Subgroups — C₃₇×#28681 #22649934

Order 9472.

698 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#446	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2584	C₂	1	2	yes	maximal
4	C₃₇×#2088	C₂	1	4	yes	maximal
5	C₃₇×#1989	C₂	1	2	yes	maximal
6	C₃₇×#1944	C₂	1	2	yes	maximal
7	C₃₇×#1944	C₂	1	2	yes	maximal
8	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₃₇	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₃₇×#499	—	2	4	no	—
12	C₃₇×#446	—	2	4	no	—
13	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#97×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#398	—	2	2	no	—
22	C₃₇×#395	—	2	2	no	—
23	C₃₇×#389	—	2	4	no	—
24	C₃₇×#389	C₂²	1	4	yes	—
25	C₃₇×#389	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
27	C₃₇×#340	C₂²	1	4	yes	—
28	C₃₇×#340	C₂²	1	4	yes	—
29	C₃₇×#340	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	#125×C₃₇	—	2	4	no	—
35	#125×C₃₇	—	2	4	no	—
36	C₄×Q₈×C₃₇	—	2	2	no	—
37	C₄×Q₈×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	4	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₃₇	—	2	2	no	—
48	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	#107×C₃₇	—	2	2	no	—
51	#107×C₃₇	—	2	2	no	—
52	#106×C₃₇	—	2	4	no	—
53	#106×C₃₇	Q₈	1	8	yes	complemented
54	C₄⋊C₈×C₃₇	—	2	4	no	—
55	C₄⋊C₈×C₃₇	—	2	2	no	—
56	#97×C₃₇	—	2	4	no	—
57	C₄×C₈×C₃₇	D₄	1	4	yes	—
58	#95×C₃₇	—	2	2	no	—
59	#95×C₃₇	—	2	4	no	—
60	#95×C₃₇	—	2	4	no	—
61	C₂×Q₈×C₃₇	—	2	2	no	—
62	C₂×Q₈×C₃₇	—	4	2	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	—	2	2	no	—
65	C₂²×C₄×C₃₇	—	2	2	no	—
66	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₃₇	—	2	2	no	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	4	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	2	2	no	—
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	—	2	2	no	—
83	C₄⋊C₄×C₃₇	—	2	2	no	—
84	C₄⋊C₄×C₃₇	—	4	8	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₄⋊C₄×C₃₇	—	4	4	no	—
87	C₄⋊C₄×C₃₇	—	8	4	no	—
88	C₄⋊C₄×C₃₇	C₂×Q₈	1	8	yes	—
89	C₄²×C₃₇	—	2	2	no	—
90	C₄²×C₃₇	—	2	2	no	—
91	C₄²×C₃₇	—	8	2	no	—
92	C₄²×C₃₇	C₂×D₄	1	2	yes	—
93	C₄²×C₃₇	#41	1	2	yes	—
94	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
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	8	no	—
101	C₂×C₄×C₃₇	—	2	4	no	—
102	C₂×C₄×C₃₇	—	2	4	no	—
103	C₂×C₄×C₃₇	—	4	4	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	2	no	—
107	C₂×C₄×C₃₇	—	4	4	no	—
108	C₂×C₄×C₃₇	—	4	2	no	—
109	C₂×C₄×C₃₇	—	4	2	no	—
110	C₂×C₄×C₃₇	#121	1	2	yes	—
111	C₂×C₄×C₃₇	#121	1	2	yes	—
112	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂²×Q₈	1	2	yes	—
114	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
116	C₈×C₃₇	—	2	2	no	—
117	C₈×C₃₇	—	4	2	no	—
118	C₈×C₃₇	C₄⋊Q₈	1	4	yes	—
119	#28681	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
121	C₂²×C₃₇	D₄×Q₈	1	2	yes	—
122	C₂²×C₃₇	#575	1	2	yes	—
123	C₂²×C₃₇	#577	1	2	yes	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	—	2	2	no	—
126	C₄×C₃₇	—	4	8	no	—
127	C₄×C₃₇	—	4	2	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	—	8	2	no	—
130	C₄×C₃₇	—	8	2	no	—
131	C₄×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₄×C₃₇	#536	1	1	yes	characteristic
133	C₄×C₃₇	#557	1	2	yes	—
134	C₂×#499	C₂×C₃₇	1	1	yes	characteristic
135	C₂×#446	C₂×C₃₇	1	1	yes	characteristic
