Subgroups — C₃₇×#29999 #22651252

Order 9472.

662 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2295	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2087	C₂	1	4	yes	maximal
4	C₃₇×#2068	C₂	1	4	yes	maximal
5	C₃₇×#1947	C₂	1	2	yes	maximal
6	C₃₇×#1842	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#1803	C₂	1	2	yes	maximal
8	C₂×#125×C₃₇	—	2	1	no	—
9	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#107×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
13	C₂×#103×C₃₇	C₂²	1	4	yes	—
14	C₃₇×#391	C₂²	1	2	yes	—
15	C₃₇×#389	—	2	2	no	—
16	C₃₇×#389	C₂²	1	2	yes	—
17	C₃₇×#389	C₂²	1	4	yes	—
18	C₃₇×#387	—	2	2	no	—
19	C₃₇×#384	C₂²	1	2	yes	—
20	C₃₇×#382	—	2	1	no	—
21	C₄×C₄⋊C₄×C₃₇	—	2	1	no	—
22	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#376	—	2	1	no	—
24	C₄³×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#340	C₂²	1	4	yes	—
26	C₃₇×#340	C₂²	1	4	yes	—
27	C₂²×Q₈×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
29	#125×C₃₇	—	4	4	no	—
30	C₄×Q₈×C₃₇	—	2	2	no	—
31	C₄×Q₈×C₃₇	Q₈	1	4	yes	—
32	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₇	—	2	1	no	—
41	C₂×C₄²×C₃₇	—	2	1	no	—
42	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	#107×C₃₇	—	4	4	no	—
48	C₄⋊C₈×C₃₇	—	2	4	no	—
49	C₄⋊C₈×C₃₇	—	4	4	no	—
50	#103×C₃₇	—	4	4	no	—
51	#95×C₃₇	—	2	2	no	—
52	#95×C₃₇	—	2	2	no	—
53	#95×C₃₇	—	2	2	no	—
54	#95×C₃₇	—	2	1	no	—
55	#95×C₃₇	—	2	4	no	—
56	C₂×Q₈×C₃₇	—	2	1	no	—
57	C₂×Q₈×C₃₇	—	2	2	no	—
58	C₂×Q₈×C₃₇	—	2	2	no	—
59	C₂×Q₈×C₃₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₃₇	—	2	1	no	—
61	C₂²×C₄×C₃₇	—	2	1	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₃₇	—	2	4	no	—
70	C₂×C₈×C₃₇	—	2	4	no	—
71	C₂×C₈×C₃₇	—	2	4	no	—
72	C₄⋊C₄×C₃₇	—	2	4	no	—
73	C₄⋊C₄×C₃₇	—	2	1	no	—
74	C₄⋊C₄×C₃₇	—	2	4	no	—
75	C₄⋊C₄×C₃₇	—	2	2	no	—
76	C₄⋊C₄×C₃₇	—	2	2	no	—
77	C₄⋊C₄×C₃₇	—	2	4	no	—
78	C₄⋊C₄×C₃₇	—	4	4	no	—
79	C₄⋊C₄×C₃₇	—	4	4	no	—
80	C₄⋊C₄×C₃₇	—	4	8	no	—
81	C₄⋊C₄×C₃₇	—	4	4	no	—
82	C₄⋊C₄×C₃₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₃₇	—	2	4	no	—
84	C₄²×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	—	2	2	no	—
86	C₄²×C₃₇	—	2	2	no	—
87	C₄²×C₃₇	—	4	2	no	—
88	C₄²×C₃₇	—	4	2	no	—
89	C₄²×C₃₇	Q₁₆	1	4	yes	—
90	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
91	C₄²×C₃₇	#41	1	2	yes	—
92	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₃₇	—	2	4	no	—
94	Q₈×C₃₇	—	4	1	no	—
95	Q₈×C₃₇	—	4	1	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	2	no	—
101	C₂×C₄×C₃₇	—	2	2	no	—
102	C₂×C₄×C₃₇	—	2	2	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	—	4	2	no	—
108	C₂×C₄×C₃₇	—	4	2	no	—
109	C₂×C₄×C₃₇	—	4	4	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
121	C₈×C₃₇	—	4	4	no	—
122	#29999	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂²×C₃₇	#537	1	2	yes	—
125	C₂²×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
126	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×C₃₇	#575	1	2	yes	—
128	C₄×C₃₇	—	2	2	no	—
129	C₄×C₃₇	—	2	4	no	—
130	C₄×C₃₇	—	4	2	no	—
131	C₄×C₃₇	—	4	2	no	—
132	C₄×C₃₇	—	4	2	no	—
133	C₄×C₃₇	—	8	2	no	—
134	C₄×C₃₇	—	8	4	no	—
135	C₄×C₃₇	#477	1	4	yes	—
136	C₄×C₃₇	#540	1	1	yes	characteristic
137	C₄×C₃₇	#541	1	1	yes	characteristic
138	C₄×C₃₇	#544	1	2	yes	—
139	#2859	C₂×C₃₇	1	1	yes	characteristic
140	#2295	C₂×C₃₇	1	1	yes	characteristic
141	#2087	C₂×C₃₇	1	4	yes	—
