Subgroups — C₃₇×#28766 #22650019

Order 9472.

746 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#495	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3140	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2046	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1979	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1978	C₂	1	2	yes	maximal
7	C₃₇×#1967	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₃₇	C₂²	1	2	yes	—
11	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#533	C₂²	1	1	yes	characteristic
13	C₂×#124×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#495	—	2	2	no	—
15	C₃₇×#495	—	2	2	no	—
16	C₃₇×#489	—	2	4	no	—
17	C₃₇×#488	—	2	4	no	—
18	C₃₇×#446	—	2	2	no	—
19	C₈×Q₈×C₃₇	—	2	2	no	—
20	C₂×#108×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#424	C₂²	1	2	yes	—
22	C₃₇×#423	C₂²	1	1	yes	characteristic
23	C₃₇×#419	C₂²	1	4	yes	—
24	C₃₇×#418	C₂²	1	4	yes	—
25	C₂×#101×C₃₇	C₂²	1	4	yes	—
26	C₂×#100×C₃₇	C₂²	1	4	yes	—
27	C₃₇×#405	C₂²	1	2	yes	—
28	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
30	C₃₇×#341	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₃₇	—	2	1	no	—
34	C₂×Q₁₆×C₃₇	—	2	1	no	—
35	C₂×Q₁₆×C₃₇	—	2	4	no	—
36	C₂×SD₁₆×C₃₇	—	2	4	no	—
37	C₂×SD₁₆×C₃₇	—	4	4	no	—
38	C₂×D₈×C₃₇	—	2	1	no	—
39	C₂×D₈×C₃₇	—	2	1	no	—
40	C₂×D₈×C₃₇	—	2	4	no	—
41	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
42	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₃₇	D₄	1	2	yes	—
46	#125×C₃₇	D₄	1	2	yes	—
47	#124×C₃₇	—	2	2	no	—
48	#124×C₃₇	—	2	2	no	—
49	#124×C₃₇	—	2	4	no	—
50	C₂²⋊Q₈×C₃₇	—	2	2	no	—
51	C₄×Q₈×C₃₇	—	2	2	no	—
52	#117×C₃₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	#108×C₃₇	—	2	4	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	#103×C₃₇	—	2	8	no	—
59	#102×C₃₇	—	2	8	no	—
60	#101×C₃₇	—	4	4	no	—
61	#100×C₃₇	—	4	4	no	—
62	#97×C₃₇	—	2	2	no	—
63	#97×C₃₇	—	2	2	no	—
64	C₄×C₈×C₃₇	—	2	2	no	—
65	C₄×C₈×C₃₇	—	2	2	no	—
66	C₄×C₈×C₃₇	D₄	1	2	yes	—
67	C₄×C₈×C₃₇	D₄	1	2	yes	—
68	C₂⁴×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	—	2	4	no	—
73	C₂×Q₈×C₃₇	—	2	4	no	—
74	C₂×D₄×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	2	no	—
76	C₂×D₄×C₃₇	—	2	4	no	—
77	C₂×D₄×C₃₇	—	2	4	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	#41	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₇	—	4	2	no	—
82	Q₁₆×C₃₇	—	4	2	no	—
83	SD₁₆×C₃₇	—	4	8	no	—
84	D₈×C₃₇	—	4	2	no	—
85	D₈×C₃₇	—	4	2	no	—
86	M₁₆×C₃₇	—	2	4	no	—
87	M₁₆×C₃₇	—	2	2	no	—
88	M₁₆×C₃₇	—	2	4	no	—
89	M₁₆×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	—	2	2	no	—
91	C₂×C₈×C₃₇	—	2	4	no	—
92	C₂×C₈×C₃₇	—	2	2	no	—
93	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₇	#41	1	4	yes	—
97	C₄⋊C₄×C₃₇	—	2	4	no	—
98	C₄⋊C₄×C₃₇	—	2	2	no	—
99	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
100	C₂²⋊C₄×C₃₇	—	2	2	no	—
101	C₂²⋊C₄×C₃₇	—	4	4	no	—
102	C₂²⋊C₄×C₃₇	—	4	4	no	—
103	C₄²×C₃₇	—	2	2	no	—
104	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₇	C₂×D₄	1	2	yes	—
106	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₃₇	—	4	2	no	—
108	C₂³×C₃₇	—	4	4	no	—
109	C₂³×C₃₇	—	8	4	no	—
110	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
111	Q₈×C₃₇	—	4	4	no	—
112	Q₈×C₃₇	—	4	2	no	—
113	Q₈×C₃₇	—	4	2	no	—
114	Q₈×C₃₇	—	4	4	no	—
115	D₄×C₃₇	—	4	2	no	—
116	D₄×C₃₇	—	4	2	no	—
117	D₄×C₃₇	—	4	4	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	4	no	—
122	C₂×C₄×C₃₇	#121	1	2	yes	—
123	C₂×C₄×C₃₇	#121	1	2	yes	—
124	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
128	C₈×C₃₇	—	2	2	no	—
129	C₈×C₃₇	—	4	2	no	—
130	C₈×C₃₇	—	4	4	no	—
131	C₈×C₃₇	#124	1	2	yes	—
132	C₈×C₃₇	#124	1	2	yes	—
133	#28766	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₃₇	—	2	2	no	—
135	C₂²×C₃₇	—	8	2	no	—
136	C₂²×C₃₇	—	8	4	no	—
137	C₂²×C₃₇	—	16	4	no	—
138	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
139	C₂²×C₃₇	#559	1	2	yes	—
140	C₄×C₃₇	—	2	2	no	—
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	4	4	no	—
143	C₄×C₃₇	—	8	2	no	—
