Subgroups — C₃₇×#2497 #17991315

Order 4736.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#400	C₂	1	2	yes	maximal, complemented
2	C₃₇×#396	C₂	1	2	yes	maximal
3	C₃₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#388	C₂	1	2	yes	maximal
6	C₃₇×#387	C₂	1	2	yes	maximal, complemented
7	C₃₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#380	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₃₇	C₂²	1	2	yes	—
28	#95×C₃₇	C₂²	1	1	yes	characteristic
29	#95×C₃₇	C₂²	1	1	yes	characteristic
30	#95×C₃₇	C₂²	1	2	yes	—
31	#95×C₃₇	C₂²	1	2	yes	—
32	#95×C₃₇	C₂²	1	2	yes	—
33	#95×C₃₇	C₂²	1	2	yes	—
34	#95×C₃₇	C₂²	1	2	yes	—
35	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₇	—	2	2	no	—
37	C₂²×C₄×C₃₇	—	2	2	no	—
38	C₂²×C₄×C₃₇	—	2	2	no	—
39	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₃₇	—	2	4	no	—
50	C₄⋊C₄×C₃₇	—	2	2	no	—
51	C₄⋊C₄×C₃₇	—	2	2	no	—
52	C₄⋊C₄×C₃₇	—	2	2	no	—
53	C₄⋊C₄×C₃₇	—	2	2	no	—
54	C₄⋊C₄×C₃₇	—	4	2	no	—
55	C₄⋊C₄×C₃₇	—	4	2	no	—
56	C₄⋊C₄×C₃₇	—	4	2	no	—
57	C₂²⋊C₄×C₃₇	—	2	2	no	—
58	C₂²⋊C₄×C₃₇	—	4	2	no	—
59	C₂²⋊C₄×C₃₇	—	4	2	no	—
60	C₂²⋊C₄×C₃₇	Q₈	1	4	yes	—
61	C₄²×C₃₇	—	2	2	no	—
62	C₄²×C₃₇	—	2	2	no	—
63	C₄²×C₃₇	—	4	2	no	—
64	C₂³×C₃₇	—	2	1	no	—
65	C₂³×C₃₇	—	2	1	no	—
66	C₂³×C₃₇	—	2	1	no	—
67	C₂³×C₃₇	—	2	1	no	—
68	C₂³×C₃₇	—	2	1	no	—
69	C₂³×C₃₇	—	2	1	no	—
70	C₂³×C₃₇	C₂×Q₈	1	2	yes	—
71	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₄×C₃₇	—	2	1	no	—
73	C₂×C₄×C₃₇	—	2	1	no	—
74	C₂×C₄×C₃₇	—	2	1	no	—
75	C₂×C₄×C₃₇	—	2	1	no	—
76	C₂×C₄×C₃₇	—	2	1	no	—
77	C₂×C₄×C₃₇	—	2	1	no	—
78	C₂×C₄×C₃₇	—	2	1	no	—
79	C₂×C₄×C₃₇	—	2	1	no	—
80	C₂×C₄×C₃₇	—	2	2	no	—
81	C₂×C₄×C₃₇	—	2	2	no	—
82	C₂×C₄×C₃₇	—	2	2	no	—
83	C₂×C₄×C₃₇	—	2	2	no	—
84	C₂×C₄×C₃₇	—	2	2	no	—
85	C₂×C₄×C₃₇	—	2	2	no	—
86	C₂×C₄×C₃₇	—	2	1	no	—
87	C₂×C₄×C₃₇	—	2	1	no	—
88	C₂×C₄×C₃₇	—	2	1	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	2	no	—
93	C₂×C₄×C₃₇	—	2	2	no	—
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	1	no	—
96	C₂×C₄×C₃₇	—	4	4	no	—
97	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₃₇	#41	1	2	yes	—
100	C₂×C₄×C₃₇	#41	1	2	yes	—
101	C₂×C₄×C₃₇	#41	1	2	yes	—
102	C₂×C₄×C₃₇	#41	1	2	yes	—
103	C₂²×C₃₇	—	2	2	no	—
104	C₂²×C₃₇	—	2	2	no	—
105	C₂²×C₃₇	—	2	2	no	—
106	C₂²×C₃₇	—	4	1	no	—
107	C₂²×C₃₇	—	4	1	no	—
108	C₂²×C₃₇	—	4	1	no	—
109	C₂²×C₃₇	—	4	1	no	—
110	C₂²×C₃₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
112	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
113	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
114	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
115	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₄×C₃₇	—	2	2	no	—
118	C₄×C₃₇	—	2	2	no	—
119	C₄×C₃₇	—	2	2	no	—
120	C₄×C₃₇	—	4	1	no	—
121	C₄×C₃₇	—	4	2	no	—
122	C₄×C₃₇	—	4	2	no	—
123	C₄×C₃₇	—	4	2	no	—
124	C₄×C₃₇	—	4	1	no	—
125	C₄×C₃₇	—	4	2	no	—
126	C₄×C₃₇	#126	1	4	yes	—
127	#2497	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₃₇	—	4	2	no	—
129	C₂×C₃₇	C₂×#126	1	1	yes	characteristic
130	C₂×C₃₇	#533	1	1	yes	characteristic
131	C₂×C₃₇	#540	1	1	yes	characteristic
132	C₂×C₃₇	#553	1	1	yes	characteristic
133	C₂×C₃₇	#554	1	1	yes	characteristic
134	C₂×C₃₇	#554	1	1	yes	characteristic
