Subgroups — C₃₇×#12807 #22634060

Order 9472.

428 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	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2126	C₂	1	2	yes	maximal
3	C₃₇×#1812	C₂	1	2	yes	maximal, complemented
4	C₃₇×#1391	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₃₇	C₄	1	1	yes	characteristic, complemented
6	C₂×#131×C₃₇	C₂²	1	1	yes	characteristic
7	C₃₇×#532	C₂²	1	1	yes	characteristic
8	C₃₇×#436	—	2	2	no	—
9	C₈×D₄×C₃₇	—	2	2	no	—
10	C₃₇×#427	—	2	2	no	—
11	C₃₇×#423	C₄	1	1	yes	characteristic, complemented
12	C₃₇×#410	C₂²	1	4	yes	—
13	C₃₇×#408	C₄	1	1	yes	characteristic, complemented
14	C₃₇×#408	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
16	C₃₇×#354	—	2	2	no	—
17	C₃₇×#353	—	2	2	no	—
18	C₃₇×H(3,ℤ/4)	—	2	2	no	—
19	C₂×#41×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#41×C₃₇	C₂³	1	1	yes	characteristic
21	#131×C₃₇	—	4	1	no	—
22	#131×C₃₇	—	4	1	no	—
23	#131×C₃₇	—	4	1	no	—
24	#131×C₃₇	—	4	1	no	—
25	C₂×M₁₆×C₃₇	—	2	1	no	—
26	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₃₇	D₄	1	2	yes	—
29	C₂²×C₈×C₃₇	—	2	1	no	—
30	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	D₄	1	2	yes	—
33	C₄⋊Q₈×C₃₇	Q₈	1	1	yes	characteristic
34	#127×C₃₇	D₄	1	1	yes	characteristic
35	#124×C₃₇	D₄	1	1	yes	characteristic
36	#124×C₃₇	Q₈	1	1	yes	characteristic
37	#121×C₃₇	—	2	2	no	—
38	C₄×D₄×C₃₇	—	2	2	no	—
39	#117×C₃₇	—	2	4	no	—
40	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
41	#107×C₃₇	—	4	2	no	—
42	#106×C₃₇	—	4	2	no	—
43	C₄⋊C₈×C₃₇	—	2	2	no	—
44	#99×C₃₇	—	2	4	no	—
45	#99×C₃₇	—	2	4	no	—
46	C₂²⋊C₈×C₃₇	—	2	2	no	—
47	C₂²⋊C₈×C₃₇	—	2	2	no	—
48	C₄×C₈×C₃₇	—	2	1	no	—
49	C₄×C₈×C₃₇	—	2	1	no	—
50	#41×C₃₇	—	4	1	no	—
51	#41×C₃₇	—	4	1	no	—
52	#41×C₃₇	—	4	1	no	—
53	#41×C₃₇	—	4	1	no	—
54	C₂×Q₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×D₄×C₃₇	—	2	1	no	—
57	C₂×D₄×C₃₇	—	2	2	no	—
58	C₂×D₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₄×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂×D₄×C₃₇	#41	1	2	yes	—
61	C₂²×C₄×C₃₇	—	2	1	no	—
62	C₂²×C₄×C₃₇	—	2	1	no	—
63	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
65	M₁₆×C₃₇	—	4	1	no	—
66	M₁₆×C₃₇	—	4	2	no	—
67	M₁₆×C₃₇	—	4	2	no	—
68	M₁₆×C₃₇	—	4	1	no	—
69	C₂×C₈×C₃₇	—	2	2	no	—
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	—	2	1	no	—
72	C₂×C₈×C₃₇	—	2	1	no	—
73	C₂×C₈×C₃₇	—	2	1	no	—
74	C₂×C₈×C₃₇	—	2	1	no	—
75	C₂×C₈×C₃₇	—	4	1	no	—
76	C₂×C₈×C₃₇	—	4	1	no	—
77	C₂×C₈×C₃₇	—	4	1	no	—
78	C₂×C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	—	4	4	no	—
83	C₂²⋊C₄×C₃₇	—	2	2	no	—
84	C₂²⋊C₄×C₃₇	—	2	2	no	—
85	C₂²⋊C₄×C₃₇	—	4	4	no	—
86	C₄²×C₃₇	—	2	1	no	—
87	C₄²×C₃₇	—	4	4	no	—
88	C₄²×C₃₇	C₄⋊C₄	1	1	yes	characteristic
89	C₄²×C₃₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₃₇	—	2	1	no	—
91	C₂³×C₃₇	#117	1	1	yes	characteristic
92	C₂³×C₃₇	#121	1	2	yes	—
93	Q₈×C₃₇	—	4	1	no	—
94	Q₈×C₃₇	—	4	1	no	—
95	D₄×C₃₇	—	4	1	no	—
96	D₄×C₃₇	—	4	2	no	—
97	D₄×C₃₇	—	4	1	no	—
98	D₄×C₃₇	—	4	4	no	—
99	D₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	—	2	1	no	—
101	C₂×C₄×C₃₇	—	2	2	no	—
102	C₂×C₄×C₃₇	—	2	1	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	2	1	no	—
105	C₂×C₄×C₃₇	—	2	1	no	—
106	C₂×C₄×C₃₇	—	2	1	no	—
107	C₂×C₄×C₃₇	—	4	2	no	—
108	C₂×C₄×C₃₇	—	4	1	no	—
109	C₂×C₄×C₃₇	—	4	1	no	—
110	C₂×C₄×C₃₇	—	8	4	no	—
111	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
114	C₈×C₃₇	—	2	2	no	—
115	C₈×C₃₇	—	2	1	no	—
116	C₈×C₃₇	—	4	1	no	—
117	C₈×C₃₇	—	4	1	no	—
118	C₈×C₃₇	#99	1	1	yes	characteristic
119	C₈×C₃₇	#99	1	1	yes	characteristic
120	#12807	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	—	2	2	no	—
122	C₂²×C₃₇	—	2	1	no	—
123	C₂²×C₃₇	—	4	1	no	—
124	C₂²×C₃₇	—	4	2	no	—
125	C₂²×C₃₇	—	4	2	no	—
126	C₂²×C₃₇	—	4	1	no	—
