Subgroups — C₃₇×#28498 #22649751

Order 9472.

1002 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3152	C₂	1	2	yes	maximal
2	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#473	C₂	1	4	yes	maximal, complemented
4	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2020	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1966	C₂	1	2	yes	maximal, complemented
7	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₃₇	C₂²	1	8	yes	—
9	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
11	C₂×#131×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#532	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#497	—	2	4	no	—
15	C₃₇×#494	—	2	2	no	—
16	C₃₇×#494	—	2	2	no	—
17	C₃₇×#494	—	2	2	no	—
18	C₃₇×#492	—	2	2	no	—
19	C₃₇×#473	—	2	16	no	—
20	C₃₇×#436	—	2	2	no	—
21	C₈×D₄×C₃₇	—	2	2	no	—
22	C₂×#108×C₃₇	C₂²	1	2	yes	—
23	C₃₇×#423	C₂²	1	1	yes	characteristic
24	C₂×#101×C₃₇	C₂²	1	8	yes	—
25	C₃₇×#408	C₂²	1	2	yes	—
26	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
28	C₃₇×#341	C₂²	1	2	yes	—
29	C₂×#41×C₃₇	C₂³	1	2	yes	—
30	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
31	#137×C₃₇	—	2	16	no	—
32	#137×C₃₇	—	4	8	no	—
33	C₂×Q₁₆×C₃₇	—	2	8	no	—
34	C₂×Q₁₆×C₃₇	—	2	8	no	—
35	C₂×Q₁₆×C₃₇	—	2	4	no	—
36	C₂×Q₁₆×C₃₇	—	2	2	no	—
37	C₂×Q₁₆×C₃₇	—	2	2	no	—
38	C₂×Q₁₆×C₃₇	—	4	4	no	—
39	C₂×SD₁₆×C₃₇	—	2	8	no	—
40	#131×C₃₇	—	2	8	no	—
41	C₂×M₁₆×C₃₇	—	2	2	no	—
42	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
43	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
44	C₂²×C₈×C₃₇	—	2	2	no	—
45	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₃₇	—	2	2	no	—
48	C₄⋊Q₈×C₃₇	—	2	4	no	—
49	C₄⋊Q₈×C₃₇	—	2	2	no	—
50	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
51	#127×C₃₇	D₄	1	1	yes	characteristic
52	#124×C₃₇	D₄	1	2	yes	—
53	#121×C₃₇	—	2	2	no	—
54	C₄×D₄×C₃₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	#108×C₃₇	D₄	1	8	yes	complemented
58	C₄⋊C₈×C₃₇	—	2	2	no	—
59	#101×C₃₇	—	2	16	no	—
60	C₂²⋊C₈×C₃₇	—	2	4	no	—
61	#97×C₃₇	—	2	2	no	—
62	C₄×C₈×C₃₇	—	2	2	no	—
63	C₄×C₈×C₃₇	D₄	1	2	yes	—
64	C₄×C₈×C₃₇	D₄	1	2	yes	—
65	#41×C₃₇	—	2	4	no	—
66	#41×C₃₇	—	2	4	no	—
67	C₂×Q₈×C₃₇	—	2	8	no	—
68	C₂×Q₈×C₃₇	—	2	4	no	—
69	C₂×Q₈×C₃₇	—	2	4	no	—
70	C₂×Q₈×C₃₇	—	2	8	no	—
71	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	—	2	2	no	—
74	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	2	4	no	—
77	C₂²×C₄×C₃₇	#41	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
79	Q₁₆×C₃₇	—	4	16	no	—
80	Q₁₆×C₃₇	—	4	8	no	—
81	Q₁₆×C₃₇	—	4	4	no	—
82	Q₁₆×C₃₇	—	4	4	no	—
83	SD₁₆×C₃₇	—	4	16	no	—
84	M₁₆×C₃₇	—	2	4	no	—
85	M₁₆×C₃₇	—	2	4	no	—
86	M₁₆×C₃₇	—	4	4	no	—
87	M₁₆×C₃₇	C₂×D₄	1	8	yes	—
88	C₂×C₈×C₃₇	—	2	4	no	—
89	C₂×C₈×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	—	2	2	no	—
91	C₂×C₈×C₃₇	—	4	4	no	—
92	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	4	4	no	—
98	C₄⋊C₄×C₃₇	—	4	4	no	—
99	C₂²⋊C₄×C₃₇	—	2	4	no	—
100	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₃₇	C₂×D₄	1	2	yes	—
102	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₃₇	—	2	2	no	—
104	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
105	Q₈×C₃₇	—	2	4	no	—
106	Q₈×C₃₇	—	4	4	no	—
107	Q₈×C₃₇	—	4	4	no	—
108	Q₈×C₃₇	—	4	8	no	—
109	D₄×C₃₇	—	2	4	no	—
110	D₄×C₃₇	—	4	4	no	—
111	D₄×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	—	2	4	no	—
113	C₂×C₄×C₃₇	—	2	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	8	no	—
118	C₂×C₄×C₃₇	#121	1	2	yes	—
119	C₂×C₄×C₃₇	#121	1	2	yes	—
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
124	C₈×C₃₇	—	2	2	no	—
125	C₈×C₃₇	—	2	8	no	—
126	C₈×C₃₇	—	4	2	no	—
127	C₈×C₃₇	#127	1	2	yes	—
128	C₈×C₃₇	#127	1	2	yes	—
129	#28498	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₃₇	—	2	2	no	—
131	C₂²×C₃₇	—	4	2	no	—
132	C₂²×C₃₇	—	4	4	no	—
133	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
134	C₂²×C₃₇	#550	1	2	yes	—
135	C₄×C₃₇	—	2	2	no	—
136	C₄×C₃₇	—	4	2	no	—
137	C₄×C₃₇	—	8	4	no	—
138	C₄×C₃₇	#473	1	4	yes	complemented
139	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
