Subgroups — C₃₇×#28255 #22649508

Order 9472.

932 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#462	C₂	1	4	yes	maximal
2	C₄×C₄⋊Q₈×C₃₇	C₂	1	2	yes	maximal
3	C₃₇×#2056	C₂	1	4	yes	maximal
4	C₃₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#1803	C₂	1	4	yes	maximal
6	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₃₇	C₂²	1	4	yes	—
9	C₃₇×#462	—	2	16	no	—
10	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
12	C₂×#103×C₃₇	C₂²	1	8	yes	—
13	C₃₇×#391	C₂²	1	4	yes	—
14	C₃₇×#389	—	2	2	no	—
15	C₃₇×#389	—	2	2	no	—
16	C₃₇×#384	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
18	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
19	C₄³×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#340	C₂²	1	4	yes	—
21	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
22	C₂×Q₁₆×C₃₇	—	2	16	no	—
23	C₂×Q₁₆×C₃₇	—	4	4	no	—
24	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
25	C₄⋊Q₈×C₃₇	—	2	4	no	—
26	C₄⋊Q₈×C₃₇	D₄	1	8	yes	—
27	C₄×Q₈×C₃₇	—	2	8	no	—
28	C₄×Q₈×C₃₇	—	2	8	no	—
29	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₃₇	—	2	4	no	—
36	C₂×C₄²×C₃₇	D₄	1	2	yes	—
37	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
41	C₄⋊C₈×C₃₇	—	2	8	no	—
42	C₄⋊C₈×C₃₇	—	4	2	no	—
43	#103×C₃₇	—	2	16	no	—
44	#95×C₃₇	—	2	4	no	—
45	#95×C₃₇	—	2	4	no	—
46	C₂×Q₈×C₃₇	—	2	4	no	—
47	C₂×Q₈×C₃₇	—	2	4	no	—
48	C₂×Q₈×C₃₇	—	2	4	no	—
49	C₂×Q₈×C₃₇	—	2	4	no	—
50	C₂×Q₈×C₃₇	—	2	4	no	—
51	C₂×Q₈×C₃₇	C₂×D₄	1	8	yes	—
52	C₂²×C₄×C₃₇	—	2	4	no	—
53	C₂²×C₄×C₃₇	—	2	4	no	—
54	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₃₇	—	4	16	no	—
60	C₂×C₈×C₃₇	—	2	8	no	—
61	C₂×C₈×C₃₇	—	2	4	no	—
62	C₄⋊C₄×C₃₇	—	2	8	no	—
63	C₄⋊C₄×C₃₇	—	2	8	no	—
64	C₄⋊C₄×C₃₇	—	2	4	no	—
65	C₄⋊C₄×C₃₇	—	2	4	no	—
66	C₄⋊C₄×C₃₇	—	4	4	no	—
67	C₄⋊C₄×C₃₇	—	4	8	no	—
68	C₄⋊C₄×C₃₇	—	4	4	no	—
69	C₄²×C₃₇	—	2	4	no	—
70	C₄²×C₃₇	—	2	1	no	—
71	C₄²×C₃₇	—	2	1	no	—
72	C₄²×C₃₇	—	4	1	no	—
73	C₄²×C₃₇	—	4	8	no	—
74	C₄²×C₃₇	Q₁₆	1	4	yes	complemented
75	C₄²×C₃₇	C₂×D₄	1	4	yes	—
76	C₄²×C₃₇	#41	1	4	yes	—
77	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₃₇	—	2	8	no	—
79	Q₈×C₃₇	—	2	8	no	—
80	Q₈×C₃₇	—	4	4	no	—
81	Q₈×C₃₇	—	4	4	no	—
82	C₂×C₄×C₃₇	—	2	1	no	—
83	C₂×C₄×C₃₇	—	2	2	no	—
84	C₂×C₄×C₃₇	—	2	2	no	—
85	C₂×C₄×C₃₇	—	2	1	no	—
86	C₂×C₄×C₃₇	—	2	8	no	—
87	C₂×C₄×C₃₇	—	2	1	no	—
88	C₂×C₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	—	4	4	no	—
90	C₂×C₄×C₃₇	—	4	4	no	—
91	C₂×C₄×C₃₇	—	4	4	no	—
92	C₂×C₄×C₃₇	—	4	4	no	—
93	C₂×C₄×C₃₇	—	4	4	no	—
94	C₂×C₄×C₃₇	#121	1	4	yes	—
95	C₂×C₄×C₃₇	#121	1	4	yes	—
96	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
98	C₂×C₄×C₃₇	#137	1	4	yes	—
99	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₈×C₃₇	—	4	4	no	—
104	#28255	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₃₇	C₂×#121	1	2	yes	—
106	C₂²×C₃₇	#537	1	1	yes	characteristic
107	C₂²×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
108	C₂²×C₃₇	C₂×#137	1	2	yes	—
109	C₂²×C₃₇	#575	1	1	yes	characteristic
110	C₄×C₃₇	—	2	2	no	—
111	C₄×C₃₇	—	2	4	no	—
112	C₄×C₃₇	—	4	1	no	—
113	C₄×C₃₇	—	4	8	no	—
114	C₄×C₃₇	—	8	4	no	—
115	C₄×C₃₇	#462	1	8	yes	complemented
116	C₄×C₃₇	D₄²	1	1	yes	characteristic
117	C₄×C₃₇	#547	1	2	yes	—
118	C₄×C₃₇	#555	1	1	yes	characteristic
119	C₂×#462	C₂×C₃₇	1	4	yes	—
120	C₄×C₄⋊Q₈	C₂×C₃₇	1	2	yes	—
121	#2056	C₂×C₃₇	1	4	yes	—
122	#1842	C₂×C₃₇	1	1	yes	characteristic
123	#1803	C₂×C₃₇	1	4	yes	—
