Subgroups — C₃₇×#29233 #22650486

Order 9472.

532 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2102	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#2050	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2031	C₂	1	2	yes	maximal, complemented
4	C₃₇×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#1932	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1906	C₂	1	2	yes	maximal
7	C₃₇×#1841	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1827	C₂	1	2	yes	maximal, complemented
9	C₃₇×#1759	C₂	1	2	yes	maximal, complemented
10	C₃₇×#575	C₂²	1	1	yes	characteristic
11	C₃₇×#533	C₂²	1	1	yes	characteristic
12	C₄×#41×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#430	C₂²	1	1	yes	characteristic
14	C₃₇×#423	C₂²	1	1	yes	characteristic
15	C₃₇×#421	C₂²	1	2	yes	—
16	C₂×#104×C₃₇	C₂²	1	2	yes	—
17	C₂×#104×C₃₇	C₂²	1	2	yes	—
18	C₃₇×#413	C₂²	1	2	yes	—
19	C₃₇×#410	C₂²	1	2	yes	—
20	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#376	C₂²	1	1	yes	characteristic
22	C₃₇×#344	C₂²	1	2	yes	—
23	C₃₇×#344	C₂²	1	2	yes	—
24	C₃₇×#339	C₂²	1	2	yes	—
25	C₃₇×#339	C₂²	1	2	yes	—
26	C₃₇×#339	C₂²	1	2	yes	—
27	C₃₇×#339	C₂²	1	2	yes	—
28	C₃₇×#338	C₂²	1	1	yes	characteristic
29	C₃₇×H(3,ℤ/4)	C₂²	1	2	yes	—
30	C₃₇×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41×C₃₇	C₂³	1	2	yes	—
32	#137×C₃₇	—	2	2	no	—
33	AGL(1,ℤ/8)×C₃₇	—	2	2	no	—
34	#135×C₃₇	—	4	1	no	—
35	C₂×M₁₆×C₃₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
37	C₂×M₁₆×C₃₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₃₇	—	2	1	no	—
40	#125×C₃₇	—	2	1	no	—
41	C₂²⋊Q₈×C₃₇	—	4	1	no	—
42	C₄×Q₈×C₃₇	—	2	2	no	—
43	C₄×Q₈×C₃₇	—	4	1	no	—
44	C₄×D₄×C₃₇	—	2	2	no	—
45	C₄×D₄×C₃₇	—	4	2	no	—
46	#117×C₃₇	—	2	2	no	—
47	#117×C₃₇	C₂³	1	2	yes	—
48	#117×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₇	—	2	2	no	—
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
52	#108×C₃₇	—	4	1	no	—
53	C₄⋊C₈×C₃₇	—	4	1	no	—
54	#104×C₃₇	—	2	2	no	—
55	#104×C₃₇	—	2	2	no	—
56	#104×C₃₇	—	2	2	no	—
57	#104×C₃₇	—	2	2	no	—
58	#104×C₃₇	—	4	2	no	—
59	#101×C₃₇	—	2	2	no	—
60	#100×C₃₇	—	2	2	no	—
61	#99×C₃₇	—	4	2	no	—
62	#97×C₃₇	—	2	2	no	—
63	C₄×C₈×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	4	1	no	—
65	#41×C₃₇	—	2	2	no	—
66	#41×C₃₇	—	2	2	no	—
67	#41×C₃₇	—	2	2	no	—
68	#41×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	4	1	no	—
70	C₂×Q₈×C₃₇	#41	1	2	yes	—
71	C₂×D₄×C₃₇	—	2	2	no	—
72	C₂×D₄×C₃₇	#41	1	2	yes	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	1	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	4	2	no	—
77	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₃₇	—	4	1	no	—
79	SD₁₆×C₃₇	—	4	2	no	—
80	D₈×C₃₇	—	4	1	no	—
81	M₁₆×C₃₇	—	2	2	no	—
82	M₁₆×C₃₇	—	2	2	no	—
83	M₁₆×C₃₇	—	2	2	no	—
84	M₁₆×C₃₇	—	2	2	no	—
85	M₁₆×C₃₇	—	2	2	no	—
86	M₁₆×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	2	2	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	—	2	1	no	—