140	C₄×C₃₇	#535	1	1	yes	characteristic
141	C₄×C₃₇	D₄²	1	2	yes	—
142	#3152	C₂×C₃₇	1	2	yes	—
143	C₂×#494	C₂×C₃₇	1	1	yes	characteristic, complemented
144	C₂×#473	C₂×C₃₇	1	4	yes	complemented
145	#2931	C₂×C₃₇	1	1	yes	characteristic
146	#2020	C₂×C₃₇	1	4	yes	complemented
147	#1966	C₂×C₃₇	1	2	yes	complemented
148	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
149	C₂×C₃₇	—	2	2	no	—
150	C₂×C₃₇	—	8	2	no	—
151	C₂×C₃₇	#2541	1	1	yes	characteristic
152	C₂×C₃₇	C₂×#473	1	2	yes	—
153	C₂×#137	C₂²×C₃₇	1	8	yes	—
154	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
155	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
156	C₂×#131	C₂²×C₃₇	1	2	yes	—
157	#532	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
159	#497	—	2	4	no	—
160	#494	—	2	2	no	—
161	#494	—	2	2	no	—
162	#494	—	2	2	no	—
163	#492	—	2	2	no	—
164	#473	—	2	16	no	—
165	#436	—	2	2	no	—
166	C₈×D₄	—	2	2	no	—
167	C₂×#108	C₂²×C₃₇	1	2	yes	—
168	#423	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×#101	C₂²×C₃₇	1	8	yes	—
170	#408	C₂²×C₃₇	1	2	yes	—
171	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
173	#341	C₂²×C₃₇	1	2	yes	—
174	C₃₇	#28498	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×#41	C₂³×C₃₇	1	2	yes	—
176	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
177	#137	—	2	16	no	—
178	#137	—	4	8	no	—
179	C₂×Q₁₆	—	2	8	no	—
180	C₂×Q₁₆	—	2	8	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	4	4	no	—
185	C₂×SD₁₆	—	2	8	no	—
186	#131	—	2	8	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
189	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
197	#127	D₄×C₃₇	1	1	yes	characteristic
198	#124	D₄×C₃₇	1	2	yes	—
199	#121	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
203	#108	D₄×C₃₇	1	8	yes	complemented
204	C₄⋊C₈	—	2	2	no	—
205	#101	—	2	16	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	#97	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	D₄×C₃₇	1	2	yes	—
210	C₄×C₈	D₄×C₃₇	1	2	yes	—
211	#41	—	2	4	no	—
212	#41	—	2	4	no	—
213	C₂×Q₈	—	2	8	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	8	no	—
217	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	#41×C₃₇	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
225	Q₁₆	—	4	16	no	—
226	Q₁₆	—	4	8	no	—
227	Q₁₆	—	4	4	no	—
228	Q₁₆	—	4	4	no	—
229	SD₁₆	—	4	16	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	4	no	—
233	M₁₆	C₂×D₄×C₃₇	1	8	yes	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₃₇	1	2	yes	—
248	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	4	4	no	—
254	Q₈	—	4	8	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	#121×C₃₇	1	2	yes	—
265	C₂×C₄	#121×C₃₇	1	2	yes	—
266	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
270	C₈	—	2	2	no	—
271	C₈	—	2	8	no	—
272	C₈	—	4	2	no	—
273	C₈	#127×C₃₇	1	2	yes	—
274	C₈	#127×C₃₇	1	2	yes	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
279	C₂²	C₃₇×#550	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₃₇×#473	1	4	yes	complemented
284	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
285	C₄	C₃₇×#535	1	1	yes	characteristic
286	C₄	D₄²×C₃₇	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	8	2	no	—
289	C₂	C₃₇×#2541	1	1	yes	characteristic
290	C₂	C₂×C₃₇×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#532	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
2. C₂×C₃₇×#494	C₂	semidirect
3. C₂×C₃₇×#473	C₂	semidirect
5. C₃₇×#2020	C₂	semidirect
6. C₃₇×#1966	C₂	semidirect
7. C₄×#108×C₃₇	C₂	semidirect
57. #108×C₃₇	D₄	semidirect
129. #28498	C₃₇	direct (N × H)
138. C₄×C₃₇	#473	semidirect
143. C₂×#494	C₂×C₃₇	semidirect
144. C₂×#473	C₂×C₃₇	semidirect
146. #2020	C₂×C₃₇	semidirect
147. #1966	C₂×C₃₇	semidirect
148. C₄×#108	C₂×C₃₇	semidirect
174. C₃₇	#28498	direct (N × H)
203. #108	D₄×C₃₇	semidirect
283. C₄	C₃₇×#473	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3152	2
C₂×C₃₇×#494	2
C₂×C₃₇×#473	2
C₃₇×#2931	2
C₃₇×#2020	2
C₃₇×#1966	2
C₄×#108×C₃₇	2
#28498	37

Subgroups — C37×#28498 #22649751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28498 #22649751