127	C₂²×C₃₇	#380	1	1	yes	characteristic
128	C₄×C₃₇	—	2	2	no	—
129	C₄×C₃₇	—	2	1	no	—
130	C₄×C₃₇	—	2	2	no	—
131	C₄×C₃₇	—	4	1	no	—
132	C₄×C₃₇	—	4	1	no	—
133	C₄×C₃₇	—	16	4	no	—
134	C₄×C₃₇	C₂×#99	1	1	yes	characteristic
135	C₄×C₃₇	#410	1	1	yes	characteristic
136	#2931	C₂×C₃₇	1	1	yes	characteristic
137	#2126	C₂×C₃₇	1	2	yes	—
138	#1812	C₂×C₃₇	1	2	yes	complemented
139	#1391	C₂×C₃₇	1	2	yes	complemented
140	C₂×C₃₇	—	2	1	no	—
141	C₂×C₃₇	—	4	2	no	—
142	C₂×C₃₇	—	4	1	no	—
143	C₂×C₃₇	—	8	1	no	—
144	C₂×C₃₇	#1799	1	1	yes	characteristic
145	C₂×#131	C₄×C₃₇	1	1	yes	characteristic, complemented
146	C₂×#131	C₂²×C₃₇	1	1	yes	characteristic
147	#532	C₂²×C₃₇	1	1	yes	characteristic
148	#436	—	2	2	no	—
149	C₈×D₄	—	2	2	no	—
150	#427	—	2	2	no	—
151	#423	C₄×C₃₇	1	1	yes	characteristic, complemented
152	#410	C₂²×C₃₇	1	4	yes	—
153	#408	C₄×C₃₇	1	1	yes	characteristic, complemented
154	#408	C₂²×C₃₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₄×C₃₇	1	1	yes	characteristic, complemented
156	#354	—	2	2	no	—
157	#353	—	2	2	no	—
158	H(3,ℤ/4)	—	2	2	no	—
159	C₃₇	#12807	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×#41	C₂×C₄×C₃₇	1	1	yes	characteristic
161	C₂×#41	C₂³×C₃₇	1	1	yes	characteristic
162	#131	—	4	1	no	—
163	#131	—	4	1	no	—
164	#131	—	4	1	no	—
165	#131	—	4	1	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
169	C₂×M₁₆	D₄×C₃₇	1	2	yes	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
173	C₂²×C₈	D₄×C₃₇	1	2	yes	—
174	C₄⋊Q₈	Q₈×C₃₇	1	1	yes	characteristic
175	#127	D₄×C₃₇	1	1	yes	characteristic
176	#124	D₄×C₃₇	1	1	yes	characteristic
177	#124	Q₈×C₃₇	1	1	yes	characteristic
178	#121	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	#117	—	2	4	no	—
181	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
182	#107	—	4	2	no	—
183	#106	—	4	2	no	—
184	C₄⋊C₈	—	2	2	no	—
185	#99	—	2	4	no	—
186	#99	—	2	4	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	#41	—	4	1	no	—
192	#41	—	4	1	no	—
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	C₄⋊C₄×C₃₇	1	1	yes	characteristic
196	C₂×Q₈	C₂×Q₈×C₃₇	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
200	C₂×D₄	C₂×D₄×C₃₇	1	1	yes	characteristic
201	C₂×D₄	#41×C₃₇	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	C₂²⋊C₄×C₃₇	1	2	yes	—
220	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₄⋊C₄×C₃₇	1	1	yes	characteristic
230	C₄²	C₄⋊C₄×C₃₇	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	#117×C₃₇	1	1	yes	characteristic
233	C₂³	#121×C₃₇	1	2	yes	—
234	Q₈	—	4	1	no	—
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	#99×C₃₇	1	1	yes	characteristic
260	C₈	#99×C₃₇	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₃₇×#380	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	16	4	no	—
274	C₄	C₂×#99×C₃₇	1	1	yes	characteristic
275	C₄	C₃₇×#410	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	4	2	no	—
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₃₇×#1799	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×#41	C₂³×C₃₇
socle	C₂×C₃₇	#1799
prime radical	C₃₇×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂×C₈×C₃₇ ◁ C₂×#131×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1812	C₂	semidirect
4. C₃₇×#1391	C₂	semidirect
5. C₂×#131×C₃₇	C₄	semidirect
11. C₃₇×#423	C₄	semidirect
13. C₃₇×#408	C₄	semidirect
15. C₂×C₄×C₈×C₃₇	C₄	semidirect
120. #12807	C₃₇	direct (N × H)
138. #1812	C₂×C₃₇	semidirect
139. #1391	C₂×C₃₇	semidirect
145. C₂×#131	C₄×C₃₇	semidirect
151. #423	C₄×C₃₇	semidirect
153. #408	C₄×C₃₇	semidirect
155. C₂×C₄×C₈	C₄×C₃₇	semidirect
159. C₃₇	#12807	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×#2931	2
C₃₇×#2126	2
C₃₇×#1812	2
C₃₇×#1391	2
#12807	37

Subgroups — C37×#12807 #22634060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#12807 #22634060