Subgroups — C₃₇×#28516 #22649769

Order 9472.

990 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#478	C₂	1	4	yes	maximal, complemented
2	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2094	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2093	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1961	C₂	1	2	yes	maximal
7	C₃₇×#1847	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₃₇	—	2	2	no	—
12	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#478	—	2	16	no	—
14	C₂×#107×C₃₇	C₂²	1	4	yes	—
15	C₂×#106×C₃₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#102×C₃₇	C₂²	1	4	yes	—
18	C₂×#102×C₃₇	C₂²	1	4	yes	—
19	C₂×#97×C₃₇	C₂²	1	2	yes	—
20	C₃₇×#390	—	2	2	no	—
21	C₃₇×#390	C₂²	1	2	yes	—
22	C₃₇×#389	—	2	2	no	—
23	C₃₇×#389	—	2	2	no	—
24	C₃₇×#384	C₂²	1	1	yes	characteristic
25	C₃₇×#384	C₂²	1	1	yes	characteristic
26	C₃₇×#384	C₂²	1	1	yes	characteristic
27	C₃₇×#382	C₂²	1	2	yes	—
28	C₄³×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×D₄×C₃₇	—	2	2	no	—
31	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₇	—	2	2	no	—
33	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
34	#127×C₃₇	—	2	2	no	—
35	#127×C₃₇	Q₈	1	4	yes	—
36	#125×C₃₇	—	2	8	no	—
37	C₄×D₄×C₃₇	—	2	8	no	—
38	C₄×D₄×C₃₇	—	4	8	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
49	C₂×C₄²×C₃₇	D₄	1	2	yes	—
50	C₂×C₄²×C₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	#107×C₃₇	—	2	8	no	—
55	#106×C₃₇	—	2	8	no	—
56	C₄⋊C₈×C₃₇	D₄	1	8	yes	complemented
57	#102×C₃₇	—	2	8	no	—
58	#102×C₃₇	—	2	8	no	—
59	#97×C₃₇	—	2	4	no	—
60	#95×C₃₇	—	2	4	no	—
61	C₂⁴×C₃₇	—	2	2	no	—
62	C₂×D₄×C₃₇	—	2	8	no	—
63	C₂×D₄×C₃₇	—	2	4	no	—
64	C₂×D₄×C₃₇	—	2	2	no	—
65	C₂×D₄×C₃₇	—	2	4	no	—
66	C₂×D₄×C₃₇	—	4	4	no	—
67	C₂×D₄×C₃₇	—	4	4	no	—
68	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₃₇	—	2	2	no	—
70	C₂²×C₄×C₃₇	—	2	2	no	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	—	4	8	no	—
73	C₂²×C₄×C₃₇	—	4	4	no	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₃₇	—	2	4	no	—
80	C₂×C₈×C₃₇	—	2	4	no	—
81	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
82	C₄⋊C₄×C₃₇	—	2	4	no	—
83	C₄⋊C₄×C₃₇	—	2	4	no	—
84	C₄⋊C₄×C₃₇	—	2	4	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₄⋊C₄×C₃₇	—	4	4	no	—
87	C₄⋊C₄×C₃₇	—	4	4	no	—
88	C₄⋊C₄×C₃₇	—	4	4	no	—
89	C₄⋊C₄×C₃₇	—	4	4	no	—
90	C₂²⋊C₄×C₃₇	—	4	4	no	—
91	C₂²⋊C₄×C₃₇	—	4	8	no	—
92	C₄²×C₃₇	—	2	4	no	—
93	C₄²×C₃₇	—	2	2	no	—
94	C₄²×C₃₇	—	4	2	no	—
95	C₄²×C₃₇	C₂×D₄	1	4	yes	—
96	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
97	C₄²×C₃₇	#41	1	2	yes	—
98	C₂³×C₃₇	—	2	4	no	—
99	C₂³×C₃₇	—	4	2	no	—
100	C₂³×C₃₇	—	4	2	no	—
101	C₂³×C₃₇	—	4	4	no	—
102	C₂³×C₃₇	—	4	4	no	—
103	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₃₇	—	2	8	no	—
105	D₄×C₃₇	—	4	16	no	—
106	D₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	2	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	2	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	2	no	—
115	C₂×C₄×C₃₇	—	4	2	no	—
116	C₂×C₄×C₃₇	—	4	4	no	—
117	C₂×C₄×C₃₇	—	4	4	no	—
118	C₂×C₄×C₃₇	—	4	4	no	—
119	C₂×C₄×C₃₇	—	8	8	no	—
120	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₃₇	#127	1	2	yes	—
122	C₂×C₄×C₃₇	#127	1	2	yes	—
123	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
124	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
125	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
128	C₈×C₃₇	—	2	8	no	—
129	#28516	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₃₇	—	4	8	no	—
131	C₂²×C₃₇	—	4	4	no	—
132	C₂²×C₃₇	—	8	2	no	—
133	C₂²×C₃₇	—	8	2	no	—
134	C₂²×C₃₇	—	8	4	no	—
135	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
137	C₂²×C₃₇	#536	1	1	yes	characteristic
138	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
139	C₂²×C₃₇	#575	1	2	yes	—
140	C₄×C₃₇	—	2	4	no	—
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	2	4	no	—
144	C₄×C₃₇	—	8	2	no	—
145	C₄×C₃₇	—	8	2	no	—
146	C₄×C₃₇	—	8	2	no	—
147	C₄×C₃₇	#547	1	2	yes	—
148	C₄×C₃₇	D₄×Q₈	1	2	yes	—
149	C₂×#478	C₂×C₃₇	1	4	yes	complemented
150	#2859	C₂×C₃₇	1	1	yes	characteristic, complemented
