Subgroups — C₃₇×#28744 #22649997

Order 9472.

794 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#446	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2568	C₂	1	2	yes	maximal
4	C₃₇×#2056	C₂	1	4	yes	maximal
5	C₃₇×#1981	C₂	1	2	yes	maximal
6	C₃₇×#1971	C₂	1	4	yes	maximal
7	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#494	—	2	4	no	—
13	C₃₇×#446	—	2	4	no	—
14	C₂×#106×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#103×C₃₇	C₂²	1	8	yes	—
18	C₂×#97×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#400	—	2	4	no	—
21	C₃₇×#391	—	2	2	no	—
22	C₃₇×#391	C₂²	1	8	yes	—
23	C₃₇×#382	—	2	2	no	—
24	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
25	C₃₇×#340	C₂²	1	2	yes	—
26	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
28	C₂×Q₁₆×C₃₇	—	2	4	no	—
29	C₂×Q₁₆×C₃₇	—	2	4	no	—
30	C₂×Q₁₆×C₃₇	—	2	8	no	—
31	C₂×Q₁₆×C₃₇	—	4	4	no	—
32	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₃₇	—	2	4	no	—
36	C₄⋊Q₈×C₃₇	—	2	4	no	—
37	C₄×Q₈×C₃₇	—	2	2	no	—
38	C₄×Q₈×C₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	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	C₄⋊C₈×C₃₇	—	2	4	no	—
49	C₄⋊C₈×C₃₇	—	2	2	no	—
50	#103×C₃₇	—	4	8	no	—
51	#97×C₃₇	—	2	4	no	—
52	C₄×C₈×C₃₇	D₄	1	4	yes	—
53	#95×C₃₇	—	2	2	no	—
54	#95×C₃₇	—	2	8	no	—
55	#95×C₃₇	—	2	8	no	—
56	C₂×Q₈×C₃₇	—	2	2	no	—
57	C₂×Q₈×C₃₇	—	2	4	no	—
58	C₂×Q₈×C₃₇	—	2	4	no	—
59	C₂×Q₈×C₃₇	—	2	4	no	—
60	C₂×Q₈×C₃₇	—	2	8	no	—
61	C₂×Q₈×C₃₇	—	2	8	no	—
62	C₂²×C₄×C₃₇	—	2	4	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₃₇	—	4	8	no	—
70	Q₁₆×C₃₇	—	4	8	no	—
71	C₂×C₈×C₃₇	—	2	2	no	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₇	#41	1	4	yes	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	4	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	2	4	no	—
81	C₄⋊C₄×C₃₇	—	4	4	no	—
82	C₄⋊C₄×C₃₇	—	4	4	no	—
83	C₄²×C₃₇	—	2	2	no	—
84	C₄²×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	—	4	4	no	—
86	C₄²×C₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₃₇	#41	1	2	yes	—
88	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₃₇	—	4	4	no	—
90	Q₈×C₃₇	—	4	4	no	—
91	Q₈×C₃₇	—	4	4	no	—
92	Q₈×C₃₇	—	4	8	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	2	4	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	4	4	no	—
99	C₂×C₄×C₃₇	—	4	4	no	—
100	C₂×C₄×C₃₇	—	4	8	no	—
101	C₂×C₄×C₃₇	—	4	4	no	—
102	C₂×C₄×C₃₇	—	4	4	no	—
103	C₂×C₄×C₃₇	#121	1	2	yes	—
104	C₂×C₄×C₃₇	#121	1	2	yes	—
105	C₂×C₄×C₃₇	#137	1	4	yes	—
106	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
110	C₈×C₃₇	—	2	2	no	—
111	C₈×C₃₇	—	4	2	no	—
112	C₈×C₃₇	#124	1	4	yes	—
113	#28744	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
115	C₂²×C₃₇	#550	1	2	yes	—
116	C₂²×C₃₇	C₂×#137	1	2	yes	—
117	C₂²×C₃₇	#578	1	2	yes	—
118	C₄×C₃₇	—	2	2	no	—
119	C₄×C₃₇	—	4	2	no	—
120	C₄×C₃₇	—	4	2	no	—
121	C₄×C₃₇	—	8	4	no	—
122	C₄×C₃₇	—	8	4	no	—
123	C₄×C₃₇	#470	1	4	yes	—
124	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
125	C₄×C₃₇	#535	1	1	yes	characteristic
126	C₄×C₃₇	#559	1	2	yes	—
127	C₂×#494	C₂×C₃₇	1	1	yes	characteristic
128	C₂×#446	C₂×C₃₇	1	1	yes	characteristic
129	#2568	C₂×C₃₇	1	2	yes	—
130	#2056	C₂×C₃₇	1	4	yes	—