144	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₃₇	#535	1	1	yes	characteristic
146	C₄×C₃₇	#550	1	2	yes	—
147	C₂×#495	C₂×C₃₇	1	1	yes	characteristic
148	#3140	C₂×C₃₇	1	2	yes	complemented
149	#2960	C₂×C₃₇	1	1	yes	characteristic, complemented
150	#2046	C₂×C₃₇	1	4	yes	complemented
151	#1979	C₂×C₃₇	1	2	yes	complemented
152	#1978	C₂×C₃₇	1	2	yes	—
153	#1967	C₂×C₃₇	1	2	yes	complemented
154	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
155	C₂×C₃₇	—	2	2	no	—
156	C₂×C₃₇	—	16	2	no	—
157	C₂×C₃₇	#2548	1	1	yes	characteristic
158	C₂×C₃₇	#3067	1	2	yes	—
159	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
160	C₂²×SD₁₆	C₂²×C₃₇	1	2	yes	—
161	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
162	#533	C₂²×C₃₇	1	1	yes	characteristic
163	C₂×#124	C₂²×C₃₇	1	2	yes	—
164	#495	—	2	2	no	—
165	#495	—	2	2	no	—
166	#489	—	2	4	no	—
167	#488	—	2	4	no	—
168	#446	—	2	2	no	—
169	C₈×Q₈	—	2	2	no	—
170	C₂×#108	C₂²×C₃₇	1	2	yes	—
171	#424	C₂²×C₃₇	1	2	yes	—
172	#423	C₂²×C₃₇	1	1	yes	characteristic
173	#419	C₂²×C₃₇	1	4	yes	—
174	#418	C₂²×C₃₇	1	4	yes	—
175	C₂×#101	C₂²×C₃₇	1	4	yes	—
176	C₂×#100	C₂²×C₃₇	1	4	yes	—
177	#405	C₂²×C₃₇	1	2	yes	—
178	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
180	#341	C₂²×C₃₇	1	2	yes	—
181	C₃₇	#28766	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
183	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	4	no	—
187	C₂×SD₁₆	—	2	4	no	—
188	C₂×SD₁₆	—	4	4	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
193	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
196	C₄⋊Q₈	D₄×C₃₇	1	2	yes	—
197	#125	D₄×C₃₇	1	2	yes	—
198	#124	—	2	2	no	—
199	#124	—	2	2	no	—
200	#124	—	2	4	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	#117	C₂³×C₃₇	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
206	#108	—	2	4	no	—
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	2	2	no	—
209	#103	—	2	8	no	—
210	#102	—	2	8	no	—
211	#101	—	4	4	no	—
212	#100	—	4	4	no	—
213	#97	—	2	2	no	—
214	#97	—	2	2	no	—
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	D₄×C₃₇	1	2	yes	—
218	C₄×C₈	D₄×C₃₇	1	2	yes	—
219	C₂⁴	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	#41×C₃₇	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
232	Q₁₆	—	4	2	no	—
233	Q₁₆	—	4	2	no	—
234	SD₁₆	—	4	8	no	—
235	D₈	—	4	2	no	—
236	D₈	—	4	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
245	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
247	C₂×C₈	#41×C₃₇	1	4	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₃₇	1	2	yes	—
257	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
258	C₂³	—	4	2	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	8	4	no	—
261	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	4	4	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	#121×C₃₇	1	2	yes	—
274	C₂×C₄	#121×C₃₇	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
278	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	4	2	no	—
281	C₈	—	4	4	no	—
282	C₈	#124×C₃₇	1	2	yes	—
283	C₈	#124×C₃₇	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
289	C₂²	C₃₇×#559	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
295	C₄	C₃₇×#535	1	1	yes	characteristic
296	C₄	C₃₇×#550	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	—	16	2	no	—
299	C₂	C₃₇×#2548	1	1	yes	characteristic
300	C₂	C₃₇×#3067	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
2. C₃₇×#3140	C₂	semidirect
3. C₃₇×#2960	C₂	semidirect
4. C₃₇×#2046	C₂	semidirect
5. C₃₇×#1979	C₂	semidirect
7. C₃₇×#1967	C₂	semidirect
8. C₄×#108×C₃₇	C₂	semidirect
133. #28766	C₃₇	direct (N × H)
148. #3140	C₂×C₃₇	semidirect
149. #2960	C₂×C₃₇	semidirect
150. #2046	C₂×C₃₇	semidirect
151. #1979	C₂×C₃₇	semidirect
153. #1967	C₂×C₃₇	semidirect
154. C₄×#108	C₂×C₃₇	semidirect
181. C₃₇	#28766	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×#495	2
C₃₇×#3140	2
C₃₇×#2960	2
C₃₇×#2046	2
C₃₇×#1979	2
C₃₇×#1978	2
C₃₇×#1967	2
C₄×#108×C₃₇	2
#28766	37

Subgroups — C37×#28766 #22650019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28766 #22650019