Subgroups — C₃₇×#8688 #22629941

Order 9472.

414 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#1547	C₂	1	2	yes	maximal
5	C₃₇×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₃₇	—	2	1	no	—
7	C₃₇×#532	C₂²	1	1	yes	characteristic
8	C₃₇×#436	—	2	1	no	—
9	C₃₇×#436	C₄	1	2	yes	—
10	C₈×D₄×C₃₇	—	2	1	no	—
11	C₈×D₄×C₃₇	C₄	1	2	yes	—
12	C₃₇×#423	C₂²	1	2	yes	—
13	C₃₇×#423	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#408	—	2	1	no	—
16	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₃₇×#341	—	2	1	no	—
18	C₃₇×#341	—	2	1	no	—
19	C₃₇×#335	—	2	1	no	—
20	C₃₇×#335	—	2	1	no	—
21	C₃₇×#332	—	2	2	no	—
22	C₃₇×#331	—	2	2	no	—
23	C₂×#41×C₃₇	—	2	1	no	—
24	#131×C₃₇	—	4	4	no	—
25	C₂×M₁₆×C₃₇	—	2	1	no	—
26	C₂×M₁₆×C₃₇	—	2	2	no	—
27	C₂×M₁₆×C₃₇	—	2	2	no	—
28	C₂²×C₈×C₃₇	—	2	2	no	—
29	C₂²×C₈×C₃₇	—	2	2	no	—
30	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
33	#127×C₃₇	D₄	1	1	yes	characteristic
34	#124×C₃₇	—	2	1	no	—
35	#121×C₃₇	—	2	1	no	—
36	#121×C₃₇	Q₈	1	2	yes	—
37	C₄×D₄×C₃₇	—	2	1	no	—
38	C₄×D₄×C₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₃₇	—	2	1	no	—
41	C₄⋊C₈×C₃₇	—	2	2	no	—
42	C₄⋊C₈×C₃₇	—	2	2	no	—
43	C₄⋊C₈×C₃₇	—	2	2	no	—
44	C₄⋊C₈×C₃₇	—	2	2	no	—
45	C₄⋊C₈×C₃₇	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₃₇	—	2	2	no	—
47	C₂²⋊C₈×C₃₇	—	2	2	no	—
48	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
52	#41×C₃₇	—	4	2	no	—
53	#41×C₃₇	—	4	2	no	—
54	C₂×Q₈×C₃₇	—	2	1	no	—
55	C₂×D₄×C₃₇	—	2	2	no	—
56	C₂×D₄×C₃₇	—	2	1	no	—
57	C₂×D₄×C₃₇	—	2	1	no	—
58	C₂×D₄×C₃₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₃₇	—	2	2	no	—
60	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
63	M₁₆×C₃₇	—	4	4	no	—
64	M₁₆×C₃₇	—	4	2	no	—
65	M₁₆×C₃₇	—	4	2	no	—
66	M₁₆×C₃₇	—	4	2	no	—
67	C₂×C₈×C₃₇	—	2	2	no	—
68	C₂×C₈×C₃₇	—	2	2	no	—
69	C₂×C₈×C₃₇	—	2	2	no	—
70	C₂×C₈×C₃₇	—	4	4	no	—
71	C₂×C₈×C₃₇	—	4	2	no	—
72	C₂×C₈×C₃₇	—	4	2	no	—
73	C₂×C₈×C₃₇	—	8	2	no	—
74	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₃₇	—	2	1	no	—
79	C₄⋊C₄×C₃₇	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₃₇	—	2	2	no	—
81	C₂²⋊C₄×C₃₇	—	2	2	no	—
82	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₃₇	#41	1	1	yes	characteristic
86	C₂³×C₃₇	—	2	2	no	—
87	C₂³×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
88	Q₈×C₃₇	—	4	2	no	—
89	D₄×C₃₇	—	4	2	no	—
90	D₄×C₃₇	—	4	4	no	—
91	D₄×C₃₇	—	4	2	no	—
92	D₄×C₃₇	—	4	1	no	—
93	D₄×C₃₇	—	4	1	no	—
94	C₂×C₄×C₃₇	—	2	1	no	—
95	C₂×C₄×C₃₇	—	2	1	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	4	1	no	—
98	C₂×C₄×C₃₇	—	4	2	no	—
99	C₂×C₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₃₇	#135	1	2	yes	—
107	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₃₇	—	2	1	no	—
109	C₈×C₃₇	—	2	1	no	—
110	C₈×C₃₇	—	2	1	no	—
111	C₈×C₃₇	—	2	1	no	—
112	C₈×C₃₇	—	4	2	no	—
113	C₈×C₃₇	—	8	2	no	—
114	C₈×C₃₇	—	8	2	no	—
115	#8688	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₃₇	—	2	1	no	—
117	C₂²×C₃₇	—	2	1	no	—
118	C₂²×C₃₇	—	4	2	no	—
119	C₂²×C₃₇	—	4	2	no	—
120	C₂²×C₃₇	—	4	2	no	—
121	C₂²×C₃₇	#380	1	1	yes	characteristic
122	C₂²×C₃₇	#478	1	2	yes	—
123	C₄×C₃₇	—	2	1	no	—
124	C₄×C₃₇	—	2	1	no	—
125	C₄×C₃₇	—	2	1	no	—
126	C₄×C₃₇	—	4	2	no	—
127	C₄×C₃₇	#416	1	1	yes	characteristic
128	C₄×C₃₇	#418	1	1	yes	characteristic
129	C₄×C₃₇	#463	1	2	yes	—
130	C₄×C₃₇	#472	1	1	yes	characteristic
131	C₄×C₃₇	#473	1	1	yes	characteristic
132	#2931	C₂×C₃₇	1	1	yes	characteristic
