Subgroups — C₃₇×#33828 #22655081

Order 9472.

488 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3175	C₂	1	2	yes	maximal, complemented
2	C₃₇×#2227	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2224	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2220	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₇×#2215	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#2211	C₂	1	2	yes	maximal, complemented
8	C₃₇×#2210	C₂	1	2	yes	maximal, complemented
9	C₃₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₇×#542	—	2	2	no	—
11	C₂×Q₃₂×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₃₇	C₂²	1	2	yes	—
13	C₂×D₁₆×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
16	C₃₇×#500	C₂²	1	2	yes	—
17	C₃₇×#495	C₂²	1	2	yes	—
18	C₃₇×#489	—	2	2	no	—
19	C₃₇×#468	—	2	2	no	—
20	C₃₇×#467	C₂²	1	2	yes	—
21	C₃₇×#466	C₂²	1	2	yes	—
22	C₃₇×#440	—	2	2	no	—
23	C₄×Q₁₆×C₃₇	C₂²	1	2	yes	—
24	C₄×D₈×C₃₇	C₂²	1	2	yes	—
25	C₃₇×#405	C₂²	1	1	yes	characteristic
26	C₃₇×#366	C₂²	1	1	yes	characteristic
27	C₃₇×#366	C₂²	1	1	yes	characteristic
28	C₃₇×#366	C₂²	1	2	yes	—
29	C₃₇×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃₇×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃₇×#358	C₂²	1	2	yes	—
32	C₃₇×#358	C₂²	1	2	yes	—
33	C₃₇×#357	C₂²	1	2	yes	—
34	C₃₇×#357	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₃₇	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₃₇	—	2	2	no	—
37	C₂×D₈×C₃₇	C₂³	1	2	yes	—
38	C₂×M₁₆×C₃₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₃₇	—	2	2	no	—
41	#126×C₃₇	—	4	2	no	—
42	#124×C₃₇	—	2	2	no	—
43	#124×C₃₇	—	4	2	no	—
44	#123×C₃₇	—	4	2	no	—
45	C₂²⋊Q₈×C₃₇	—	2	2	no	—
46	C₂²⋊Q₈×C₃₇	—	4	2	no	—
47	#121×C₃₇	—	2	2	no	—
48	C₄×Q₈×C₃₇	—	2	2	no	—
49	C₄×D₄×C₃₇	—	2	2	no	—
50	#117×C₃₇	D₄	1	1	yes	characteristic
51	Q₃₂×C₃₇	—	4	1	no	—
52	SD₃₂×C₃₇	—	4	2	no	—
53	D₁₆×C₃₇	—	4	1	no	—
54	M₃₂×C₃₇	—	4	1	no	—
55	C₂×C₁₆×C₃₇	—	4	1	no	—
56	C₂×C₁₆×C₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₃₇	C₂³	1	2	yes	—
58	C₂×C₁₆×C₃₇	C₂³	1	1	yes	characteristic
59	#107×C₃₇	—	2	2	no	—
60	#107×C₃₇	C₂³	1	2	yes	—
61	#107×C₃₇	C₂³	1	2	yes	—
62	#103×C₃₇	—	2	2	no	—
63	#103×C₃₇	—	2	2	no	—
64	#102×C₃₇	—	2	2	no	—
65	#102×C₃₇	—	2	2	no	—
66	#97×C₃₇	D₄	1	2	yes	—
67	C₄×C₈×C₃₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	4	2	no	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	4	2	no	—
72	C₂²×C₄×C₃₇	—	4	2	no	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₃₇	—	2	2	no	—
75	Q₁₆×C₃₇	—	2	2	no	—
76	SD₁₆×C₃₇	—	4	4	no	—
77	D₈×C₃₇	—	2	2	no	—
78	D₈×C₃₇	—	2	2	no	—
79	M₁₆×C₃₇	—	4	1	no	—
80	C₂×C₈×C₃₇	—	2	1	no	—
81	C₂×C₈×C₃₇	—	2	1	no	—
82	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₃₇	—	2	2	no	—
86	C₄⋊C₄×C₃₇	—	2	2	no	—
87	C₄⋊C₄×C₃₇	—	4	2	no	—
88	C₄⋊C₄×C₃₇	—	4	2	no	—
89	C₄⋊C₄×C₃₇	—	4	2	no	—
