Subgroups — C₃₇×#28518 #22649771

Order 9472.

822 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#479	C₂	1	4	yes	maximal
2	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2096	C₂	1	2	yes	maximal
5	C₃₇×#2095	C₂	1	2	yes	maximal
6	C₃₇×#1960	C₂	1	2	yes	maximal
7	C₃₇×#1847	C₂	1	2	yes	maximal
8	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#479	—	2	16	no	—
13	C₂×#107×C₃₇	C₂²	1	4	yes	—
14	C₂×#106×C₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₃₇	C₂²	1	4	yes	—
17	C₂×#103×C₃₇	C₂²	1	4	yes	—
18	C₂×#97×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#391	C₂²	1	2	yes	—
20	C₃₇×#389	—	2	2	no	—
21	C₃₇×#389	—	2	2	no	—
22	C₃₇×#389	—	2	2	no	—
23	C₃₇×#384	C₂²	1	1	yes	characteristic
24	C₃₇×#384	C₂²	1	1	yes	characteristic
25	C₃₇×#384	C₂²	1	1	yes	characteristic
26	C₃₇×#382	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
28	C₄³×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₃₇	—	2	2	no	—
33	C₄⋊Q₈×C₃₇	Q₈	1	4	yes	—
34	#125×C₃₇	—	2	8	no	—
35	C₄×Q₈×C₃₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₇	—	2	2	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	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₇	—	2	2	no	—
47	C₂×C₄²×C₃₇	D₄	1	2	yes	—
48	C₂×C₄²×C₃₇	D₄	1	2	yes	—
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	#107×C₃₇	—	2	8	no	—
53	#106×C₃₇	—	2	8	no	—
54	C₄⋊C₈×C₃₇	D₄	1	8	yes	—
55	#103×C₃₇	—	2	8	no	—
56	#103×C₃₇	—	2	8	no	—
57	#97×C₃₇	—	2	4	no	—
58	#95×C₃₇	—	2	4	no	—
59	#95×C₃₇	—	2	4	no	—
60	C₂×Q₈×C₃₇	—	2	2	no	—
61	C₂×Q₈×C₃₇	—	2	4	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₃₇	—	2	2	no	—
65	C₂²×C₄×C₃₇	—	2	2	no	—
66	C₂²×C₄×C₃₇	—	2	2	no	—
67	C₂²×C₄×C₃₇	—	2	2	no	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
76	C₄⋊C₄×C₃₇	—	2	8	no	—
77	C₄⋊C₄×C₃₇	—	2	4	no	—
78	C₄⋊C₄×C₃₇	—	2	4	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	4	4	no	—
81	C₄⋊C₄×C₃₇	—	4	4	no	—
82	C₄⋊C₄×C₃₇	—	4	4	no	—
83	C₄⋊C₄×C₃₇	—	4	4	no	—
84	C₄⋊C₄×C₃₇	—	4	4	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₄²×C₃₇	—	2	4	no	—
87	C₄²×C₃₇	—	2	2	no	—
88	C₄²×C₃₇	—	4	2	no	—
89	C₄²×C₃₇	—	4	4	no	—
90	C₄²×C₃₇	C₂×D₄	1	4	yes	—
91	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
92	C₄²×C₃₇	#41	1	2	yes	—
93	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₃₇	—	2	8	no	—
95	Q₈×C₃₇	—	4	4	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	4	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	2	no	—
107	C₂×C₄×C₃₇	—	4	2	no	—
108	C₂×C₄×C₃₇	—	4	4	no	—
109	C₂×C₄×C₃₇	—	4	4	no	—
110	C₂×C₄×C₃₇	—	4	4	no	—
111	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₃₇	#127	1	2	yes	—
113	C₂×C₄×C₃₇	#127	1	2	yes	—
114	C₂×C₄×C₃₇	#137	1	4	yes	—
115	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
119	C₈×C₃₇	—	2	8	no	—
120	#28518	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
123	C₂²×C₃₇	#536	1	1	yes	characteristic
124	C₂²×C₃₇	C₂×#137	1	2	yes	—
125	C₂²×C₃₇	#575	1	2	yes	—
126	C₄×C₃₇	—	2	4	no	—
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	—	2	2	no	—
129	C₄×C₃₇	—	2	4	no	—
130	C₄×C₃₇	—	4	4	no	—
131	C₄×C₃₇	—	8	2	no	—
132	C₄×C₃₇	—	8	2	no	—
133	C₄×C₃₇	—	8	2	no	—
134	C₄×C₃₇	#547	1	2	yes	—
135	C₄×C₃₇	D₄×Q₈	1	2	yes	—
136	C₂×#479	C₂×C₃₇	1	4	yes	—
137	#2859	C₂×C₃₇	1	1	yes	characteristic
