Subgroups — C₃₇×#22025 #22643278

Order 9472.

586 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1985	C₂	1	2	yes	maximal
3	C₃₇×#1940	C₂	1	2	yes	maximal
4	C₃₇×#1917	C₂	1	4	yes	maximal
5	C₃₇×#1840	C₂	1	2	yes	maximal
6	C₃₇×#1832	C₂	1	2	yes	maximal
7	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
12	C₂×#97×C₃₇	C₂²	1	2	yes	—
13	C₂×#97×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#391	C₄	1	2	yes	—
15	C₃₇×#389	C₄	1	2	yes	—
16	C₃₇×#387	C₄	1	4	yes	—
17	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
19	C₃₇×#376	C₂²	1	2	yes	—
20	C₄³×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#336	C₂²	1	1	yes	characteristic
22	C₃₇×#336	C₂²	1	4	yes	—
23	C₃₇×#336	C₂²	1	4	yes	—
24	C₃₇×#336	C₂²	1	2	yes	—
25	C₃₇×#336	C₂²	1	2	yes	—
26	C₃₇×#336	C₂²	1	4	yes	—
27	C₃₇×#336	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
32	C₄×Q₈×C₃₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₃₇	—	2	4	no	—
45	C₄⋊C₈×C₃₇	—	4	4	no	—
46	#97×C₃₇	—	2	4	no	—
47	#97×C₃₇	—	4	2	no	—
48	#95×C₃₇	C₂×C₄	1	4	yes	—
49	#95×C₃₇	C₂×C₄	1	4	yes	—
50	#95×C₃₇	C₂×C₄	1	2	yes	—
51	#95×C₃₇	C₂×C₄	1	4	yes	—
52	C₂×Q₈×C₃₇	—	2	2	no	—
53	C₂×Q₈×C₃₇	—	4	2	no	—
54	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₃₇	—	2	2	no	—
63	C₂×C₈×C₃₇	—	2	2	no	—
64	C₂×C₈×C₃₇	—	2	4	no	—
65	C₂×C₈×C₃₇	—	2	4	no	—
66	C₂×C₈×C₃₇	—	2	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	—	2	4	no	—
71	C₄⋊C₄×C₃₇	—	2	2	no	—
72	C₄⋊C₄×C₃₇	—	2	4	no	—
73	C₄⋊C₄×C₃₇	—	4	2	no	—
74	C₄⋊C₄×C₃₇	—	4	4	no	—
75	C₄²×C₃₇	—	2	4	no	—
76	C₄²×C₃₇	—	2	1	no	—
77	C₄²×C₃₇	—	2	2	no	—
78	C₄²×C₃₇	—	2	2	no	—
79	C₄²×C₃₇	—	2	1	no	—
80	C₄²×C₃₇	—	2	2	no	—
81	C₄²×C₃₇	—	4	2	no	—
82	C₄²×C₃₇	—	4	2	no	—
83	C₄²×C₃₇	#41	1	4	yes	—
84	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₃₇	—	4	4	no	—
86	C₂×C₄×C₃₇	—	2	2	no	—
87	C₂×C₄×C₃₇	—	2	2	no	—
88	C₂×C₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	—	2	2	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	—	2	4	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	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	—	2	4	no	—
101	C₂×C₄×C₃₇	#117	1	2	yes	—
102	C₂×C₄×C₃₇	#117	1	2	yes	—
103	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
104	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
106	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
107	C₈×C₃₇	—	4	2	no	—
108	C₈×C₃₇	—	4	2	no	—
109	C₈×C₃₇	—	4	4	no	—
110	#22025	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₇	C₂×#117	1	1	yes	characteristic
112	C₂²×C₃₇	#519	1	1	yes	characteristic
113	C₂²×C₃₇	#520	1	1	yes	characteristic
114	C₂²×C₃₇	#568	1	2	yes	—
115	C₂²×C₃₇	#568	1	2	yes	—
116	C₄×C₃₇	—	2	2	no	—
117	C₄×C₃₇	—	2	2	no	—
118	C₄×C₃₇	—	4	2	no	—
119	C₄×C₃₇	—	4	2	no	—
120	C₄×C₃₇	—	4	2	no	—
121	C₄×C₃₇	—	4	2	no	—
122	C₄×C₃₇	—	4	4	no	—
123	C₄×C₃₇	#541	1	2	yes	—
124	C₄×C₃₇	#542	1	2	yes	—
125	#2295	C₂×C₃₇	1	1	yes	characteristic
126	#1985	C₂×C₃₇	1	2	yes	—
127	#1940	C₂×C₃₇	1	2	yes	—
128	#1917	C₂×C₃₇	1	4	yes	—