90	C₄⋊C₄×C₃₇	D₈	1	1	yes	characteristic
91	C₄⋊C₄×C₃₇	D₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₃₇	—	4	2	no	—
93	C₂²⋊C₄×C₃₇	—	4	2	no	—
94	C₂²⋊C₄×C₃₇	—	4	2	no	—
95	C₂²⋊C₄×C₃₇	—	4	2	no	—
96	C₂²⋊C₄×C₃₇	—	4	2	no	—
97	C₂²⋊C₄×C₃₇	D₈	1	2	yes	—
98	C₄²×C₃₇	—	4	2	no	—
99	C₄²×C₃₇	C₂×D₄	1	2	yes	—
100	C₁₆×C₃₇	—	2	1	no	—
101	C₁₆×C₃₇	—	2	1	no	—
102	C₁₆×C₃₇	—	2	2	no	—
103	C₂³×C₃₇	—	4	2	no	—
104	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₃₇	—	4	2	no	—
106	Q₈×C₃₇	—	4	2	no	—
107	Q₈×C₃₇	—	16	2	no	—
108	D₄×C₃₇	—	4	2	no	—
109	D₄×C₃₇	—	4	2	no	—
110	D₄×C₃₇	—	16	2	no	—
111	C₂×C₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	2	no	—
115	C₂×C₄×C₃₇	—	4	2	no	—
116	C₂×C₄×C₃₇	—	16	2	no	—
117	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₇	—	2	1	no	—
122	C₈×C₃₇	—	2	2	no	—
123	C₈×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
125	#33828	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₃₇	—	2	1	no	—
127	C₂²×C₃₇	—	2	1	no	—
128	C₂²×C₃₇	—	2	1	no	—
129	C₂²×C₃₇	—	8	2	no	—
130	C₂²×C₃₇	—	8	2	no	—
131	C₂²×C₃₇	—	8	2	no	—
132	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₃₇	—	2	2	no	—
134	C₄×C₃₇	—	2	2	no	—
135	C₄×C₃₇	—	2	1	no	—
136	C₄×C₃₇	—	8	2	no	—
137	C₄×C₃₇	—	8	2	no	—
138	C₄×C₃₇	—	8	2	no	—
139	C₄×C₃₇	#537	1	1	yes	characteristic
140	C₄×C₃₇	#575	1	1	yes	characteristic
141	#3175	C₂×C₃₇	1	2	yes	complemented
142	#2227	C₂×C₃₇	1	2	yes	complemented
143	#2224	C₂×C₃₇	1	2	yes	complemented
144	#2220	C₂×C₃₇	1	2	yes	complemented
145	#2216	C₂×C₃₇	1	1	yes	characteristic, complemented
146	#2215	C₂×C₃₇	1	1	yes	characteristic
147	#2211	C₂×C₃₇	1	2	yes	complemented
148	#2210	C₂×C₃₇	1	2	yes	complemented
149	#2151	C₂×C₃₇	1	1	yes	characteristic, complemented
150	C₂×C₃₇	—	4	1	no	—
151	C₂×C₃₇	—	16	2	no	—
152	C₂×C₃₇	#3245	1	1	yes	characteristic
153	C₂×C₃₇	#3415	1	1	yes	characteristic
154	C₂×C₃₇	#3417	1	1	yes	characteristic
155	#542	—	2	2	no	—
156	C₂×Q₃₂	C₂²×C₃₇	1	1	yes	characteristic
157	C₂×SD₃₂	C₂²×C₃₇	1	2	yes	—
158	C₂×D₁₆	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×M₃₂	C₂²×C₃₇	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
161	#500	C₂²×C₃₇	1	2	yes	—
162	#495	C₂²×C₃₇	1	2	yes	—
163	#489	—	2	2	no	—
164	#468	—	2	2	no	—
165	#467	C₂²×C₃₇	1	2	yes	—
166	#466	C₂²×C₃₇	1	2	yes	—
167	#440	—	2	2	no	—
168	C₄×Q₁₆	C₂²×C₃₇	1	2	yes	—
169	C₄×D₈	C₂²×C₃₇	1	2	yes	—
170	#405	C₂²×C₃₇	1	1	yes	characteristic
171	#366	C₂²×C₃₇	1	1	yes	characteristic
172	#366	C₂²×C₃₇	1	1	yes	characteristic
173	#366	C₂²×C₃₇	1	2	yes	—
174	C₄⋊C₁₆	C₂²×C₃₇	1	2	yes	—
175	C₄⋊C₁₆	C₂²×C₃₇	1	2	yes	—
176	#358	C₂²×C₃₇	1	2	yes	—
177	#358	C₂²×C₃₇	1	2	yes	—
178	#357	C₂²×C₃₇	1	2	yes	—
179	#357	C₂²×C₃₇	1	2	yes	—
180	C₃₇	#33828	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×Q₁₆	C₂³×C₃₇	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×D₈	C₂³×C₃₇	1	2	yes	—
