Subgroups — C₃₇×#29810 #22651063

Order 9472.

1024 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2619	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2054	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1999	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1961	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1835	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₇	C₂²	1	4	yes	—
9	C₂×#127×C₃₇	C₂²	1	2	yes	—
10	C₂×#121×C₃₇	C₂²	1	4	yes	—
11	C₂×#117×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#493	—	2	4	no	—
13	C₃₇×#433	—	2	4	no	—
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₂×#102×C₃₇	C₂²	1	8	yes	—
16	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₃₇×#394	—	2	4	no	—
20	C₃₇×#392	—	2	4	no	—
21	C₃₇×#390	C₂²	1	4	yes	—
22	C₃₇×#382	C₂²	1	2	yes	—
23	C₃₇×#340	C₂²	1	4	yes	—
24	C₂²×D₄×C₃₇	—	2	4	no	—
25	C₂²×D₄×C₃₇	—	2	4	no	—
26	C₂²×D₄×C₃₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂×D₈×C₃₇	—	2	8	no	—
29	C₂×D₈×C₃₇	—	4	8	no	—
30	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
32	#127×C₃₇	—	2	4	no	—
33	#127×C₃₇	—	2	4	no	—
34	#121×C₃₇	—	4	8	no	—
35	#117×C₃₇	—	2	2	no	—
36	#117×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₃₇	—	2	8	no	—
41	C₂×C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
42	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₃₇	—	2	4	no	—
45	#102×C₃₇	—	4	8	no	—
46	C₂²⋊C₈×C₃₇	—	2	4	no	—
47	#97×C₃₇	—	2	2	no	—
48	C₄×C₈×C₃₇	D₄	1	4	yes	complemented
49	#95×C₃₇	—	2	4	no	—
50	C₂⁴×C₃₇	—	2	4	no	—
51	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×D₄×C₃₇	—	2	4	no	—
53	C₂×D₄×C₃₇	—	2	4	no	—
54	C₂×D₄×C₃₇	—	2	8	no	—
55	C₂×D₄×C₃₇	—	2	8	no	—
56	C₂×D₄×C₃₇	—	4	8	no	—
57	C₂×D₄×C₃₇	—	4	8	no	—
58	C₂×D₄×C₃₇	—	8	8	no	—
59	C₂×D₄×C₃₇	—	8	8	no	—
60	C₂²×C₄×C₃₇	—	2	1	no	—
61	C₂²×C₄×C₃₇	—	2	1	no	—
62	C₂²×C₄×C₃₇	—	2	4	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
68	D₈×C₃₇	—	4	16	no	—
69	C₂×C₈×C₃₇	—	2	4	no	—
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	—	2	4	no	—
72	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
73	C₄⋊C₄×C₃₇	—	2	4	no	—
74	C₄⋊C₄×C₃₇	—	2	8	no	—
75	C₂²⋊C₄×C₃₇	—	2	4	no	—
76	C₂²⋊C₄×C₃₇	—	4	8	no	—
77	C₂²⋊C₄×C₃₇	—	8	8	no	—
78	C₄²×C₃₇	—	2	1	no	—
79	C₄²×C₃₇	—	2	1	no	—
80	C₄²×C₃₇	C₂×D₄	1	2	yes	—
81	C₄²×C₃₇	#41	1	2	yes	—
82	C₂³×C₃₇	—	2	1	no	—
83	C₂³×C₃₇	—	2	2	no	—
84	C₂³×C₃₇	—	2	4	no	—
85	C₂³×C₃₇	—	4	4	no	—
86	C₂³×C₃₇	—	4	4	no	—
87	C₂³×C₃₇	—	4	4	no	—
88	C₂³×C₃₇	—	4	8	no	—
89	C₂³×C₃₇	—	4	8	no	—
90	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
91	D₄×C₃₇	—	4	8	no	—
92	D₄×C₃₇	—	4	8	no	—
93	D₄×C₃₇	—	8	16	no	—
94	D₄×C₃₇	—	8	16	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	1	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	4	no	—
100	C₂×C₄×C₃₇	—	4	2	no	—
101	C₂×C₄×C₃₇	—	4	2	no	—
102	C₂×C₄×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	—	4	8	no	—
104	C₂×C₄×C₃₇	#121	1	4	yes	—
105	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₃₇	H(5,𝔽₂)	1	4	yes	—
108	C₈×C₃₇	—	2	4	no	—
109	C₈×C₃₇	—	4	2	no	—
110	#29810	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₇	—	4	1	no	—
112	C₂²×C₃₇	—	4	4	no	—
113	C₂²×C₃₇	—	4	2	no	—
114	C₂²×C₃₇	—	8	4	no	—
115	C₂²×C₃₇	—	8	4	no	—
116	C₂²×C₃₇	—	8	4	no	—
117	C₂²×C₃₇	—	8	8	no	—
118	C₂²×C₃₇	—	8	8	no	—
119	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
120	C₂²×C₃₇	#535	1	2	yes	—
121	C₂²×C₃₇	#576	1	4	yes	—
122	C₄×C₃₇	—	2	4	no	—
123	C₄×C₃₇	—	4	1	no	—
124	C₄×C₃₇	—	4	2	no	—
125	C₄×C₃₇	—	8	4	no	—
126	C₄×C₃₇	#535	1	2	yes	—
127	C₄×C₃₇	#540	1	2	yes	—
128	C₂×#493	C₂×C₃₇	1	1	yes	characteristic, complemented
129	C₂×#433	C₂×C₃₇	1	1	yes	characteristic, complemented
130	#2619	C₂×C₃₇	1	2	yes	complemented
131	#2054	C₂×C₃₇	1	4	yes	complemented
132	#1999	C₂×C₃₇	1	4	yes	complemented
