Subgroups — C₃₇×#29820 #22651073

Order 9472.

780 conjugacy classes of subgroups.

The subgroup lattice has 258 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, complemented
2	C₃₇×#3039	C₂	1	2	yes	maximal
3	C₃₇×#2922	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2047	C₂	1	4	yes	maximal, complemented
5	C₃₇×#2020	C₂	1	4	yes	maximal
6	C₃₇×#1966	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₃₇	C₂²	1	4	yes	—
9	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₄×#41×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#494	—	2	2	no	—
13	C₃₇×#494	—	2	2	no	—
14	C₃₇×#462	—	2	4	no	—
15	C₃₇×#461	—	2	4	no	—
16	C₃₇×#433	—	2	2	no	—
17	C₃₇×#431	—	2	2	no	—
18	C₃₇×#424	C₂²	1	2	yes	—
19	C₃₇×#423	C₂²	1	2	yes	—
20	C₃₇×#419	C₂²	1	4	yes	—
21	C₂×#101×C₃₇	C₂²	1	8	yes	—
22	C₃₇×#408	C₂²	1	2	yes	—
23	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#341	C₂²	1	4	yes	—
26	C₂×#41×C₃₇	C₂³	1	1	yes	characteristic
27	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
28	#137×C₃₇	—	4	8	no	—
29	C₂×Q₁₆×C₃₇	—	2	4	no	—
30	C₂×Q₁₆×C₃₇	—	2	8	no	—
31	C₂×Q₁₆×C₃₇	—	4	8	no	—
32	C₂×SD₁₆×C₃₇	—	2	4	no	—
33	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
34	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₃₇	—	2	4	no	—
37	C₄⋊Q₈×C₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₃₇	—	2	2	no	—
39	C₄×Q₈×C₃₇	D₄	1	2	yes	—
40	C₄×D₄×C₃₇	—	2	2	no	—
41	C₄×D₄×C₃₇	D₄	1	2	yes	—
42	#117×C₃₇	—	2	1	no	—
43	#117×C₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
45	C₂×C₄²×C₃₇	—	2	1	no	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₃₇	—	2	4	no	—
48	C₄⋊C₈×C₃₇	—	2	4	no	—
49	#103×C₃₇	—	2	8	no	—
50	#101×C₃₇	—	4	8	no	—
51	C₂²⋊C₈×C₃₇	—	2	4	no	—
52	#97×C₃₇	—	2	1	no	—
53	C₄×C₈×C₃₇	—	2	1	no	—
54	C₄×C₈×C₃₇	D₄	1	2	yes	—
55	C₄×C₈×C₃₇	D₄	1	2	yes	—
56	#41×C₃₇	—	2	4	no	—
57	C₂×Q₈×C₃₇	—	2	4	no	—
58	C₂×Q₈×C₃₇	—	2	4	no	—
59	C₂×Q₈×C₃₇	—	2	8	no	—
60	C₂×Q₈×C₃₇	—	2	8	no	—
61	C₂×Q₈×C₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂×D₄×C₃₇	—	2	1	no	—
63	C₂×D₄×C₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	—	2	2	no	—
65	C₂²×C₄×C₃₇	—	2	1	no	—
66	C₂²×C₄×C₃₇	—	2	4	no	—
67	C₂²×C₄×C₃₇	#41	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₃₇	—	4	16	no	—
70	Q₁₆×C₃₇	—	4	8	no	—
71	SD₁₆×C₃₇	—	4	8	no	—
72	M₁₆×C₃₇	—	2	4	no	—
73	M₁₆×C₃₇	—	2	8	no	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	—	2	2	no	—
76	C₂×C₈×C₃₇	—	2	4	no	—
77	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₃₇	—	2	2	no	—
79	C₄⋊C₄×C₃₇	—	4	4	no	—
80	C₄⋊C₄×C₃₇	—	4	4	no	—
81	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²⋊C₄×C₃₇	—	2	1	no	—
83	C₂²⋊C₄×C₃₇	—	2	2	no	—
84	C₄²×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	—	2	1	no	—
86	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂×D₄	1	2	yes	—
89	C₂³×C₃₇	—	2	1	no	—
90	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
91	Q₈×C₃₇	—	2	2	no	—
92	Q₈×C₃₇	—	4	4	no	—
93	Q₈×C₃₇	—	4	8	no	—
94	Q₈×C₃₇	—	4	4	no	—
95	D₄×C₃₇	—	2	2	no	—
96	D₄×C₃₇	—	4	2	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	1	no	—
100	C₂×C₄×C₃₇	—	4	2	no	—
101	C₂×C₄×C₃₇	—	4	4	no	—
102	C₂×C₄×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	—	4	8	no	—
104	C₂×C₄×C₃₇	#121	1	2	yes	—
105	C₂×C₄×C₃₇	#121	1	2	yes	—
106	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₇	H(5,𝔽₂)	1	4	yes	—
109	C₈×C₃₇	—	2	4	no	—
110	C₈×C₃₇	—	4	2	no	—
111	C₈×C₃₇	—	4	4	no	—
112	#29820	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₃₇	—	2	2	no	—
114	C₂²×C₃₇	—	4	1	no	—
115	C₂²×C₃₇	—	4	2	no	—
116	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
117	C₂²×C₃₇	#540	1	2	yes	—
118	C₄×C₃₇	—	2	2	no	—
119	C₄×C₃₇	—	4	1	no	—
120	C₄×C₃₇	—	4	2	no	—
121	C₄×C₃₇	—	8	4	no	—
122	C₄×C₃₇	#473	1	4	yes	—