184	C₂×M₁₆	D₄×C₃₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	2	no	—
187	#126	—	4	2	no	—
188	#124	—	2	2	no	—
189	#124	—	4	2	no	—
190	#123	—	4	2	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₂²⋊Q₈	—	4	2	no	—
193	#121	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	D₄×C₃₇	1	1	yes	characteristic
197	Q₃₂	—	4	1	no	—
198	SD₃₂	—	4	2	no	—
199	D₁₆	—	4	1	no	—
200	M₃₂	—	4	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	C₂³×C₃₇	1	1	yes	characteristic
203	C₂×C₁₆	C₂³×C₃₇	1	2	yes	—
204	C₂×C₁₆	C₂³×C₃₇	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	C₂³×C₃₇	1	2	yes	—
207	#107	C₂³×C₃₇	1	2	yes	—
208	#103	—	2	2	no	—
209	#103	—	2	2	no	—
210	#102	—	2	2	no	—
211	#102	—	2	2	no	—
212	#97	D₄×C₃₇	1	2	yes	—
213	C₄×C₈	C₂³×C₃₇	1	2	yes	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	SD₁₆	—	4	4	no	—
223	D₈	—	2	2	no	—
224	D₈	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	D₈×C₃₇	1	1	yes	characteristic
237	C₄⋊C₄	D₈×C₃₇	1	1	yes	characteristic
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	D₈×C₃₇	1	2	yes	—
244	C₄²	—	4	2	no	—
245	C₄²	C₂×D₄×C₃₇	1	2	yes	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	16	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	16	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	16	2	no	—
263	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
264	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
265	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	2	no	—
269	C₈	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
270	C₈	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	C₃₇×#537	1	1	yes	characteristic
285	C₄	C₃₇×#575	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	16	2	no	—
288	C₂	C₃₇×#3245	1	1	yes	characteristic
289	C₂	C₃₇×#3415	1	1	yes	characteristic
290	C₂	C₃₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
prime radical	C₃₇×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ #117×C₃₇ ◁ C₃₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3175	C₂	semidirect
2. C₃₇×#2227	C₂	semidirect
3. C₃₇×#2224	C₂	semidirect
4. C₃₇×#2220	C₂	semidirect
5. C₃₇×#2216	C₂	semidirect
7. C₃₇×#2211	C₂	semidirect
8. C₃₇×#2210	C₂	semidirect
9. C₃₇×#2151	C₂	semidirect
125. #33828	C₃₇	direct (N × H)
141. #3175	C₂×C₃₇	semidirect
142. #2227	C₂×C₃₇	semidirect
143. #2224	C₂×C₃₇	semidirect
144. #2220	C₂×C₃₇	semidirect
145. #2216	C₂×C₃₇	semidirect
147. #2211	C₂×C₃₇	semidirect
148. #2210	C₂×C₃₇	semidirect
149. #2151	C₂×C₃₇	semidirect
180. C₃₇	#33828	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×#3175	2
C₃₇×#2227	2
C₃₇×#2224	2
C₃₇×#2220	2
C₃₇×#2216	2
C₃₇×#2215	2
C₃₇×#2211	2
C₃₇×#2210	2
C₃₇×#2151	2
#33828	37

Subgroups — C37×#33828 #22655081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33828 #22655081