90	C₂×C₈×C₃₇	—	2	1	no	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	4	2	no	—
94	C₄⋊C₄×C₃₇	—	4	2	no	—
95	C₄⋊C₄×C₃₇	—	4	1	no	—
96	C₂²⋊C₄×C₃₇	—	2	2	no	—
97	C₂²⋊C₄×C₃₇	—	2	2	no	—
98	C₂²⋊C₄×C₃₇	—	4	2	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	1	no	—
101	C₄²×C₃₇	—	2	2	no	—
102	C₄²×C₃₇	—	2	2	no	—
103	C₄²×C₃₇	—	4	2	no	—
104	C₄²×C₃₇	#41	1	2	yes	—
105	C₄²×C₃₇	#41	1	2	yes	—
106	C₄²×C₃₇	#41	1	1	yes	characteristic
107	C₄²×C₃₇	#41	1	1	yes	characteristic
108	C₂³×C₃₇	—	2	2	no	—
109	C₂³×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
110	Q₈×C₃₇	—	2	2	no	—
111	Q₈×C₃₇	—	2	2	no	—
112	Q₈×C₃₇	—	8	2	no	—
113	D₄×C₃₇	—	2	2	no	—
114	D₄×C₃₇	—	2	2	no	—
115	D₄×C₃₇	—	4	2	no	—
116	D₄×C₃₇	—	4	2	no	—
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	4	2	no	—
123	C₂×C₄×C₃₇	—	4	2	no	—
124	C₂×C₄×C₃₇	—	4	2	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	—	8	2	no	—
127	C₂×C₄×C₃₇	—	8	1	no	—
128	C₂×C₄×C₃₇	—	8	4	no	—
129	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₈×C₃₇	—	4	2	no	—
134	C₈×C₃₇	—	4	1	no	—
135	C₈×C₃₇	—	4	1	no	—
136	C₈×C₃₇	—	4	2	no	—
137	#29233	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₃₇	—	4	1	no	—
139	C₂²×C₃₇	—	4	2	no	—
140	C₂²×C₃₇	—	8	2	no	—
141	C₂²×C₃₇	#538	1	1	yes	characteristic
142	C₂²×C₃₇	#551	1	2	yes	—
143	C₄×C₃₇	—	4	1	no	—
144	C₄×C₃₇	—	4	2	no	—
145	C₄×C₃₇	—	4	2	no	—
146	C₄×C₃₇	—	4	1	no	—
147	C₄×C₃₇	—	4	2	no	—
148	C₄×C₃₇	—	8	2	no	—
149	C₄×C₃₇	—	8	2	no	—
150	C₄×C₃₇	#529	1	2	yes	—
151	C₄×C₃₇	#551	1	1	yes	characteristic
152	C₄×C₃₇	#555	1	1	yes	characteristic
153	#2102	C₂×C₃₇	1	1	yes	characteristic, complemented
154	#2050	C₂×C₃₇	1	1	yes	characteristic
155	#2031	C₂×C₃₇	1	2	yes	complemented
156	#1995	C₂×C₃₇	1	1	yes	characteristic, complemented
157	#1932	C₂×C₃₇	1	2	yes	complemented
158	#1906	C₂×C₃₇	1	2	yes	—
159	#1841	C₂×C₃₇	1	2	yes	complemented
160	#1827	C₂×C₃₇	1	2	yes	complemented
161	#1759	C₂×C₃₇	1	2	yes	complemented
162	C₂×C₃₇	—	2	1	no	—
163	C₂×C₃₇	—	2	2	no	—
164	C₂×C₃₇	—	8	2	no	—
165	C₂×C₃₇	#2572	1	1	yes	characteristic
166	#575	C₂²×C₃₇	1	1	yes	characteristic
167	#533	C₂²×C₃₇	1	1	yes	characteristic
168	C₄×#41	C₂²×C₃₇	1	2	yes	—
169	#430	C₂²×C₃₇	1	1	yes	characteristic
170	#423	C₂²×C₃₇	1	1	yes	characteristic
171	#421	C₂²×C₃₇	1	2	yes	—
172	C₂×#104	C₂²×C₃₇	1	2	yes	—
173	C₂×#104	C₂²×C₃₇	1	2	yes	—
174	#413	C₂²×C₃₇	1	2	yes	—
175	#410	C₂²×C₃₇	1	2	yes	—
176	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
177	#376	C₂²×C₃₇	1	1	yes	characteristic
178	#344	C₂²×C₃₇	1	2	yes	—
179	#344	C₂²×C₃₇	1	2	yes	—
180	#339	C₂²×C₃₇	1	2	yes	—
181	#339	C₂²×C₃₇	1	2	yes	—
182	#339	C₂²×C₃₇	1	2	yes	—
183	#339	C₂²×C₃₇	1	2	yes	—
184	#338	C₂²×C₃₇	1	1	yes	characteristic
185	H(3,ℤ/4)	C₂²×C₃₇	1	1	yes	characteristic
186	H(3,ℤ/4)	C₂²×C₃₇	1	2	yes	—
187	C₃₇	#29233	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×#41	C₂³×C₃₇	1	2	yes	—