135	C₂×C₃₇	#555	1	1	yes	characteristic
136	#400	C₂×C₃₇	1	2	yes	complemented
137	#396	C₂×C₃₇	1	2	yes	—
138	#389	C₂×C₃₇	1	1	yes	characteristic, complemented
139	#389	C₂×C₃₇	1	1	yes	characteristic, complemented
140	#388	C₂×C₃₇	1	2	yes	—
141	#387	C₂×C₃₇	1	2	yes	complemented
142	#384	C₂×C₃₇	1	1	yes	characteristic, complemented
143	#380	C₂×C₃₇	1	1	yes	characteristic
144	#380	C₂×C₃₇	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic, complemented
146	C₄×C₂²⋊C₄	C₂×C₃₇	1	1	yes	characteristic
147	C₃₇	#2497	1	1	yes	characteristic, direct product factor, Sylow
148	C₂³×C₄	C₂²×C₃₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
162	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×C₃₇	1	2	yes	—
164	#95	C₂²×C₃₇	1	1	yes	characteristic
165	#95	C₂²×C₃₇	1	1	yes	characteristic
166	#95	C₂²×C₃₇	1	2	yes	—
167	#95	C₂²×C₃₇	1	2	yes	—
168	#95	C₂²×C₃₇	1	2	yes	—
169	#95	C₂²×C₃₇	1	2	yes	—
170	#95	C₂²×C₃₇	1	2	yes	—
171	C₂⁴	C₂³×C₃₇	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
178	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
179	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
180	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
181	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	Q₈×C₃₇	1	4	yes	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	4	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	C₂×Q₈×C₃₇	1	2	yes	—
207	C₂³	C₂⁴×C₃₇	1	1	yes	characteristic
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	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	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₂×Q₈×C₃₇	1	2	yes	—
234	C₂×C₄	C₂×Q₈×C₃₇	1	2	yes	—
235	C₂×C₄	#41×C₃₇	1	2	yes	—
236	C₂×C₄	#41×C₃₇	1	2	yes	—
237	C₂×C₄	#41×C₃₇	1	2	yes	—
238	C₂×C₄	#41×C₃₇	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	C₂²×Q₈×C₃₇	1	1	yes	characteristic
247	C₂²	C₂×#41×C₃₇	1	1	yes	characteristic
248	C₂²	C₂×#41×C₃₇	1	1	yes	characteristic
249	C₂²	C₂×#41×C₃₇	1	1	yes	characteristic
250	C₂²	C₂×#41×C₃₇	1	1	yes	characteristic
251	C₂²	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
252	C₂²	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	2	no	—
262	C₄	#126×C₃₇	1	4	yes	—
263	C₂	—	4	2	no	—
264	C₂	C₂×#126×C₃₇	1	1	yes	characteristic
265	C₂	C₃₇×#533	1	1	yes	characteristic
266	C₂	C₃₇×#540	1	1	yes	characteristic
267	C₂	C₃₇×#553	1	1	yes	characteristic
268	C₂	C₃₇×#554	1	1	yes	characteristic
269	C₂	C₃₇×#554	1	1	yes	characteristic
270	C₂	C₃₇×#555	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₇×#400	C₂	semidirect
3. C₃₇×#389	C₂	semidirect
4. C₃₇×#389	C₂	semidirect
6. C₃₇×#387	C₂	semidirect
7. C₃₇×#384	C₂	semidirect
10. C₄×C₄⋊C₄×C₃₇	C₂	semidirect
127. #2497	C₃₇	direct (N × H)
136. #400	C₂×C₃₇	semidirect
138. #389	C₂×C₃₇	semidirect
139. #389	C₂×C₃₇	semidirect
141. #387	C₂×C₃₇	semidirect
142. #384	C₂×C₃₇	semidirect
145. C₄×C₄⋊C₄	C₂×C₃₇	semidirect
147. C₃₇	#2497	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×#400	2
C₃₇×#396	2
C₃₇×#389	2
C₃₇×#388	2
C₃₇×#387	2
C₃₇×#384	2
C₃₇×#380	2
C₄×C₄⋊C₄×C₃₇	2
C₄×C₂²⋊C₄×C₃₇	2
#2497	37

Subgroups — C37×#2497 #17991315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#2497 #17991315