124	C₂×C₃₇	#2537	1	1	yes	characteristic
125	C₂×C₃₇	C₂×#462	1	2	yes	—
126	C₂×C₃₇	#3037	1	2	yes	—
127	C₂×C₃₇	#3249	1	1	yes	characteristic
128	C₂×C₃₇	#3255	1	1	yes	characteristic
129	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
130	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
131	C₂×C₄×Q₈	C₂²×C₃₇	1	4	yes	—
132	#462	—	2	16	no	—
133	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
134	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
135	C₂×#103	C₂²×C₃₇	1	8	yes	—
136	#391	C₂²×C₃₇	1	4	yes	—
137	#389	—	2	2	no	—
138	#389	—	2	2	no	—
139	#384	C₂²×C₃₇	1	2	yes	—
140	C₄×C₄⋊C₄	—	2	2	no	—
141	C₄×C₄⋊C₄	—	2	2	no	—
142	C₄³	C₂²×C₃₇	1	1	yes	characteristic
143	#340	C₂²×C₃₇	1	4	yes	—
144	C₃₇	#28255	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
146	C₂×Q₁₆	—	2	16	no	—
147	C₂×Q₁₆	—	4	4	no	—
148	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	D₄×C₃₇	1	8	yes	—
151	C₄×Q₈	—	2	8	no	—
152	C₄×Q₈	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
159	C₂×C₄²	—	2	4	no	—
160	C₂×C₄²	D₄×C₃₇	1	2	yes	—
161	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
165	C₄⋊C₈	—	2	8	no	—
166	C₄⋊C₈	—	4	2	no	—
167	#103	—	2	16	no	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	C₂×D₄×C₃₇	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
183	Q₁₆	—	4	16	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	4	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	8	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	8	no	—
198	C₄²	Q₁₆×C₃₇	1	4	yes	complemented
199	C₄²	C₂×D₄×C₃₇	1	4	yes	—
200	C₄²	#41×C₃₇	1	4	yes	—
201	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
202	Q₈	—	2	8	no	—
203	Q₈	—	2	8	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	4	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	#121×C₃₇	1	4	yes	—
219	C₂×C₄	#121×C₃₇	1	4	yes	—
220	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
221	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
222	C₂×C₄	#137×C₃₇	1	4	yes	—
223	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
224	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
225	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
226	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
227	C₈	—	4	4	no	—
228	C₂²	C₂×#121×C₃₇	1	2	yes	—
229	C₂²	C₃₇×#537	1	1	yes	characteristic
230	C₂²	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
231	C₂²	C₂×#137×C₃₇	1	2	yes	—
232	C₂²	C₃₇×#575	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	8	no	—
237	C₄	—	8	4	no	—
238	C₄	C₃₇×#462	1	8	yes	complemented
239	C₄	D₄²×C₃₇	1	1	yes	characteristic
240	C₄	C₃₇×#547	1	2	yes	—
241	C₄	C₃₇×#555	1	1	yes	characteristic
242	C₂	C₃₇×#2537	1	1	yes	characteristic
243	C₂	C₂×C₃₇×#462	1	2	yes	—
244	C₂	C₃₇×#3037	1	2	yes	—
245	C₂	C₃₇×#3249	1	1	yes	characteristic
246	C₂	C₃₇×#3255	1	1	yes	characteristic

	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 ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
74. C₄²×C₃₇	Q₁₆	semidirect
104. #28255	C₃₇	direct (N × H)
115. C₄×C₃₇	#462	semidirect
144. C₃₇	#28255	direct (N × H)
198. C₄²	Q₁₆×C₃₇	semidirect
238. C₄	C₃₇×#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#462	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2056	2
C₃₇×#1842	2
C₃₇×#1803	2
#28255	37

Subgroups — C37×#28255 #22649508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28255 #22649508