151	C₄×#127	C₂×C₃₇	1	1	yes	characteristic
152	#2094	C₂×C₃₇	1	2	yes	complemented
153	#2093	C₂×C₃₇	1	2	yes	complemented
154	#1961	C₂×C₃₇	1	2	yes	—
155	#1847	C₂×C₃₇	1	2	yes	complemented
156	#1767	C₂×C₃₇	1	1	yes	characteristic, complemented
157	C₂×C₃₇	—	8	4	no	—
158	C₂×C₃₇	#2542	1	1	yes	characteristic
159	C₂×C₃₇	#3077	1	2	yes	—
160	C₂×C₃₇	#3150	1	1	yes	characteristic
161	C₂×C₃₇	#3151	1	1	yes	characteristic
162	C₂×C₃₇	#3234	1	2	yes	—
163	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
164	C₂×#125	C₂²×C₃₇	1	2	yes	—
165	C₂×C₄×D₄	—	2	2	no	—
166	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
167	#478	—	2	16	no	—
168	C₂×#107	C₂²×C₃₇	1	4	yes	—
169	C₂×#106	C₂²×C₃₇	1	4	yes	—
170	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
171	C₂×#102	C₂²×C₃₇	1	4	yes	—
172	C₂×#102	C₂²×C₃₇	1	4	yes	—
173	C₂×#97	C₂²×C₃₇	1	2	yes	—
174	#390	—	2	2	no	—
175	#390	C₂²×C₃₇	1	2	yes	—
176	#389	—	2	2	no	—
177	#389	—	2	2	no	—
178	#384	C₂²×C₃₇	1	1	yes	characteristic
179	#384	C₂²×C₃₇	1	1	yes	characteristic
180	#384	C₂²×C₃₇	1	1	yes	characteristic
181	#382	C₂²×C₃₇	1	2	yes	—
182	C₄³	C₂²×C₃₇	1	1	yes	characteristic
183	#336	C₂²×C₃₇	1	2	yes	—
184	C₃₇	#28516	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
189	#127	—	2	2	no	—
190	#127	Q₈×C₃₇	1	4	yes	—
191	#125	—	2	8	no	—
192	C₄×D₄	—	2	8	no	—
193	C₄×D₄	—	4	8	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	D₄×C₃₇	1	2	yes	—
205	C₂×C₄²	D₄×C₃₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
209	#107	—	2	8	no	—
210	#106	—	2	8	no	—
211	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
212	#102	—	2	8	no	—
213	#102	—	2	8	no	—
214	#97	—	2	4	no	—
215	#95	—	2	4	no	—
216	C₂⁴	—	2	2	no	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	8	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	8	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×D₄×C₃₇	1	4	yes	—
251	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
252	C₄²	#41×C₃₇	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
259	D₄	—	2	8	no	—
260	D₄	—	4	16	no	—
261	D₄	—	4	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	8	no	—
275	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
276	C₂×C₄	#127×C₃₇	1	2	yes	—
277	C₂×C₄	#127×C₃₇	1	2	yes	—
278	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
279	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
280	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
282	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
283	C₈	—	2	8	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	8	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	4	no	—
289	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
290	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
291	C₂²	C₃₇×#536	1	1	yes	characteristic
292	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
293	C₂²	C₃₇×#575	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	2	4	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₃₇×#547	1	2	yes	—
302	C₄	D₄×Q₈×C₃₇	1	2	yes	—
303	C₂	—	8	4	no	—
304	C₂	C₃₇×#2542	1	1	yes	characteristic
305	C₂	C₃₇×#3077	1	2	yes	—
306	C₂	C₃₇×#3150	1	1	yes	characteristic
307	C₂	C₃₇×#3151	1	1	yes	characteristic
308	C₂	C₃₇×#3234	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
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
1. C₂×C₃₇×#478	C₂	semidirect
2. C₃₇×#2859	C₂	semidirect
4. C₃₇×#2094	C₂	semidirect
5. C₃₇×#2093	C₂	semidirect
7. C₃₇×#1847	C₂	semidirect
8. C₃₇×#1767	C₂	semidirect
56. C₄⋊C₈×C₃₇	D₄	semidirect
129. #28516	C₃₇	direct (N × H)
149. C₂×#478	C₂×C₃₇	semidirect
150. #2859	C₂×C₃₇	semidirect
152. #2094	C₂×C₃₇	semidirect
153. #2093	C₂×C₃₇	semidirect
155. #1847	C₂×C₃₇	semidirect
156. #1767	C₂×C₃₇	semidirect
184. C₃₇	#28516	direct (N × H)
211. C₄⋊C₈	D₄×C₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₇×#478	2
C₃₇×#2859	2
C₄×#127×C₃₇	2
C₃₇×#2094	2
C₃₇×#2093	2
C₃₇×#1961	2
C₃₇×#1847	2
C₃₇×#1767	2
#28516	37

Subgroups — C37×#28516 #22649769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28516 #22649769