Subgroups — C₃₇×#28809 #22650062

Order 9472.

822 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#498	C₂	1	2	yes	maximal
2	C₃₇×#2551	C₂	1	2	yes	maximal
3	C₃₇×#2058	C₂	1	4	yes	maximal
4	C₃₇×#1948	C₂	1	2	yes	maximal
5	C₃₇×#1941	C₂	1	2	yes	maximal
6	C₃₇×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₃₇	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	4	yes	—
11	C₃₇×#498	—	2	8	no	—
12	C₂×#106×C₃₇	C₂²	1	2	yes	—
13	C₂×#106×C₃₇	C₂²	1	4	yes	—
14	C₂×#106×C₃₇	C₂²	1	2	yes	—
15	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#395	—	2	4	no	—
21	C₃₇×#389	—	2	2	no	—
22	C₃₇×#389	C₂²	1	4	yes	—
23	C₃₇×#384	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#340	C₂²	1	4	yes	—
27	C₃₇×#340	C₂²	1	2	yes	—
28	C₃₇×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₇	—	2	4	no	—
30	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₃₇	—	2	8	no	—
34	C₄⋊Q₈×C₃₇	—	4	4	no	—
35	C₄⋊Q₈×C₃₇	—	4	8	no	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₇	—	2	2	no	—
45	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	#106×C₃₇	—	2	4	no	—
48	#106×C₃₇	—	2	8	no	—
49	#106×C₃₇	—	4	2	no	—
50	#106×C₃₇	D₄	1	4	yes	—
51	#106×C₃₇	Q₈	1	4	yes	complemented
52	C₄⋊C₈×C₃₇	Q₈	1	4	yes	complemented
53	C₄×C₈×C₃₇	—	2	4	no	—
54	C₄×C₈×C₃₇	Q₈	1	4	yes	complemented
55	#95×C₃₇	—	2	2	no	—
56	#95×C₃₇	—	2	4	no	—
57	#95×C₃₇	D₄	1	2	yes	—
58	C₂×Q₈×C₃₇	—	4	8	no	—
59	C₂×Q₈×C₃₇	—	4	8	no	—
60	C₂×Q₈×C₃₇	—	8	4	no	—
61	C₂²×C₄×C₃₇	—	2	2	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₃₇	—	2	2	no	—
69	C₂×C₈×C₃₇	—	2	1	no	—
70	C₂×C₈×C₃₇	—	2	1	no	—
71	C₂×C₈×C₃₇	—	2	2	no	—
72	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	2	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	2	4	no	—
81	C₄⋊C₄×C₃₇	—	4	8	no	—
82	C₄⋊C₄×C₃₇	—	4	8	no	—
83	C₄⋊C₄×C₃₇	—	4	4	no	—
84	C₄⋊C₄×C₃₇	—	8	2	no	—
85	C₄⋊C₄×C₃₇	SD₁₆	1	8	yes	—
86	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₃₇	C₂×Q₈	1	4	yes	—
88	C₄²×C₃₇	—	2	2	no	—
89	C₄²×C₃₇	—	2	2	no	—
90	C₄²×C₃₇	—	4	4	no	—
91	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
92	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
93	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₃₇	—	8	16	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	2	4	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	2	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	4	4	no	—
106	C₂×C₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₃₇	C₂×SD₁₆	1	8	yes	—
110	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₈×C₃₇	—	2	2	no	—
118	C₈×C₃₇	—	2	4	no	—
119	C₈×C₃₇	C₄⋊Q₈	1	4	yes	complemented
120	#28809	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₃₇	D₄²	1	1	yes	characteristic
123	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
124	C₂²×C₃₇	C₂²×SD₁₆	1	2	yes	—
125	C₂²×C₃₇	#577	1	2	yes	—
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	—	2	8	no	—
128	C₄×C₃₇	—	4	4	no	—
129	C₄×C₃₇	—	4	4	no	—
130	C₄×C₃₇	—	8	4	no	—
131	C₄×C₃₇	#498	1	4	yes	complemented
132	C₄×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₄×C₃₇	#544	1	1	yes	characteristic
134	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
135	C₄×C₃₇	Q₈²	1	1	yes	characteristic
136	C₂×#498	C₂×C₃₇	1	2	yes	—
137	#2551	C₂×C₃₇	1	2	yes	—
138	#2058	C₂×C₃₇	1	4	yes	—
139	#1948	C₂×C₃₇	1	2	yes	—
140	#1941	C₂×C₃₇	1	2	yes	—
141	#1844	C₂×C₃₇	1	1	yes	characteristic
142	C₄×#106	C₂×C₃₇	1	1	yes	characteristic
143	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
144	C₂×C₃₇	#2551	1	1	yes	characteristic
145	C₂×C₃₇	C₂×#498	1	1	yes	characteristic
146	C₂×C₃₇	#3163	1	1	yes	characteristic
147	C₂×C₃₇	D₄×SD₁₆	1	2	yes	—
148	C₂×C₃₇	Q₈×SD₁₆	1	2	yes	—
149	C₂×C₄⋊Q₈	—	2	2	no	—
150	C₂×C₄⋊Q₈	C₂²×C₃₇	1	4	yes	—
