Subgroups — C₃₇×#8179 #22629432

Order 9472.

562 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2293	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₇×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₃₇×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
7	C₂×#117×C₃₇	—	2	1	no	—
8	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
9	C₂×#97×C₃₇	—	2	1	no	—
10	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
11	C₃₇×#390	C₄	1	2	yes	—
12	C₃₇×#389	C₄	1	2	yes	—
13	C₃₇×#388	—	2	2	no	—
14	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₃₇	—	2	1	no	—
16	C₄³×C₃₇	C₂²	1	1	yes	characteristic
17	C₃₇×#336	—	2	2	no	—
18	C₃₇×#336	—	2	1	no	—
19	C₃₇×D₄⋊C₈	C₄	1	8	yes	complemented
20	C₂²×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₇	—	2	1	no	—
22	C₂²×C₈×C₃₇	—	2	2	no	—
23	C₂²×C₈×C₃₇	—	2	2	no	—
24	C₄×D₄×C₃₇	—	2	2	no	—
25	C₄×D₄×C₃₇	C₂×C₄	1	4	yes	—
26	#117×C₃₇	—	4	4	no	—
27	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
33	C₂×C₄²×C₃₇	—	2	1	no	—
34	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₇	D₄	1	2	yes	—
38	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₃₇	—	2	2	no	—
40	C₄⋊C₈×C₃₇	C₂×C₄	1	4	yes	—
41	#97×C₃₇	—	4	2	no	—
42	C₄×C₈×C₃₇	—	2	2	no	—
43	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
44	#95×C₃₇	—	2	2	no	—
45	#95×C₃₇	—	2	1	no	—
46	C₂⁴×C₃₇	—	2	1	no	—
47	C₂×D₄×C₃₇	—	2	1	no	—
48	C₂×D₄×C₃₇	—	2	4	no	—
49	C₂×D₄×C₃₇	C₄²	1	2	yes	—
50	C₂²×C₄×C₃₇	—	2	1	no	—
51	C₂²×C₄×C₃₇	—	2	4	no	—
52	C₂²×C₄×C₃₇	—	2	2	no	—
53	C₂²×C₄×C₃₇	—	4	2	no	—
54	C₂²×C₄×C₃₇	—	4	1	no	—
55	C₂²×C₄×C₃₇	—	4	1	no	—
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	1	yes	characteristic
59	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₇	—	2	4	no	—
62	C₂×C₈×C₃₇	—	2	4	no	—
63	C₂×C₈×C₃₇	—	4	4	no	—
64	C₂×C₈×C₃₇	—	4	2	no	—
65	C₂×C₈×C₃₇	—	4	2	no	—
66	C₄⋊C₄×C₃₇	—	2	1	no	—
67	C₄⋊C₄×C₃₇	—	2	4	no	—
68	C₄⋊C₄×C₃₇	—	4	4	no	—
69	C₄⋊C₄×C₃₇	C₄²	1	2	yes	—
70	C₂²⋊C₄×C₃₇	—	2	4	no	—
71	C₂²⋊C₄×C₃₇	—	4	4	no	—
72	C₂²⋊C₄×C₃₇	—	4	4	no	—
73	C₄²×C₃₇	—	2	4	no	—
74	C₄²×C₃₇	—	2	1	no	—
75	C₄²×C₃₇	—	2	2	no	—
76	C₄²×C₃₇	—	2	1	no	—
77	C₄²×C₃₇	—	4	2	no	—
78	C₄²×C₃₇	—	4	2	no	—
79	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₃₇	#41	1	2	yes	—
81	C₂³×C₃₇	—	2	2	no	—
82	C₂³×C₃₇	—	4	2	no	—
83	C₂³×C₃₇	—	4	1	no	—
84	C₂³×C₃₇	—	4	1	no	—
85	C₂³×C₃₇	—	4	2	no	—
86	C₂³×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	D₄×C₃₇	—	2	4	no	—
88	D₄×C₃₇	—	4	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	2	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	—	4	1	no	—
98	C₂×C₄×C₃₇	—	4	1	no	—
99	C₂×C₄×C₃₇	—	4	4	no	—
100	C₂×C₄×C₃₇	—	4	4	no	—
101	C₂×C₄×C₃₇	—	4	2	no	—
102	C₂×C₄×C₃₇	—	4	2	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₃₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₃₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₃₇	#131	1	4	yes	—
109	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₃₇	#137	1	1	yes	characteristic
112	C₈×C₃₇	—	4	4	no	—
113	C₈×C₃₇	—	4	4	no	—
114	#8179	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₃₇	—	4	2	no	—
116	C₂²×C₃₇	—	4	2	no	—
117	C₂²×C₃₇	—	4	2	no	—
118	C₂²×C₃₇	—	8	2	no	—
119	C₂²×C₃₇	—	8	2	no	—
120	C₂²×C₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₃₇	#408	1	2	yes	—
122	C₂²×C₃₇	#417	1	1	yes	characteristic
123	C₂²×C₃₇	#418	1	1	yes	characteristic
124	C₂²×C₃₇	#421	1	2	yes	—
125	C₄×C₃₇	—	2	2	no	—
126	C₄×C₃₇	—	4	2	no	—
