Subgroups — C₃₇×#28762 #22650015

Order 9472.

726 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3139	C₂	1	2	yes	maximal
3	C₃₇×#2960	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2047	C₂	1	4	yes	maximal
5	C₃₇×#1979	C₂	1	4	yes	maximal
6	C₃₇×#1966	C₂	1	2	yes	maximal
7	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
10	C₃₇×#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#494	—	2	2	no	—
13	C₃₇×#494	—	2	2	no	—
14	C₃₇×#491	—	2	4	no	—
15	C₃₇×#487	—	2	4	no	—
16	C₃₇×#446	—	2	2	no	—
17	C₈×Q₈×C₃₇	—	2	2	no	—
18	C₂×#108×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#424	C₂²	1	2	yes	—
20	C₃₇×#423	C₂²	1	1	yes	characteristic
21	C₃₇×#419	C₂²	1	8	yes	—
22	C₂×#101×C₃₇	C₂²	1	8	yes	—
23	C₃₇×#405	C₂²	1	2	yes	—
24	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#341	C₂²	1	2	yes	—
27	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
28	C₂×Q₁₆×C₃₇	—	2	4	no	—
29	C₂×Q₁₆×C₃₇	—	2	2	no	—
30	C₂×Q₁₆×C₃₇	—	2	2	no	—
31	C₂×Q₁₆×C₃₇	—	2	8	no	—
32	C₂×Q₁₆×C₃₇	—	4	4	no	—
33	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
34	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₃₇	—	2	2	no	—
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₃₇	D₄	1	2	yes	—
41	#125×C₃₇	D₄	1	2	yes	—
42	C₂²⋊Q₈×C₃₇	—	2	2	no	—
43	C₄×Q₈×C₃₇	—	2	2	no	—
44	#117×C₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	#108×C₃₇	—	2	4	no	—
48	C₄⋊C₈×C₃₇	—	2	4	no	—
49	C₄⋊C₈×C₃₇	—	2	2	no	—
50	#103×C₃₇	—	2	16	no	—
51	#101×C₃₇	—	4	8	no	—
52	#97×C₃₇	—	2	2	no	—
53	#97×C₃₇	—	2	2	no	—
54	C₄×C₈×C₃₇	—	2	2	no	—
55	C₄×C₈×C₃₇	—	2	2	no	—
56	C₄×C₈×C₃₇	D₄	1	2	yes	—
57	C₄×C₈×C₃₇	D₄	1	2	yes	—
58	C₂×Q₈×C₃₇	—	2	2	no	—
59	C₂×Q₈×C₃₇	—	2	4	no	—
60	C₂×Q₈×C₃₇	—	2	4	no	—
61	C₂×Q₈×C₃₇	—	2	8	no	—
62	C₂×Q₈×C₃₇	—	2	8	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	#41	1	2	yes	—
65	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
66	Q₁₆×C₃₇	—	4	8	no	—
67	Q₁₆×C₃₇	—	4	4	no	—
68	Q₁₆×C₃₇	—	4	4	no	—
69	M₁₆×C₃₇	—	2	4	no	—
70	M₁₆×C₃₇	—	2	4	no	—
71	M₁₆×C₃₇	—	2	4	no	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	—	2	2	no	—
75	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₃₇	#41	1	4	yes	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	2	2	no	—
81	C₄⋊C₄×C₃₇	—	4	4	no	—
82	C₄⋊C₄×C₃₇	—	4	4	no	—
83	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
84	C₂²⋊C₄×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	—	2	2	no	—
86	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
90	Q₈×C₃₇	—	4	4	no	—
91	Q₈×C₃₇	—	4	4	no	—
92	Q₈×C₃₇	—	4	4	no	—
93	Q₈×C₃₇	—	4	8	no	—
94	C₂×C₄×C₃₇	—	2	4	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	4	4	no	—
97	C₂×C₄×C₃₇	—	4	8	no	—
98	C₂×C₄×C₃₇	#121	1	2	yes	—
99	C₂×C₄×C₃₇	#121	1	2	yes	—
100	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
104	C₈×C₃₇	—	2	2	no	—
105	C₈×C₃₇	—	4	2	no	—
106	C₈×C₃₇	—	4	4	no	—
107	C₈×C₃₇	#124	1	2	yes	—
108	C₈×C₃₇	#124	1	2	yes	—
109	#28762	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₃₇	—	2	2	no	—
111	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
112	C₂²×C₃₇	#559	1	2	yes	—
113	C₄×C₃₇	—	2	2	no	—
114	C₄×C₃₇	—	4	4	no	—
115	C₄×C₃₇	—	8	4	no	—
116	C₄×C₃₇	#473	1	4	yes	—
117	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
118	C₄×C₃₇	#535	1	1	yes	characteristic
119	C₄×C₃₇	#550	1	2	yes	—