189	#137	—	2	2	no	—
190	AGL(1,ℤ/8)	—	2	2	no	—
191	#135	—	4	1	no	—
192	C₂×M₁₆	C₂³×C₃₇	1	1	yes	characteristic
193	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
194	C₂×M₁₆	C₂³×C₃₇	1	1	yes	characteristic
195	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
196	C₄⋊Q₈	—	2	1	no	—
197	#125	—	2	1	no	—
198	C₂²⋊Q₈	—	4	1	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	4	2	no	—
203	#117	—	2	2	no	—
204	#117	C₂³×C₃₇	1	2	yes	—
205	#117	C₂³×C₃₇	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
208	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
209	#108	—	4	1	no	—
210	C₄⋊C₈	—	4	1	no	—
211	#104	—	2	2	no	—
212	#104	—	2	2	no	—
213	#104	—	2	2	no	—
214	#104	—	2	2	no	—
215	#104	—	4	2	no	—
216	#101	—	2	2	no	—
217	#100	—	2	2	no	—
218	#99	—	4	2	no	—
219	#97	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	#95	—	4	1	no	—
222	#41	—	2	2	no	—
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	#41×C₃₇	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	#41×C₃₇	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
235	Q₁₆	—	4	1	no	—
236	SD₁₆	—	4	2	no	—
237	D₈	—	4	1	no	—
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	#41×C₃₇	1	2	yes	—
262	C₄²	#41×C₃₇	1	2	yes	—
263	C₄²	#41×C₃₇	1	1	yes	characteristic
264	C₄²	#41×C₃₇	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	Q₈	—	2	2	no	—
269	Q₈	—	8	2	no	—
270	D₄	—	2	2	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
288	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
289	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
290	C₈	—	4	2	no	—
291	C₈	—	4	1	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	C₃₇×#538	1	1	yes	characteristic
298	C₂²	C₃₇×#551	1	2	yes	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₃₇×#529	1	2	yes	—
307	C₄	C₃₇×#551	1	1	yes	characteristic
308	C₄	C₃₇×#555	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	8	2	no	—
312	C₂	C₃₇×#2572	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₇	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle	C₂×C₃₇	#2572
prime radical	C₃₇×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#2102	C₂	semidirect
3. C₃₇×#2031	C₂	semidirect
4. C₃₇×#1995	C₂	semidirect
5. C₃₇×#1932	C₂	semidirect
7. C₃₇×#1841	C₂	semidirect
8. C₃₇×#1827	C₂	semidirect
9. C₃₇×#1759	C₂	semidirect
137. #29233	C₃₇	direct (N × H)
153. #2102	C₂×C₃₇	semidirect
155. #2031	C₂×C₃₇	semidirect
156. #1995	C₂×C₃₇	semidirect
157. #1932	C₂×C₃₇	semidirect
159. #1841	C₂×C₃₇	semidirect
160. #1827	C₂×C₃₇	semidirect
161. #1759	C₂×C₃₇	semidirect
187. C₃₇	#29233	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2102	2
C₃₇×#2050	2
C₃₇×#2031	2
C₃₇×#1995	2
C₃₇×#1932	2
C₃₇×#1906	2
C₃₇×#1841	2
C₃₇×#1827	2
C₃₇×#1759	2
#29233	37

Subgroups — C37×#29233 #22650486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29233 #22650486