Subgroups — C₃₇×#29818 #22651071

Order 9472.

884 conjugacy classes of subgroups.

The subgroup lattice has 272 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	C₃₇×#3038	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2045	C₂	1	4	yes	maximal, complemented
5	C₃₇×#2018	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1965	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₃₇	C₂²	1	4	yes	—
9	C₂²×D₈×C₃₇	C₂²	1	4	yes	—
10	C₂×#127×C₃₇	C₂²	1	2	yes	—
11	C₄×#41×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#493	—	2	2	no	—
13	C₃₇×#493	—	2	2	no	—
14	C₃₇×#460	—	2	4	no	—
15	C₃₇×#459	—	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×#418	C₂²	1	4	yes	—
21	C₂×#100×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×D₄×C₃₇	—	2	4	no	—
28	C₂²×D₄×C₃₇	C₂³	1	4	yes	—
29	AGL(1,ℤ/8)×C₃₇	—	4	8	no	—
30	C₂×SD₁₆×C₃₇	—	2	4	no	—
31	C₂×D₈×C₃₇	—	2	4	no	—
32	C₂×D₈×C₃₇	—	2	8	no	—
33	C₂×D₈×C₃₇	—	4	8	no	—
34	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
35	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	#127×C₃₇	—	2	4	no	—
38	#127×C₃₇	—	2	2	no	—
39	#127×C₃₇	—	2	2	no	—
40	C₄×Q₈×C₃₇	D₄	1	2	yes	—
41	C₄×D₄×C₃₇	—	2	2	no	—
42	C₄×D₄×C₃₇	D₄	1	2	yes	—
43	#117×C₃₇	—	2	1	no	—
44	#117×C₃₇	C₂³	1	1	yes	characteristic
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	#102×C₃₇	—	2	8	no	—
50	#100×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	C₂⁴×C₃₇	—	2	4	no	—
57	#41×C₃₇	—	2	4	no	—
58	C₂×Q₈×C₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×D₄×C₃₇	—	2	1	no	—
60	C₂×D₄×C₃₇	—	2	4	no	—
61	C₂×D₄×C₃₇	—	2	4	no	—
62	C₂×D₄×C₃₇	—	2	8	no	—
63	C₂×D₄×C₃₇	—	2	8	no	—
64	C₂×D₄×C₃₇	—	4	4	no	—
65	C₂×D₄×C₃₇	—	4	4	no	—
66	C₂×D₄×C₃₇	—	8	8	no	—
67	C₂×D₄×C₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₇	—	2	2	no	—
69	C₂²×C₄×C₃₇	—	2	1	no	—
70	C₂²×C₄×C₃₇	#41	1	2	yes	—
71	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
72	SD₁₆×C₃₇	—	4	8	no	—
73	D₈×C₃₇	—	4	16	no	—
74	D₈×C₃₇	—	4	8	no	—
75	M₁₆×C₃₇	—	2	4	no	—
76	M₁₆×C₃₇	—	2	8	no	—
77	C₂×C₈×C₃₇	—	2	4	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	4	no	—
80	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²⋊C₄×C₃₇	—	2	1	no	—
84	C₂²⋊C₄×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	—	2	2	no	—
86	C₄²×C₃₇	—	2	1	no	—
87	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₇	C₂×D₄	1	2	yes	—
90	C₂³×C₃₇	—	2	1	no	—
91	C₂³×C₃₇	—	4	4	no	—
92	C₂³×C₃₇	—	4	8	no	—
93	C₂³×C₃₇	—	8	8	no	—
94	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
95	Q₈×C₃₇	—	2	2	no	—
96	D₄×C₃₇	—	2	2	no	—
97	D₄×C₃₇	—	4	2	no	—
98	D₄×C₃₇	—	4	8	no	—
99	D₄×C₃₇	—	4	4	no	—
100	D₄×C₃₇	—	4	4	no	—
101	D₄×C₃₇	—	8	8	no	—
102	D₄×C₃₇	—	8	8	no	—
103	C₂×C₄×C₃₇	—	2	4	no	—
104	C₂×C₄×C₃₇	—	2	2	no	—
105	C₂×C₄×C₃₇	—	2	1	no	—
106	C₂×C₄×C₃₇	—	4	2	no	—
107	C₂×C₄×C₃₇	—	4	4	no	—
108	C₂×C₄×C₃₇	#121	1	2	yes	—
109	C₂×C₄×C₃₇	#121	1	2	yes	—
110	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	H(5,𝔽₂)	1	4	yes	—
113	C₈×C₃₇	—	2	4	no	—
114	C₈×C₃₇	—	4	2	no	—
115	C₈×C₃₇	—	4	4	no	—
116	#29818	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₃₇	—	2	2	no	—
118	C₂²×C₃₇	—	4	1	no	—
119	C₂²×C₃₇	—	4	2	no	—
120	C₂²×C₃₇	—	8	4	no	—
121	C₂²×C₃₇	—	8	8	no	—
122	C₂²×C₃₇	—	16	8	no	—
123	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
124	C₂²×C₃₇	#540	1	2	yes	—
125	C₄×C₃₇	—	2	2	no	—
126	C₄×C₃₇	—	4	1	no	—
127	C₄×C₃₇	—	4	2	no	—
128	C₄×C₃₇	#472	1	4	yes	—
129	C₄×C₃₇	#535	1	2	yes	—
130	C₄×C₃₇	#535	1	2	yes	—
131	C₂×#493	C₂×C₃₇	1	1	yes	characteristic, complemented
132	#3038	C₂×C₃₇	1	2	yes	complemented
133	#2922	C₂×C₃₇	1	1	yes	characteristic, complemented
134	#2045	C₂×C₃₇	1	4	yes	complemented
135	#2018	C₂×C₃₇	1	4	yes	complemented