151	#498	—	2	8	no	—
152	C₂×#106	C₂²×C₃₇	1	4	yes	—
153	C₂×#106	C₂²×C₃₇	1	2	yes	—
154	C₂×#106	C₂²×C₃₇	1	2	yes	—
155	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
156	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
160	#395	—	2	4	no	—
161	#389	—	2	2	no	—
162	#389	C₂²×C₃₇	1	4	yes	—
163	#384	C₂²×C₃₇	1	2	yes	—
164	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
166	#340	C₂²×C₃₇	1	2	yes	—
167	#340	C₂²×C₃₇	1	4	yes	—
168	#336	C₂²×C₃₇	1	2	yes	—
169	C₃₇	#28809	1	1	yes	characteristic, direct product factor, Sylow
170	C₂²×Q₈	—	2	4	no	—
171	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	8	no	—
175	C₄⋊Q₈	—	4	4	no	—
176	C₄⋊Q₈	—	4	8	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
187	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
188	#106	—	2	4	no	—
189	#106	—	2	8	no	—
190	#106	—	4	2	no	—
191	#106	D₄×C₃₇	1	4	yes	—
192	#106	Q₈×C₃₇	1	4	yes	complemented
193	C₄⋊C₈	Q₈×C₃₇	1	4	yes	complemented
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	Q₈×C₃₇	1	4	yes	complemented
196	#95	—	2	2	no	—
197	#95	—	2	4	no	—
198	#95	D₄×C₃₇	1	2	yes	—
199	C₂×Q₈	—	4	8	no	—
200	C₂×Q₈	—	4	8	no	—
201	C₂×Q₈	—	8	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
214	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
215	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
216	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
217	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	8	2	no	—
226	C₄⋊C₄	SD₁₆×C₃₇	1	8	yes	—
227	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
228	C₄⋊C₄	C₂×Q₈×C₃₇	1	4	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	4	no	—
232	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
233	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
234	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
235	Q₈	—	8	16	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
249	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
250	C₂×C₄	C₂×SD₁₆×C₃₇	1	8	yes	—
251	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
252	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
256	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
257	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	4	no	—
260	C₈	C₄⋊Q₈×C₃₇	1	4	yes	complemented
261	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
262	C₂²	D₄²×C₃₇	1	1	yes	characteristic
263	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
264	C₂²	C₂²×SD₁₆×C₃₇	1	2	yes	—
265	C₂²	C₃₇×#577	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	8	no	—
268	C₄	—	4	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₃₇×#498	1	4	yes	complemented
272	C₄	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
273	C₄	C₃₇×#544	1	1	yes	characteristic
274	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
275	C₄	Q₈²×C₃₇	1	1	yes	characteristic
276	C₂	C₃₇×#2551	1	1	yes	characteristic
277	C₂	C₂×C₃₇×#498	1	1	yes	characteristic
278	C₂	C₃₇×#3163	1	1	yes	characteristic
279	C₂	D₄×SD₁₆×C₃₇	1	2	yes	—
280	C₂	Q₈×SD₁₆×C₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
51. #106×C₃₇	Q₈	semidirect
52. C₄⋊C₈×C₃₇	Q₈	semidirect
54. C₄×C₈×C₃₇	Q₈	semidirect
119. C₈×C₃₇	C₄⋊Q₈	semidirect
120. #28809	C₃₇	direct (N × H)
131. C₄×C₃₇	#498	semidirect
169. C₃₇	#28809	direct (N × H)
192. #106	Q₈×C₃₇	semidirect
193. C₄⋊C₈	Q₈×C₃₇	semidirect
195. C₄×C₈	Q₈×C₃₇	semidirect
260. C₈	C₄⋊Q₈×C₃₇	semidirect
271. C₄	C₃₇×#498	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#498	2
C₃₇×#2551	2
C₃₇×#2058	2
C₃₇×#1948	2
C₃₇×#1941	2
C₃₇×#1844	2
C₄×#106×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
#28809	37

Subgroups — C37×#28809 #22650062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28809 #22650062