127	C₄×C₃₇	—	4	2	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	—	8	2	no	—
130	C₄×C₃₇	#405	1	2	yes	—
131	C₄×C₃₇	#433	1	2	yes	—
132	C₄×C₃₇	#440	1	2	yes	—
133	C₄×C₃₇	#442	1	2	yes	—
134	#2293	C₂×C₃₇	1	1	yes	characteristic
135	#1767	C₂×C₃₇	1	1	yes	characteristic, complemented
136	#1744	C₂×C₃₇	1	1	yes	characteristic, complemented
137	C₂×D₄⋊C₈	C₂×C₃₇	1	2	yes	—
138	#1279	C₂×C₃₇	1	2	yes	complemented
139	C₂×C₃₇	—	8	2	no	—
140	C₂×C₃₇	#1495	1	2	yes	—
141	C₂×C₃₇	#1496	1	2	yes	—
142	C₂×C₃₇	#1750	1	1	yes	characteristic
143	C₂×C₃₇	#1759	1	1	yes	characteristic
144	C₂×C₃₇	#1762	1	1	yes	characteristic
145	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
146	C₂×#117	—	2	1	no	—
147	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
150	#390	C₄×C₃₇	1	2	yes	—
151	#389	C₄×C₃₇	1	2	yes	—
152	#388	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
154	C₄×C₂²⋊C₄	—	2	1	no	—
155	C₄³	C₂²×C₃₇	1	1	yes	characteristic
156	#336	—	2	2	no	—
157	#336	—	2	1	no	—
158	D₄⋊C₈	C₄×C₃₇	1	8	yes	complemented
159	C₃₇	#8179	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×D₄	C₂×C₄×C₃₇	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	C₂×C₄×C₃₇	1	4	yes	—
166	#117	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
175	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₃₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂×C₄×C₃₇	1	4	yes	—
181	#97	—	4	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
184	#95	—	2	2	no	—
185	#95	—	2	1	no	—
186	C₂⁴	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	C₄²×C₃₇	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	C₂²⋊C₄×C₃₇	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₃₇	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	C₄²×C₃₇	1	2	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	4	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
220	C₄²	#41×C₃₇	1	2	yes	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×C₄²×C₃₇	1	1	yes	characteristic
245	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₃₇	1	2	yes	—
247	C₂×C₄	C₄×D₄×C₃₇	1	2	yes	—
248	C₂×C₄	#131×C₃₇	1	4	yes	—
249	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
250	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic
251	C₂×C₄	#137×C₃₇	1	1	yes	characteristic
252	C₈	—	4	4	no	—
253	C₈	—	4	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₄×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
260	C₂²	C₃₇×#408	1	2	yes	—
261	C₂²	C₃₇×#417	1	1	yes	characteristic
262	C₂²	C₃₇×#418	1	1	yes	characteristic
263	C₂²	C₃₇×#421	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	C₃₇×#405	1	2	yes	—
270	C₄	C₃₇×#433	1	2	yes	—
271	C₄	C₃₇×#440	1	2	yes	—
272	C₄	C₃₇×#442	1	2	yes	—
273	C₂	—	8	2	no	—
274	C₂	C₃₇×#1495	1	2	yes	—
275	C₂	C₃₇×#1496	1	2	yes	—
276	C₂	C₃₇×#1750	1	1	yes	characteristic
277	C₂	C₃₇×#1759	1	1	yes	characteristic
278	C₂	C₃₇×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
socle	C₂³×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#1767	C₂	semidirect
3. C₃₇×#1744	C₂	semidirect
5. C₃₇×#1279	C₂	semidirect
19. C₃₇×D₄⋊C₈	C₄	semidirect
114. #8179	C₃₇	direct (N × H)
135. #1767	C₂×C₃₇	semidirect
136. #1744	C₂×C₃₇	semidirect
138. #1279	C₂×C₃₇	semidirect
158. D₄⋊C₈	C₄×C₃₇	semidirect
159. C₃₇	#8179	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×#2293	2
C₃₇×#1767	2
C₃₇×#1744	2
C₂×C₃₇×D₄⋊C₈	2
C₃₇×#1279	2
#8179	37

Subgroups — C37×#8179 #22629432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8179 #22629432