Subgroups — C₂²×C₃₇×#391 #22655274

Order 9472.

6174 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄²×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₇×#391	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#95×C₃₇	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₃₇	—	2	8	no	—
7	C₂³×Q₈×C₃₇	—	2	8	no	—
8	C₂³×Q₈×C₃₇	C₄	1	16	yes	complemented
9	C₂³×Q₈×C₃₇	C₂²	1	12	yes	—
10	C₂⁴×C₄×C₃₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₃₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₃₇	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₃₇	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₃₇	C₂²	1	12	yes	—
16	C₂²×C₄²×C₃₇	—	2	8	no	—
17	C₂²×C₄²×C₃₇	C₂²	1	12	yes	—
18	C₃₇×#391	C₂²	1	64	yes	direct product factor
19	C₂×#95×C₃₇	C₂²	1	48	yes	—
20	C₂⁵×C₃₇	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₃₇	—	2	128	no	—
22	C₂²×Q₈×C₃₇	—	2	48	no	—
23	C₂²×Q₈×C₃₇	—	2	48	no	—
24	C₂²×Q₈×C₃₇	C₂×C₄	1	96	yes	complemented
25	C₂²×Q₈×C₃₇	C₂³	1	16	yes	—
26	C₂³×C₄×C₃₇	—	2	8	no	—
27	C₂³×C₄×C₃₇	—	2	8	no	—
28	C₂³×C₄×C₃₇	—	2	8	no	—
29	C₂³×C₄×C₃₇	—	2	8	no	—
30	C₂³×C₄×C₃₇	—	2	8	no	—
31	C₂³×C₄×C₃₇	—	2	16	no	—
32	C₂³×C₄×C₃₇	C₂×C₄	1	16	yes	—
33	C₂³×C₄×C₃₇	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₃₇	D₄	1	8	yes	—
35	C₂³×C₄×C₃₇	D₄	1	8	yes	—
36	C₂³×C₄×C₃₇	Q₈	1	16	yes	complemented
37	C₂³×C₄×C₃₇	C₂³	1	12	yes	—
38	C₂³×C₄×C₃₇	C₂³	1	12	yes	—
39	C₂³×C₄×C₃₇	C₂³	1	12	yes	—
40	C₂³×C₄×C₃₇	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₃₇	—	2	48	no	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	16	yes	—
43	C₂×C₄²×C₃₇	—	2	48	no	—
44	C₂×C₄²×C₃₇	C₂³	1	16	yes	—
45	#95×C₃₇	C₂³	1	64	yes	—
46	C₂⁴×C₃₇	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₃₇	C₂×D₄	1	4	yes	—
48	C₂⁴×C₃₇	C₂×D₄	1	4	yes	—
49	C₂⁴×C₃₇	C₂×Q₈	1	8	yes	—
50	C₂⁴×C₃₇	#41	1	8	yes	—
51	C₂⁴×C₃₇	C₂⁴	1	3	yes	—
52	C₂×Q₈×C₃₇	—	2	384	no	—
53	C₂×Q₈×C₃₇	—	2	32	no	—
54	C₂×Q₈×C₃₇	—	2	32	no	—
55	C₂×Q₈×C₃₇	C₂²×C₄	1	64	yes	complemented
56	C₂²×C₄×C₃₇	—	2	64	no	—
57	C₂²×C₄×C₃₇	—	2	64	no	—
58	C₂²×C₄×C₃₇	—	2	24	no	—
59	C₂²×C₄×C₃₇	—	2	24	no	—
60	C₂²×C₄×C₃₇	—	2	24	no	—
61	C₂²×C₄×C₃₇	—	2	24	no	—
62	C₂²×C₄×C₃₇	—	2	24	no	—
63	C₂²×C₄×C₃₇	—	2	48	no	—
64	C₂²×C₄×C₃₇	—	4	32	no	—
65	C₂²×C₄×C₃₇	C₂²⋊C₄	1	64	yes	—
66	C₂²×C₄×C₃₇	C₂²×C₄	1	48	yes	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	24	yes	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	24	yes	—
69	C₂²×C₄×C₃₇	C₂×D₄	1	24	yes	—
