Subgroups — C₃₇×#28299 #22649552

Order 9472.

730 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3077	C₂	1	2	yes	maximal, complemented
2	C₃₇×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#442	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2063	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1978	C₂	1	2	yes	maximal
6	C₃₇×#1951	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1800	C₂	1	2	yes	maximal, complemented
9	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#533	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#478	—	2	4	no	—
13	C₃₇×#476	—	2	2	no	—
14	C₃₇×#474	—	2	2	no	—
15	C₃₇×#446	—	2	4	no	—
16	C₃₇×#442	—	2	4	no	—
17	C₂×#108×C₃₇	C₂²	1	2	yes	—
18	C₂×#108×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#428	C₂²	1	4	yes	—
20	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#424	C₂²	1	2	yes	—
22	C₃₇×#423	C₂²	1	2	yes	—
23	C₃₇×#423	C₂²	1	1	yes	characteristic
24	C₃₇×#418	C₂²	1	4	yes	—
25	C₂×#102×C₃₇	C₂²	1	2	yes	—
26	C₂×#100×C₃₇	C₂²	1	4	yes	—
27	C₃₇×#405	C₂²	1	2	yes	—
28	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#343	C₂²	1	4	yes	—
30	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₃₇	—	2	2	no	—
32	C₂×D₈×C₃₇	—	2	2	no	—
33	C₂×D₈×C₃₇	—	2	2	no	—
34	C₂×D₈×C₃₇	Q₈	1	4	yes	complemented
35	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
36	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₃₇	D₄	1	2	yes	—
40	#125×C₃₇	D₄	1	2	yes	—
41	C₂²⋊Q₈×C₃₇	—	2	2	no	—
42	C₄×Q₈×C₃₇	—	2	2	no	—
43	C₄×D₄×C₃₇	—	2	2	no	—
44	C₄×D₄×C₃₇	—	2	2	no	—
45	C₄×D₄×C₃₇	—	2	2	no	—
46	C₄×D₄×C₃₇	—	2	2	no	—
47	#117×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	#108×C₃₇	—	2	4	no	—
52	#108×C₃₇	—	4	2	no	—
53	#107×C₃₇	—	2	4	no	—
54	#106×C₃₇	—	2	2	no	—
55	#106×C₃₇	—	2	4	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	C₄⋊C₈×C₃₇	—	2	2	no	—
59	C₄⋊C₈×C₃₇	—	2	2	no	—
60	#102×C₃₇	—	2	4	no	—
61	#102×C₃₇	—	2	4	no	—
62	#102×C₃₇	—	2	4	no	—
63	#100×C₃₇	—	4	4	no	—
64	#97×C₃₇	—	2	2	no	—
65	#97×C₃₇	D₄	1	4	yes	—
66	C₄×C₈×C₃₇	—	2	2	no	—
67	C₂⁴×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×D₄×C₃₇	—	2	2	no	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	2	2	no	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	—	2	2	no	—
74	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	4	2	no	—
77	C₂²×C₄×C₃₇	—	4	2	no	—
78	C₂²×C₄×C₃₇	—	8	4	no	—
79	C₂²×C₄×C₃₇	#41	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	D₈×C₃₇	—	2	8	no	—
82	M₁₆×C₃₇	—	2	4	no	—
83	M₁₆×C₃₇	—	2	2	no	—
84	M₁₆×C₃₇	—	2	4	no	—
85	M₁₆×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	1	no	—
87	C₂×C₈×C₃₇	—	2	1	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	—	2	4	no	—
90	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₃₇	#41	1	2	yes	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	2	no	—
98	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
99	C₂²⋊C₄×C₃₇	—	2	2	no	—
100	C₂²⋊C₄×C₃₇	—	4	2	no	—
101	C₂²⋊C₄×C₃₇	—	4	2	no	—
102	C₄²×C₃₇	—	2	2	no	—
103	C₄²×C₃₇	C₂×D₄	1	2	yes	—
104	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₃₇	—	2	4	no	—
107	C₂³×C₃₇	—	4	2	no	—
108	C₂³×C₃₇	—	4	2	no	—
109	C₂³×C₃₇	—	8	2	no	—
110	C₂³×C₃₇	—	8	2	no	—
111	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
112	Q₈×C₃₇	—	8	2	no	—
113	D₄×C₃₇	—	2	4	no	—
114	D₄×C₃₇	—	2	4	no	—
115	D₄×C₃₇	—	4	2	no	—
116	D₄×C₃₇	—	4	2	no	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	8	4	no	—
121	C₂×C₄×C₃₇	—	8	4	no	—
122	C₂×C₄×C₃₇	—	8	2	no	—
123	C₂×C₄×C₃₇	#121	1	2	yes	—
124	C₂×C₄×C₃₇	#121	1	2	yes	—
125	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
131	C₈×C₃₇	—	2	2	no	—
132	C₈×C₃₇	—	4	2	no	—
133	C₈×C₃₇	—	4	4	no	—
134	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
135	#28299	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₃₇	—	2	2	no	—
137	C₂²×C₃₇	—	4	4	no	—
138	C₂²×C₃₇	—	8	4	no	—
139	C₂²×C₃₇	—	8	4	no	—
140	C₂²×C₃₇	—	8	2	no	—
141	C₂²×C₃₇	—	8	2	no	—
142	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
143	C₂²×C₃₇	#554	1	1	yes	characteristic
144	C₂²×C₃₇	#556	1	1	yes	characteristic
145	C₄×C₃₇	—	2	2	no	—
146	C₄×C₃₇	—	2	2	no	—
147	C₄×C₃₇	—	4	4	no	—
148	C₄×C₃₇	—	8	2	no	—
149	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₃₇	#537	1	1	yes	characteristic