142	#2068	C₂×C₃₇	1	4	yes	—
143	#1947	C₂×C₃₇	1	2	yes	—
144	#1842	C₂×C₃₇	1	1	yes	characteristic
145	#1803	C₂×C₃₇	1	2	yes	—
146	C₂×C₃₇	#2629	1	1	yes	characteristic
147	C₂×C₃₇	C₂×#477	1	1	yes	characteristic
148	C₂×C₃₇	#3080	1	1	yes	characteristic
149	C₂×C₃₇	#3249	1	2	yes	—
150	C₂×C₃₇	#3251	1	2	yes	—
151	C₂×#125	—	2	1	no	—
152	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
153	C₂×#107	C₂²×C₃₇	1	4	yes	—
154	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
156	C₂×#103	C₂²×C₃₇	1	4	yes	—
157	#391	C₂²×C₃₇	1	2	yes	—
158	#389	—	2	2	no	—
159	#389	C₂²×C₃₇	1	4	yes	—
160	#389	C₂²×C₃₇	1	2	yes	—
161	#387	—	2	2	no	—
162	#384	C₂²×C₃₇	1	2	yes	—
163	#382	—	2	1	no	—
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
166	#376	—	2	1	no	—
167	C₄³	C₂²×C₃₇	1	1	yes	characteristic
168	#340	C₂²×C₃₇	1	4	yes	—
169	#340	C₂²×C₃₇	1	4	yes	—
170	C₃₇	#29999	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×Q₈	C₂³×C₃₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
173	#125	—	4	4	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	Q₈×C₃₇	1	4	yes	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₃₇	1	2	yes	—
188	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
191	#107	—	4	4	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	4	4	no	—
194	#103	—	4	4	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	1	no	—
199	#95	—	2	4	no	—
200	C₂×Q₈	—	2	1	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	C₂×Q₈×C₃₇	1	2	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	C₂×Q₈×C₃₇	1	2	yes	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	4	2	no	—
233	C₄²	Q₁₆×C₃₇	1	4	yes	—
234	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
235	C₄²	#41×C₃₇	1	2	yes	—
236	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
237	Q₈	—	2	4	no	—
238	Q₈	—	4	1	no	—
239	Q₈	—	4	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
257	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
258	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
259	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
260	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
261	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
263	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
264	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
265	C₈	—	4	4	no	—
266	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
267	C₂²	C₃₇×#537	1	2	yes	—
268	C₂²	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
269	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic
270	C₂²	C₃₇×#575	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	4	no	—
278	C₄	C₃₇×#477	1	4	yes	—
279	C₄	C₃₇×#540	1	1	yes	characteristic
280	C₄	C₃₇×#541	1	1	yes	characteristic
281	C₄	C₃₇×#544	1	2	yes	—
282	C₂	C₃₇×#2629	1	1	yes	characteristic
283	C₂	C₂×C₃₇×#477	1	1	yes	characteristic
284	C₂	C₃₇×#3080	1	1	yes	characteristic
285	C₂	C₃₇×#3249	1	2	yes	—
286	C₂	C₃₇×#3251	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 ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
122. #29999	C₃₇	direct (N × H)
170. C₃₇	#29999	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×#2859	2
C₃₇×#2295	2
C₃₇×#2087	2
C₃₇×#2068	2
C₃₇×#1947	2
C₃₇×#1842	2
C₃₇×#1803	2
#29999	37

Subgroups — C37×#29999 #22651252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29999 #22651252