133	#1961	C₂×C₃₇	1	2	yes	complemented
134	#1835	C₂×C₃₇	1	1	yes	characteristic, complemented
135	C₂×C₃₇	—	8	1	no	—
136	C₂×C₃₇	—	16	4	no	—
137	C₂×C₃₇	#2619	1	1	yes	characteristic
138	C₂×C₃₇	#3058	1	2	yes	—
139	C₂×C₃₇	#3220	1	4	yes	—
140	C₂²×D₈	C₂²×C₃₇	1	4	yes	—
141	C₂×#127	C₂²×C₃₇	1	2	yes	—
142	C₂×#121	C₂²×C₃₇	1	4	yes	—
143	C₂×#117	C₂²×C₃₇	1	1	yes	characteristic
144	#493	—	2	4	no	—
145	#433	—	2	4	no	—
146	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
147	C₂×#102	C₂²×C₃₇	1	8	yes	—
148	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
149	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
151	#394	—	2	4	no	—
152	#392	—	2	4	no	—
153	#390	C₂²×C₃₇	1	4	yes	—
154	#382	C₂²×C₃₇	1	2	yes	—
155	#340	C₂²×C₃₇	1	4	yes	—
156	C₃₇	#29810	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×D₄	—	2	4	no	—
158	C₂²×D₄	—	2	4	no	—
159	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
160	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
161	C₂×D₈	—	2	8	no	—
162	C₂×D₈	—	4	8	no	—
163	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
164	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
165	#127	—	2	4	no	—
166	#127	—	2	4	no	—
167	#121	—	4	8	no	—
168	#117	—	2	2	no	—
169	#117	—	2	2	no	—
170	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
171	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₂²⋊C₄	—	2	8	no	—
174	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
175	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	#102	—	4	8	no	—
179	C₂²⋊C₈	—	2	4	no	—
180	#97	—	2	2	no	—
181	C₄×C₈	D₄×C₃₇	1	4	yes	complemented
182	#95	—	2	4	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	8	no	—
188	C₂×D₄	—	2	8	no	—
189	C₂×D₄	—	4	8	no	—
190	C₂×D₄	—	4	8	no	—
191	C₂×D₄	—	8	8	no	—
192	C₂×D₄	—	8	8	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
200	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
201	D₈	—	4	16	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₂²⋊C₄	—	8	8	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂×D₄×C₃₇	1	2	yes	—
214	C₄²	#41×C₃₇	1	2	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	8	no	—
222	C₂³	—	4	8	no	—
223	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
224	D₄	—	4	8	no	—
225	D₄	—	4	8	no	—
226	D₄	—	8	16	no	—
227	D₄	—	8	16	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	#121×C₃₇	1	4	yes	—
238	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
240	C₂×C₄	H(5,𝔽₂)×C₃₇	1	4	yes	—
241	C₈	—	2	4	no	—
242	C₈	—	4	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	8	no	—
250	C₂²	—	8	8	no	—
251	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
252	C₂²	C₃₇×#535	1	2	yes	—
253	C₂²	C₃₇×#576	1	4	yes	—
254	C₄	—	2	4	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	8	4	no	—
258	C₄	C₃₇×#535	1	2	yes	—
259	C₄	C₃₇×#540	1	2	yes	—
260	C₂	—	8	1	no	—
261	C₂	—	16	4	no	—
262	C₂	C₃₇×#2619	1	1	yes	characteristic
263	C₂	C₃₇×#3058	1	2	yes	—
264	C₂	C₃₇×#3220	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×#117×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#493	C₂	semidirect
2. C₂×C₃₇×#433	C₂	semidirect
3. C₃₇×#2619	C₂	semidirect
4. C₃₇×#2054	C₂	semidirect
5. C₃₇×#1999	C₂	semidirect
6. C₃₇×#1961	C₂	semidirect
7. C₃₇×#1835	C₂	semidirect
48. C₄×C₈×C₃₇	D₄	semidirect
110. #29810	C₃₇	direct (N × H)
128. C₂×#493	C₂×C₃₇	semidirect
129. C₂×#433	C₂×C₃₇	semidirect
130. #2619	C₂×C₃₇	semidirect
131. #2054	C₂×C₃₇	semidirect
132. #1999	C₂×C₃₇	semidirect
133. #1961	C₂×C₃₇	semidirect
134. #1835	C₂×C₃₇	semidirect
156. C₃₇	#29810	direct (N × H)
181. C₄×C₈	D₄×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×#493	2
C₂×C₃₇×#433	2
C₃₇×#2619	2
C₃₇×#2054	2
C₃₇×#1999	2
C₃₇×#1961	2
C₃₇×#1835	2
#29810	37

Subgroups — C37×#29810 #22651063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29810 #22651063