151	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
152	C₄×C₃₇	#550	1	1	yes	characteristic
153	#3077	C₂×C₃₇	1	2	yes	complemented
154	#2961	C₂×C₃₇	1	1	yes	characteristic, complemented
155	C₂×#442	C₂×C₃₇	1	1	yes	characteristic
156	#2063	C₂×C₃₇	1	4	yes	complemented
157	#1978	C₂×C₃₇	1	2	yes	—
158	#1951	C₂×C₃₇	1	2	yes	complemented
159	#1849	C₂×C₃₇	1	1	yes	characteristic, complemented
160	#1800	C₂×C₃₇	1	2	yes	complemented
161	C₂×C₃₇	—	2	1	no	—
162	C₂×C₃₇	—	2	1	no	—
163	C₂×C₃₇	—	8	4	no	—
164	C₂×C₃₇	#2538	1	1	yes	characteristic
165	C₂×C₃₇	#3068	1	2	yes	—
166	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
167	#533	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
169	#478	—	2	4	no	—
170	#476	—	2	2	no	—
171	#474	—	2	2	no	—
172	#446	—	2	4	no	—
173	#442	—	2	4	no	—
174	C₂×#108	C₂²×C₃₇	1	2	yes	—
175	C₂×#108	C₂²×C₃₇	1	2	yes	—
176	#428	C₂²×C₃₇	1	4	yes	—
177	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
178	#424	C₂²×C₃₇	1	2	yes	—
179	#423	C₂²×C₃₇	1	2	yes	—
180	#423	C₂²×C₃₇	1	1	yes	characteristic
181	#418	C₂²×C₃₇	1	4	yes	—
182	C₂×#102	C₂²×C₃₇	1	2	yes	—
183	C₂×#100	C₂²×C₃₇	1	4	yes	—
184	#405	C₂²×C₃₇	1	2	yes	—
185	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
186	#343	C₂²×C₃₇	1	4	yes	—
187	C₃₇	#28299	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
189	C₂³×C₄	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	Q₈×C₃₇	1	4	yes	complemented
193	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
194	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
195	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
197	C₄⋊Q₈	D₄×C₃₇	1	2	yes	—
198	#125	D₄×C₃₇	1	2	yes	—
199	C₂²⋊Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	#117	C₂³×C₃₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
209	#108	—	2	4	no	—
210	#108	—	4	2	no	—
211	#107	—	2	4	no	—
212	#106	—	2	2	no	—
213	#106	—	2	4	no	—
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	#102	—	2	4	no	—
219	#102	—	2	4	no	—
220	#102	—	2	4	no	—
221	#100	—	4	4	no	—
222	#97	—	2	2	no	—
223	#97	D₄×C₃₇	1	4	yes	—
224	C₄×C₈	—	2	2	no	—
225	C₂⁴	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	8	4	no	—
237	C₂²×C₄	#41×C₃₇	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
239	D₈	—	2	8	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
250	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
251	C₂×C₈	#41×C₃₇	1	2	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×D₄×C₃₇	1	2	yes	—
262	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
264	C₂³	—	2	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	—	8	2	no	—
269	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
270	Q₈	—	8	2	no	—
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	#121×C₃₇	1	2	yes	—
282	C₂×C₄	#121×C₃₇	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
287	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
288	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	4	no	—
292	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
300	C₂²	C₃₇×#554	1	1	yes	characteristic
301	C₂²	C₃₇×#556	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
307	C₄	C₃₇×#537	1	1	yes	characteristic
308	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
309	C₄	C₃₇×#550	1	1	yes	characteristic
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	8	4	no	—
313	C₂	C₃₇×#2538	1	1	yes	characteristic
314	C₂	C₃₇×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₈×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3077	C₂	semidirect
2. C₃₇×#2961	C₂	semidirect
4. C₃₇×#2063	C₂	semidirect
6. C₃₇×#1951	C₂	semidirect
7. C₃₇×#1849	C₂	semidirect
8. C₃₇×#1800	C₂	semidirect
34. C₂×D₈×C₃₇	Q₈	semidirect
135. #28299	C₃₇	direct (N × H)
153. #3077	C₂×C₃₇	semidirect
154. #2961	C₂×C₃₇	semidirect
156. #2063	C₂×C₃₇	semidirect
158. #1951	C₂×C₃₇	semidirect
159. #1849	C₂×C₃₇	semidirect
160. #1800	C₂×C₃₇	semidirect
187. C₃₇	#28299	direct (N × H)
192. C₂×D₈	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3077	2
C₃₇×#2961	2
C₂×C₃₇×#442	2
C₃₇×#2063	2
C₃₇×#1978	2
C₃₇×#1951	2
C₃₇×#1849	2
C₃₇×#1800	2
#28299	37

Subgroups — C37×#28299 #22649552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28299 #22649552