136	#1965	C₂×C₃₇	1	2	yes	complemented
137	#1837	C₂×C₃₇	1	1	yes	characteristic, complemented
138	C₂×C₃₇	—	2	2	no	—
139	C₂×C₃₇	—	8	1	no	—
140	C₂×C₃₇	—	16	4	no	—
141	C₂×C₃₇	#2619	1	1	yes	characteristic
142	C₂×C₃₇	C₂×#472	1	2	yes	—
143	C₂×AGL(1,ℤ/8)	C₂²×C₃₇	1	4	yes	—
144	C₂²×D₈	C₂²×C₃₇	1	4	yes	—
145	C₂×#127	C₂²×C₃₇	1	2	yes	—
146	C₄×#41	C₂²×C₃₇	1	1	yes	characteristic
147	#493	—	2	2	no	—
148	#493	—	2	2	no	—
149	#460	—	2	4	no	—
150	#459	—	2	4	no	—
151	#433	—	2	2	no	—
152	#431	—	2	2	no	—
153	#424	C₂²×C₃₇	1	2	yes	—
154	#423	C₂²×C₃₇	1	2	yes	—
155	#418	C₂²×C₃₇	1	4	yes	—
156	C₂×#100	C₂²×C₃₇	1	8	yes	—
157	#408	C₂²×C₃₇	1	2	yes	—
158	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
160	#341	C₂²×C₃₇	1	4	yes	—
161	C₃₇	#29818	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#41	C₂³×C₃₇	1	1	yes	characteristic
163	C₂²×D₄	—	2	4	no	—
164	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
165	AGL(1,ℤ/8)	—	4	8	no	—
166	C₂×SD₁₆	—	2	4	no	—
167	C₂×D₈	—	2	4	no	—
168	C₂×D₈	—	2	8	no	—
169	C₂×D₈	—	4	8	no	—
170	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
171	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
173	#127	—	2	4	no	—
174	#127	—	2	2	no	—
175	#127	—	2	2	no	—
176	C₄×Q₈	D₄×C₃₇	1	2	yes	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	D₄×C₃₇	1	2	yes	—
179	#117	—	2	1	no	—
180	#117	C₂³×C₃₇	1	1	yes	characteristic
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	4	no	—
185	#102	—	2	8	no	—
186	#100	—	4	8	no	—
187	C₂²⋊C₈	—	2	4	no	—
188	#97	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	D₄×C₃₇	1	2	yes	—
191	C₄×C₈	D₄×C₃₇	1	2	yes	—
192	C₂⁴	—	2	4	no	—
193	#41	—	2	4	no	—
194	C₂×Q₈	C₂×D₄×C₃₇	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	8	8	no	—
203	C₂×D₄	C₂×D₄×C₃₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	#41×C₃₇	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
208	SD₁₆	—	4	8	no	—
209	D₈	—	4	16	no	—
210	D₈	—	4	8	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₃₇	1	2	yes	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	8	no	—
229	C₂³	—	8	8	no	—
230	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	8	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	8	8	no	—
238	D₄	—	8	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	#121×C₃₇	1	2	yes	—
245	C₂×C₄	#121×C₃₇	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
248	C₂×C₄	H(5,𝔽₂)×C₃₇	1	4	yes	—
249	C₈	—	2	4	no	—
250	C₈	—	4	4	no	—
251	C₈	—	4	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	8	no	—
257	C₂²	—	16	8	no	—
258	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
259	C₂²	C₃₇×#540	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	2	no	—
263	C₄	C₃₇×#472	1	4	yes	—
264	C₄	C₃₇×#535	1	2	yes	—
265	C₄	C₃₇×#535	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	1	no	—
268	C₂	—	16	4	no	—
269	C₂	C₃₇×#2619	1	1	yes	characteristic
270	C₂	C₂×C₃₇×#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
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×#493	C₂	semidirect
2. C₃₇×#3038	C₂	semidirect
3. C₃₇×#2922	C₂	semidirect
4. C₃₇×#2045	C₂	semidirect
5. C₃₇×#2018	C₂	semidirect
6. C₃₇×#1965	C₂	semidirect
7. C₃₇×#1837	C₂	semidirect
116. #29818	C₃₇	direct (N × H)
131. C₂×#493	C₂×C₃₇	semidirect
132. #3038	C₂×C₃₇	semidirect
133. #2922	C₂×C₃₇	semidirect
134. #2045	C₂×C₃₇	semidirect
135. #2018	C₂×C₃₇	semidirect
136. #1965	C₂×C₃₇	semidirect
137. #1837	C₂×C₃₇	semidirect
161. C₃₇	#29818	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×#493	2
C₃₇×#3038	2
C₃₇×#2922	2
C₃₇×#2045	2
C₃₇×#2018	2
C₃₇×#1965	2
C₃₇×#1837	2
#29818	37

Subgroups — C37×#29818 #22651071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29818 #22651071