136	#2584	C₂×C₃₇	1	2	yes	—
137	#2088	C₂×C₃₇	1	4	yes	—
138	#1989	C₂×C₃₇	1	2	yes	—
139	#1944	C₂×C₃₇	1	2	yes	—
140	#1944	C₂×C₃₇	1	2	yes	—
141	C₄×#106	C₂×C₃₇	1	1	yes	characteristic
142	C₂×C₃₇	#2546	1	1	yes	characteristic
143	C₂×C₃₇	#3054	1	1	yes	characteristic
144	C₂×C₃₇	#3060	1	1	yes	characteristic
145	C₂×C₃₇	#3388	1	4	yes	—
146	C₂×#125	C₂²×C₃₇	1	2	yes	—
147	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
148	#499	—	2	4	no	—
149	#446	—	2	4	no	—
150	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
151	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
152	C₂×#106	C₂²×C₃₇	1	2	yes	—
153	C₂×#106	C₂²×C₃₇	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
156	C₂×#97	C₂²×C₃₇	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
158	#398	—	2	2	no	—
159	#395	—	2	2	no	—
160	#389	—	2	4	no	—
161	#389	C₂²×C₃₇	1	4	yes	—
162	#389	C₂²×C₃₇	1	4	yes	—
163	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
164	#340	C₂²×C₃₇	1	2	yes	—
165	#340	C₂²×C₃₇	1	4	yes	—
166	#340	C₂²×C₃₇	1	4	yes	—
167	C₃₇	#28681	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
170	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
172	#125	—	2	4	no	—
173	#125	—	2	4	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
181	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
187	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
188	#107	—	2	2	no	—
189	#107	—	2	2	no	—
190	#106	—	2	4	no	—
191	#106	Q₈×C₃₇	1	8	yes	complemented
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	2	no	—
194	#97	—	2	4	no	—
195	C₄×C₈	D₄×C₃₇	1	4	yes	—
196	#95	—	2	2	no	—
197	#95	—	2	4	no	—
198	#95	—	2	4	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
214	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	8	4	no	—
226	C₄⋊C₄	C₂×Q₈×C₃₇	1	8	yes	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	8	2	no	—
230	C₄²	C₂×D₄×C₃₇	1	2	yes	—
231	C₄²	#41×C₃₇	1	2	yes	—
232	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
233	Q₈	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#121×C₃₇	1	2	yes	—
249	C₂×C₄	#121×C₃₇	1	2	yes	—
250	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
251	C₂×C₄	C₂²×Q₈×C₃₇	1	2	yes	—
252	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
253	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₈	C₄⋊Q₈×C₃₇	1	4	yes	—
257	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
258	C₂²	D₄×Q₈×C₃₇	1	2	yes	—
259	C₂²	C₃₇×#575	1	2	yes	—
260	C₂²	C₃₇×#577	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	4	8	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
269	C₄	C₃₇×#536	1	1	yes	characteristic
270	C₄	C₃₇×#557	1	2	yes	—
271	C₂	C₃₇×#2546	1	1	yes	characteristic
272	C₂	C₃₇×#3054	1	1	yes	characteristic
273	C₂	C₃₇×#3060	1	1	yes	characteristic
274	C₂	C₃₇×#3388	1	4	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 ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
53. #106×C₃₇	Q₈	semidirect
119. #28681	C₃₇	direct (N × H)
167. C₃₇	#28681	direct (N × H)
191. #106	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#499	2
C₂×C₃₇×#446	2
C₃₇×#2584	2
C₃₇×#2088	2
C₃₇×#1989	2
C₃₇×#1944	2
C₄×#106×C₃₇	2
#28681	37

Subgroups — C37×#28681 #22649934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28681 #22649934