133	#1837	C₂×C₃₇	1	1	yes	characteristic, complemented
134	C₂×#335	C₂×C₃₇	1	1	yes	characteristic, complemented
135	#1547	C₂×C₃₇	1	2	yes	—
136	#1292	C₂×C₃₇	1	2	yes	complemented
137	C₂×C₃₇	—	2	2	no	—
138	C₂×C₃₇	—	8	2	no	—
139	C₂×C₃₇	#1804	1	1	yes	characteristic
140	C₂×C₃₇	#1813	1	1	yes	characteristic
141	C₂×C₃₇	#1816	1	1	yes	characteristic
142	C₂×#131	—	2	1	no	—
143	#532	C₂²×C₃₇	1	1	yes	characteristic
144	#436	—	2	1	no	—
145	#436	C₄×C₃₇	1	2	yes	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	C₄×C₃₇	1	2	yes	—
148	#423	C₂²×C₃₇	1	2	yes	—
149	#423	C₂²×C₃₇	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
151	#408	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
153	#341	—	2	1	no	—
154	#341	—	2	1	no	—
155	#335	—	2	1	no	—
156	#335	—	2	1	no	—
157	#332	—	2	2	no	—
158	#331	—	2	2	no	—
159	C₃₇	#8688	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×#41	—	2	1	no	—
161	#131	—	4	4	no	—
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
169	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
170	#127	D₄×C₃₇	1	1	yes	characteristic
171	#124	—	2	1	no	—
172	#121	—	2	1	no	—
173	#121	Q₈×C₃₇	1	2	yes	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	C₂×C₄×C₃₇	1	2	yes	—
176	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂×C₄×C₃₇	1	2	yes	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
187	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
188	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
189	#41	—	4	2	no	—
190	#41	—	4	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₄⋊C₄×C₃₇	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
199	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	8	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₄⋊C₄×C₃₇	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
222	C₄²	#41×C₃₇	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	1	no	—
230	D₄	—	4	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
239	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
240	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
241	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
243	C₂×C₄	#135×C₃₇	1	2	yes	—
244	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	8	2	no	—
251	C₈	—	8	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₃₇×#380	1	1	yes	characteristic
258	C₂²	C₃₇×#478	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	C₃₇×#416	1	1	yes	characteristic
264	C₄	C₃₇×#418	1	1	yes	characteristic
265	C₄	C₃₇×#463	1	2	yes	—
266	C₄	C₃₇×#472	1	1	yes	characteristic
267	C₄	C₃₇×#473	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	2	no	—
270	C₂	C₃₇×#1804	1	1	yes	characteristic
271	C₂	C₃₇×#1813	1	1	yes	characteristic
272	C₂	C₃₇×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	#380
commutator	C₄²	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
prime radical	C₃₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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
2. C₃₇×#1837	C₂	semidirect
3. C₂×C₃₇×#335	C₂	semidirect
5. C₃₇×#1292	C₂	semidirect
115. #8688	C₃₇	direct (N × H)
133. #1837	C₂×C₃₇	semidirect
134. C₂×#335	C₂×C₃₇	semidirect
136. #1292	C₂×C₃₇	semidirect
159. C₃₇	#8688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2931	2
C₃₇×#1837	2
C₂×C₃₇×#335	2
C₃₇×#1547	2
C₃₇×#1292	2
#8688	37

Subgroups — C37×#8688 #22629941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8688 #22629941