₂	C₃₇×#3162	1	1	yes	characteristic
257	C₂	C₃₇×#3166	1	1	yes	characteristic
258	C₂	C₃₇×#3240	1	2	yes	—

	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 ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
47. C₄⋊C₈×C₃₇	Q₈	semidirect
110. #28695	C₃₇	direct (N × H)
156. C₃₇	#28695	direct (N × H)
177. C₄⋊C₈	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#501	2
C₃₇×#2862	2
C₄×#125×C₃₇	2
C₃₇×#2085	2
C₃₇×#1993	2
C₃₇×#1949	2
C₃₇×#1767	2
#28695	37

Subgroups — C37×#28695 #22649948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28695 #22649948