70	C₂²×C₄×C₃₇	C₂×Q₈	1	48	yes	complemented
71	C₂²×C₄×C₃₇	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₃₇	C₂⁴	1	8	yes	—
73	C₂²×C₄×C₃₇	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₃₇	C₂⁴	1	4	yes	—
75	C₄⋊C₄×C₃₇	—	2	32	no	—
76	C₄²×C₃₇	—	2	32	no	—
77	C₂³×C₃₇	C₂×C₂²⋊C₄	1	16	yes	—
78	C₂³×C₃₇	C₄×Q₈	1	32	yes	—
79	C₂³×C₃₇	C₂²⋊Q₈	1	32	yes	—
80	C₂³×C₃₇	#124	1	16	yes	—
81	C₂³×C₃₇	C₄⋊Q₈	1	16	yes	—
82	C₂³×C₃₇	C₂³×C₄	1	6	yes	—
83	C₂³×C₃₇	C₂²×D₄	1	6	yes	—
84	C₂³×C₃₇	C₂²×D₄	1	6	yes	—
85	C₂³×C₃₇	C₂²×Q₈	1	12	yes	—
86	C₂³×C₃₇	C₂×#41	1	12	yes	—
87	C₂³×C₃₇	C₂⁵	1	1	yes	characteristic
88	Q₈×C₃₇	—	2	128	no	—
89	C₂×C₄×C₃₇	—	2	96	no	—
90	C₂×C₄×C₃₇	—	2	96	no	—
91	C₂×C₄×C₃₇	—	2	8	no	—
92	C₂×C₄×C₃₇	—	2	8	no	—
93	C₂×C₄×C₃₇	—	2	8	no	—
94	C₂×C₄×C₃₇	—	2	8	no	—
95	C₂×C₄×C₃₇	—	2	8	no	—
96	C₂×C₄×C₃₇	—	2	16	no	—
97	C₂×C₄×C₃₇	—	4	48	no	—
98	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	96	yes	—
99	C₂×C₄×C₃₇	C₂³×C₄	1	16	yes	—
100	C₂×C₄×C₃₇	C₂³×C₄	1	8	yes	—
101	C₂×C₄×C₃₇	C₂²×D₄	1	8	yes	—
102	C₂×C₄×C₃₇	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₃₇	C₂²×Q₈	1	16	yes	complemented
104	C₂²×#391	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₃₇	#391	1	64	yes	direct product factor
106	C₂²×C₃₇	C₂²×C₂²⋊C₄	1	12	yes	—
107	C₂²×C₃₇	C₂×C₄×Q₈	1	24	yes	—
108	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	24	yes	—
109	C₂²×C₃₇	C₂×#124	1	12	yes	—
110	C₂²×C₃₇	C₂×C₄⋊Q₈	1	12	yes	—
111	C₂²×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
112	C₂²×C₃₇	C₂³×D₄	1	1	yes	characteristic
113	C₂²×C₃₇	C₂³×D₄	1	1	yes	characteristic
114	C₂²×C₃₇	C₂³×Q₈	1	2	yes	—
115	C₂²×C₃₇	C₂²×#41	1	2	yes	—
116	C₄×C₃₇	—	2	16	no	—
117	C₄×C₃₇	—	2	16	no	—
118	C₄×C₃₇	—	4	8	no	—
119	C₄×C₃₇	C₂²×C₂²⋊C₄	1	16	yes	—
120	C₂⁴×Q₈	C₂×C₃₇	1	1	yes	characteristic
121	C₂³×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
122	C₂³×C₄²	C₂×C₃₇	1	1	yes	characteristic
123	C₂×#391	C₂×C₃₇	1	24	yes	direct product factor
124	C₂²×#95	C₂×C₃₇	1	4	yes	—
125	C₂×C₃₇	C₂×#391	1	24	yes	direct product factor
126	C₂×C₃₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₃₇	C₂²×C₄×Q₈	1	2	yes	—
128	C₂×C₃₇	C₂²×C₂²⋊Q₈	1	2	yes	—
129	C₂×C₃₇	C₂²×#124	1	1	yes	characteristic
130	C₂×C₃₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
131	C₂³×Q₈	—	2	8	no	—
132	C₂³×Q₈	—	2	8	no	—
133	C₂³×Q₈	C₄×C₃₇	1	16	yes	complemented
134	C₂³×Q₈	C₂²×C₃₇	1	12	yes	—
135	C₂⁴×C₄	C₂²×C₃₇	1	2	yes	—
136	C₂⁴×C₄	C₂²×C₃₇	1	2	yes	—
137	C₂⁴×C₄	C₂²×C₃₇	1	2	yes	—