123	C₄×C₃₇	#535	1	2	yes	—
124	C₄×C₃₇	#535	1	2	yes	—
125	C₂×#494	C₂×C₃₇	1	1	yes	characteristic, complemented
126	#3039	C₂×C₃₇	1	2	yes	—
127	#2922	C₂×C₃₇	1	1	yes	characteristic
128	#2047	C₂×C₃₇	1	4	yes	complemented
129	#2020	C₂×C₃₇	1	4	yes	—
130	#1966	C₂×C₃₇	1	2	yes	complemented
131	#1837	C₂×C₃₇	1	1	yes	characteristic, complemented
132	C₂×C₃₇	—	2	2	no	—
133	C₂×C₃₇	—	8	1	no	—
134	C₂×C₃₇	#2619	1	1	yes	characteristic
135	C₂×C₃₇	C₂×#473	1	2	yes	—
136	C₂×#137	C₂²×C₃₇	1	4	yes	—
137	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
138	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
139	C₄×#41	C₂²×C₃₇	1	1	yes	characteristic
140	#494	—	2	2	no	—
141	#494	—	2	2	no	—
142	#462	—	2	4	no	—
143	#461	—	2	4	no	—
144	#433	—	2	2	no	—
145	#431	—	2	2	no	—
146	#424	C₂²×C₃₇	1	2	yes	—
147	#423	C₂²×C₃₇	1	2	yes	—
148	#419	C₂²×C₃₇	1	4	yes	—
149	C₂×#101	C₂²×C₃₇	1	8	yes	—
150	#408	C₂²×C₃₇	1	2	yes	—
151	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
153	#341	C₂²×C₃₇	1	4	yes	—
154	C₃₇	#29820	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#41	C₂³×C₃₇	1	1	yes	characteristic
156	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
157	#137	—	4	8	no	—
158	C₂×Q₁₆	—	2	4	no	—
159	C₂×Q₁₆	—	2	8	no	—
160	C₂×Q₁₆	—	4	8	no	—
161	C₂×SD₁₆	—	2	4	no	—
162	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
163	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
164	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
165	C₄⋊Q₈	—	2	4	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄×Q₈	D₄×C₃₇	1	2	yes	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	D₄×C₃₇	1	2	yes	—
171	#117	—	2	1	no	—
172	#117	C₂³×C₃₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	4	no	—
178	#103	—	2	8	no	—
179	#101	—	4	8	no	—
180	C₂²⋊C₈	—	2	4	no	—
181	#97	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	D₄×C₃₇	1	2	yes	—
184	C₄×C₈	D₄×C₃₇	1	2	yes	—
185	#41	—	2	4	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	2	8	no	—
189	C₂×Q₈	—	2	8	no	—
190	C₂×Q₈	C₂×D₄×C₃₇	1	1	yes	characteristic
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₂×D₄×C₃₇	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	#41×C₃₇	1	2	yes	—
197	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
198	Q₁₆	—	4	16	no	—
199	Q₁₆	—	4	8	no	—
200	SD₁₆	—	4	8	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
217	C₄²	C₂×D₄×C₃₇	1	2	yes	—
218	C₂³	—	2	1	no	—
219	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	8	no	—
223	Q₈	—	4	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	#121×C₃₇	1	2	yes	—
234	C₂×C₄	#121×C₃₇	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
237	C₂×C₄	H(5,𝔽₂)×C₃₇	1	4	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	4	4	no	—
240	C₈	—	4	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
245	C₂²	C₃₇×#540	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	8	4	no	—
250	C₄	C₃₇×#473	1	4	yes	—
251	C₄	C₃₇×#535	1	2	yes	—
252	C₄	C₃₇×#535	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	8	1	no	—
255	C₂	C₃₇×#2619	1	1	yes	characteristic
256	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	C₂×#41×C₃₇	C₂³
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
1. C₂×C₃₇×#494	C₂	semidirect
4. C₃₇×#2047	C₂	semidirect
6. C₃₇×#1966	C₂	semidirect
7. C₃₇×#1837	C₂	semidirect
112. #29820	C₃₇	direct (N × H)
125. C₂×#494	C₂×C₃₇	semidirect
128. #2047	C₂×C₃₇	semidirect
130. #1966	C₂×C₃₇	semidirect
131. #1837	C₂×C₃₇	semidirect
154. C₃₇	#29820	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×#3039	2
C₃₇×#2922	2
C₃₇×#2047	2
C₃₇×#2020	2
C₃₇×#1966	2
C₃₇×#1837	2
#29820	37

Subgroups — C37×#29820 #22651073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29820 #22651073