120	C₂×#494	C₂×C₃₇	1	1	yes	characteristic
121	#3139	C₂×C₃₇	1	2	yes	—
122	#2960	C₂×C₃₇	1	1	yes	characteristic
123	#2047	C₂×C₃₇	1	4	yes	—
124	#1979	C₂×C₃₇	1	4	yes	—
125	#1966	C₂×C₃₇	1	2	yes	—
126	C₄×#108	C₂×C₃₇	1	1	yes	characteristic
127	C₂×C₃₇	—	2	2	no	—
128	C₂×C₃₇	#2548	1	1	yes	characteristic
129	C₂×C₃₇	C₂×#473	1	2	yes	—
130	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
131	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
132	#533	C₂²×C₃₇	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
134	#494	—	2	2	no	—
135	#494	—	2	2	no	—
136	#491	—	2	4	no	—
137	#487	—	2	4	no	—
138	#446	—	2	2	no	—
139	C₈×Q₈	—	2	2	no	—
140	C₂×#108	C₂²×C₃₇	1	2	yes	—
141	#424	C₂²×C₃₇	1	2	yes	—
142	#423	C₂²×C₃₇	1	1	yes	characteristic
143	#419	C₂²×C₃₇	1	8	yes	—
144	C₂×#101	C₂²×C₃₇	1	8	yes	—
145	#405	C₂²×C₃₇	1	2	yes	—
146	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
148	#341	C₂²×C₃₇	1	2	yes	—
149	C₃₇	#28762	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	8	no	—
155	C₂×Q₁₆	—	4	4	no	—
156	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
157	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
158	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	D₄×C₃₇	1	2	yes	—
164	#125	D₄×C₃₇	1	2	yes	—
165	C₂²⋊Q₈	—	2	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	#117	C₂³×C₃₇	1	2	yes	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
170	#108	—	2	4	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#103	—	2	16	no	—
174	#101	—	4	8	no	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	D₄×C₃₇	1	2	yes	—
180	C₄×C₈	D₄×C₃₇	1	2	yes	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	—	2	8	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	#41×C₃₇	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
189	Q₁₆	—	4	8	no	—
190	Q₁₆	—	4	4	no	—
191	Q₁₆	—	4	4	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
201	C₂×C₈	#41×C₃₇	1	4	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₃₇	1	2	yes	—
211	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
212	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
213	Q₈	—	4	4	no	—
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	4	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	#121×C₃₇	1	2	yes	—
222	C₂×C₄	#121×C₃₇	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
225	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
226	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	4	2	no	—
229	C₈	—	4	4	no	—
230	C₈	#124×C₃₇	1	2	yes	—
231	C₈	#124×C₃₇	1	2	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
234	C₂²	C₃₇×#559	1	2	yes	—
235	C₄	—	2	2	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	4	no	—
238	C₄	C₃₇×#473	1	4	yes	—
239	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
240	C₄	C₃₇×#535	1	1	yes	characteristic
241	C₄	C₃₇×#550	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	C₃₇×#2548	1	1	yes	characteristic
244	C₂	C₂×C₃₇×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
109. #28762	C₃₇	direct (N × H)
149. C₃₇	#28762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#494	2
C₃₇×#3139	2
C₃₇×#2960	2
C₃₇×#2047	2
C₃₇×#1979	2
C₃₇×#1966	2
C₄×#108×C₃₇	2
#28762	37

Subgroups — C37×#28762 #22650015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28762 #22650015