138	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
139	#2096	C₂×C₃₇	1	2	yes	—
140	#2095	C₂×C₃₇	1	2	yes	—
141	#1960	C₂×C₃₇	1	2	yes	—
142	#1847	C₂×C₃₇	1	2	yes	—
143	#1767	C₂×C₃₇	1	1	yes	characteristic
144	C₂×C₃₇	#2542	1	1	yes	characteristic
145	C₂×C₃₇	#3078	1	2	yes	—
146	C₂×C₃₇	#3150	1	1	yes	characteristic
147	C₂×C₃₇	#3152	1	1	yes	characteristic
148	C₂×C₃₇	#3236	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
150	C₂×#125	C₂²×C₃₇	1	2	yes	—
151	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
152	#479	—	2	16	no	—
153	C₂×#107	C₂²×C₃₇	1	4	yes	—
154	C₂×#106	C₂²×C₃₇	1	4	yes	—
155	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
156	C₂×#103	C₂²×C₃₇	1	4	yes	—
157	C₂×#103	C₂²×C₃₇	1	4	yes	—
158	C₂×#97	C₂²×C₃₇	1	2	yes	—
159	#391	C₂²×C₃₇	1	2	yes	—
160	#389	—	2	2	no	—
161	#389	—	2	2	no	—
162	#389	—	2	2	no	—
163	#384	C₂²×C₃₇	1	1	yes	characteristic
164	#384	C₂²×C₃₇	1	1	yes	characteristic
165	#384	C₂²×C₃₇	1	1	yes	characteristic
166	#382	C₂²×C₃₇	1	2	yes	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄³	C₂²×C₃₇	1	1	yes	characteristic
169	#336	C₂²×C₃₇	1	2	yes	—
170	C₃₇	#28518	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	Q₈×C₃₇	1	4	yes	—
175	#125	—	2	8	no	—
176	C₄×Q₈	—	2	8	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	D₄×C₃₇	1	2	yes	—
189	C₂×C₄²	D₄×C₃₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
193	#107	—	2	8	no	—
194	#106	—	2	8	no	—
195	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
196	#103	—	2	8	no	—
197	#103	—	2	8	no	—
198	#97	—	2	4	no	—
199	#95	—	2	4	no	—
200	#95	—	2	4	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
213	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂×D₄×C₃₇	1	4	yes	—
232	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
233	C₄²	#41×C₃₇	1	2	yes	—
234	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	4	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
253	C₂×C₄	#127×C₃₇	1	2	yes	—
254	C₂×C₄	#127×C₃₇	1	2	yes	—
255	C₂×C₄	#137×C₃₇	1	4	yes	—
256	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
257	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
259	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
260	C₈	—	2	8	no	—
261	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
262	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
263	C₂²	C₃₇×#536	1	1	yes	characteristic
264	C₂²	C₂×#137×C₃₇	1	2	yes	—
265	C₂²	C₃₇×#575	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	2	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	8	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	2	no	—
274	C₄	C₃₇×#547	1	2	yes	—
275	C₄	D₄×Q₈×C₃₇	1	2	yes	—
276	C₂	C₃₇×#2542	1	1	yes	characteristic
277	C₂	C₃₇×#3078	1	2	yes	—
278	C₂	C₃₇×#3150	1	1	yes	characteristic
279	C₂	C₃₇×#3152	1	1	yes	characteristic
280	C₂	C₃₇×#3236	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
120. #28518	C₃₇	direct (N × H)
170. C₃₇	#28518	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#479	2
C₃₇×#2859	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2096	2
C₃₇×#2095	2
C₃₇×#1960	2
C₃₇×#1847	2
C₃₇×#1767	2
#28518	37

Subgroups — C37×#28518 #22649771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28518 #22649771