131	#1981	C₂×C₃₇	1	2	yes	—
132	#1971	C₂×C₃₇	1	4	yes	—
133	C₄×#106	C₂×C₃₇	1	1	yes	characteristic
134	C₂×C₃₇	#2548	1	1	yes	characteristic
135	C₂×C₃₇	C₂×#470	1	1	yes	characteristic
136	C₂×C₃₇	#3059	1	1	yes	characteristic
137	C₂×C₃₇	#3398	1	4	yes	—
138	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
139	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
140	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
141	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
142	#494	—	2	4	no	—
143	#446	—	2	4	no	—
144	C₂×#106	C₂²×C₃₇	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
147	C₂×#103	C₂²×C₃₇	1	8	yes	—
148	C₂×#97	C₂²×C₃₇	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
150	#400	—	2	4	no	—
151	#391	—	2	2	no	—
152	#391	C₂²×C₃₇	1	8	yes	—
153	#382	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
155	#340	C₂²×C₃₇	1	2	yes	—
156	C₃₇	#28744	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
158	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	8	no	—
162	C₂×Q₁₆	—	4	4	no	—
163	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
164	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	4	no	—
167	C₄⋊Q₈	—	2	4	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
177	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
178	#106	—	2	4	no	—
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	#103	—	4	8	no	—
182	#97	—	2	4	no	—
183	C₄×C₈	D₄×C₃₇	1	4	yes	—
184	#95	—	2	2	no	—
185	#95	—	2	8	no	—
186	#95	—	2	8	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
199	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
200	Q₁₆	—	4	8	no	—
201	Q₁₆	—	4	8	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
207	C₂×C₈	#41×C₃₇	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂×D₄×C₃₇	1	2	yes	—
218	C₄²	#41×C₃₇	1	2	yes	—
219	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
220	Q₈	—	4	4	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#121×C₃₇	1	2	yes	—
235	C₂×C₄	#121×C₃₇	1	2	yes	—
236	C₂×C₄	#137×C₃₇	1	4	yes	—
237	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
239	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
240	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	4	2	no	—
243	C₈	#124×C₃₇	1	4	yes	—
244	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
245	C₂²	C₃₇×#550	1	2	yes	—
246	C₂²	C₂×#137×C₃₇	1	2	yes	—
247	C₂²	C₃₇×#578	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	8	4	no	—
252	C₄	—	8	4	no	—
253	C₄	C₃₇×#470	1	4	yes	—
254	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
255	C₄	C₃₇×#535	1	1	yes	characteristic
256	C₄	C₃₇×#559	1	2	yes	—
257	C₂	C₃₇×#2548	1	1	yes	characteristic
258	C₂	C₂×C₃₇×#470	1	1	yes	characteristic
259	C₂	C₃₇×#3059	1	1	yes	characteristic
260	C₂	C₃₇×#3398	1	4	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 ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
113. #28744	C₃₇	direct (N × H)
156. C₃₇	#28744	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×#494	2
C₂×C₃₇×#446	2
C₃₇×#2568	2
C₃₇×#2056	2
C₃₇×#1981	2
C₃₇×#1971	2
C₄×#106×C₃₇	2
#28744	37

Subgroups — C37×#28744 #22649997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28744 #22649997