129	#1840	C₂×C₃₇	1	2	yes	—
130	#1832	C₂×C₃₇	1	2	yes	—
131	#1767	C₂×C₃₇	1	1	yes	characteristic
132	#1745	C₂×C₃₇	1	1	yes	characteristic
133	C₂×C₃₇	#2320	1	1	yes	characteristic
134	C₂×C₃₇	#2921	1	1	yes	characteristic
135	C₂×C₃₇	#2923	1	1	yes	characteristic
136	C₂×C₃₇	#2987	1	2	yes	—
137	C₂×C₃₇	#2995	1	2	yes	—
138	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
141	C₂×#97	C₂²×C₃₇	1	2	yes	—
142	C₂×#97	C₂²×C₃₇	1	2	yes	—
143	#391	C₄×C₃₇	1	2	yes	—
144	#389	C₄×C₃₇	1	2	yes	—
145	#387	C₄×C₃₇	1	4	yes	—
146	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
147	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
148	#376	C₂²×C₃₇	1	2	yes	—
149	C₄³	C₂²×C₃₇	1	1	yes	characteristic
150	#336	C₂²×C₃₇	1	1	yes	characteristic
151	#336	C₂²×C₃₇	1	2	yes	—
152	#336	C₂²×C₃₇	1	4	yes	—
153	#336	C₂²×C₃₇	1	1	yes	characteristic
154	#336	C₂²×C₃₇	1	4	yes	—
155	#336	C₂²×C₃₇	1	4	yes	—
156	#336	C₂²×C₃₇	1	2	yes	—
157	C₃₇	#22025	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×Q₈	C₂×C₄×C₃₇	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
160	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
161	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
169	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
170	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
173	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	4	4	no	—
176	#97	—	2	4	no	—
177	#97	—	4	2	no	—
178	#95	C₂×C₄×C₃₇	1	4	yes	—
179	#95	C₂×C₄×C₃₇	1	4	yes	—
180	#95	C₂×C₄×C₃₇	1	2	yes	—
181	#95	C₂×C₄×C₃₇	1	4	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
191	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	#41×C₃₇	1	4	yes	—
214	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
215	Q₈	—	4	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	#117×C₃₇	1	2	yes	—
232	C₂×C₄	#117×C₃₇	1	2	yes	—
233	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
234	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
235	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
236	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
237	C₈	—	4	2	no	—
238	C₈	—	4	2	no	—
239	C₈	—	4	4	no	—
240	C₂²	C₂×#117×C₃₇	1	1	yes	characteristic
241	C₂²	C₃₇×#519	1	1	yes	characteristic
242	C₂²	C₃₇×#520	1	1	yes	characteristic
243	C₂²	C₃₇×#568	1	2	yes	—
244	C₂²	C₃₇×#568	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	4	no	—
252	C₄	C₃₇×#541	1	2	yes	—
253	C₄	C₃₇×#542	1	2	yes	—
254	C₂	C₃₇×#2320	1	1	yes	characteristic
255	C₂	C₃₇×#2921	1	1	yes	characteristic
256	C₂	C₃₇×#2923	1	1	yes	characteristic
257	C₂	C₃₇×#2987	1	2	yes	—
258	C₂	C₃₇×#2995	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
110. #22025	C₃₇	direct (N × H)
157. C₃₇	#22025	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×#2295	2
C₃₇×#1985	2
C₃₇×#1940	2
C₃₇×#1917	2
C₃₇×#1840	2
C₃₇×#1832	2
C₃₇×#1767	2
C₃₇×#1745	2
#22025	37

Subgroups — C37×#22025 #22643278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#22025 #22643278