138	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	2	8	no	—
140	C₂²×C₄⋊C₄	C₂²×C₃₇	1	12	yes	—
141	C₂²×C₄²	—	2	8	no	—
142	C₂²×C₄²	C₂²×C₃₇	1	12	yes	—
143	#391	C₂²×C₃₇	1	64	yes	direct product factor
144	C₂×#95	C₂²×C₃₇	1	48	yes	—
145	C₃₇	C₂²×#391	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁵	C₂³×C₃₇	1	1	yes	characteristic
147	C₂²×Q₈	—	2	128	no	—
148	C₂²×Q₈	—	2	48	no	—
149	C₂²×Q₈	—	2	48	no	—
150	C₂²×Q₈	C₂×C₄×C₃₇	1	96	yes	complemented
151	C₂²×Q₈	C₂³×C₃₇	1	16	yes	—
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	2	16	no	—
158	C₂³×C₄	C₂×C₄×C₃₇	1	16	yes	—
159	C₂³×C₄	C₂×C₄×C₃₇	1	8	yes	—
160	C₂³×C₄	D₄×C₃₇	1	8	yes	—
161	C₂³×C₄	D₄×C₃₇	1	8	yes	—
162	C₂³×C₄	Q₈×C₃₇	1	16	yes	complemented
163	C₂³×C₄	C₂³×C₃₇	1	12	yes	—
164	C₂³×C₄	C₂³×C₃₇	1	12	yes	—
165	C₂³×C₄	C₂³×C₃₇	1	12	yes	—
166	C₂³×C₄	C₂³×C₃₇	1	6	yes	—
167	C₂×C₄⋊C₄	—	2	48	no	—
168	C₂×C₄⋊C₄	C₂³×C₃₇	1	16	yes	—
169	C₂×C₄²	—	2	48	no	—
170	C₂×C₄²	C₂³×C₃₇	1	16	yes	—
171	#95	C₂³×C₃₇	1	64	yes	—
172	C₂⁴	C₂²×C₄×C₃₇	1	4	yes	—
173	C₂⁴	C₂×D₄×C₃₇	1	4	yes	—
174	C₂⁴	C₂×D₄×C₃₇	1	4	yes	—
175	C₂⁴	C₂×Q₈×C₃₇	1	8	yes	—
176	C₂⁴	#41×C₃₇	1	8	yes	—
177	C₂⁴	C₂⁴×C₃₇	1	3	yes	—
178	C₂×Q₈	—	2	384	no	—
179	C₂×Q₈	—	2	32	no	—
180	C₂×Q₈	—	2	32	no	—
181	C₂×Q₈	C₂²×C₄×C₃₇	1	64	yes	complemented
182	C₂²×C₄	—	2	64	no	—
183	C₂²×C₄	—	2	64	no	—
184	C₂²×C₄	—	2	24	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	24	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	2	48	no	—
190	C₂²×C₄	—	4	32	no	—
191	C₂²×C₄	C₂²⋊C₄×C₃₇	1	64	yes	—
192	C₂²×C₄	C₂²×C₄×C₃₇	1	48	yes	—
193	C₂²×C₄	C₂²×C₄×C₃₇	1	24	yes	—
194	C₂²×C₄	C₂×D₄×C₃₇	1	24	yes	—
195	C₂²×C₄	C₂×D₄×C₃₇	1	24	yes	—
196	C₂²×C₄	C₂×Q₈×C₃₇	1	48	yes	complemented
197	C₂²×C₄	C₂⁴×C₃₇	1	8	yes	—
198	C₂²×C₄	C₂⁴×C₃₇	1	8	yes	—
199	C₂²×C₄	C₂⁴×C₃₇	1	8	yes	—
200	C₂²×C₄	C₂⁴×C₃₇	1	4	yes	—
201	C₄⋊C₄	—	2	32	no	—
202	C₄²	—	2	32	no	—
203	C₂³	C₂×C₂²⋊C₄×C₃₇	1	16	yes	—
204	C₂³	C₄×Q₈×C₃₇	1	32	yes	—
205	C₂³	C₂²⋊Q₈×C₃₇	1	32	yes	—
206	C₂³	#124×C₃₇	1	16	yes	—
207	C₂³	C₄⋊Q₈×C₃₇	1	16	yes	—
208	C₂³	C₂³×C₄×C₃₇	1	6	yes	—
209	C₂³	C₂²×D₄×C₃₇	1	6	yes	—
210	C₂³	C₂²×D₄×C₃₇	1	6	yes	—
211	C₂³	C₂²×Q₈×C₃₇	1	12	yes	—
212	C₂³	C₂×#41×C₃₇	1	12	yes	—
213	C₂³	C₂⁵×C₃₇	1	1	yes	characteristic
214	Q₈	—	2	128	no	—
215	C₂×C₄	—	2	96	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	96	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	4	48	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	96	yes	—
225	C₂×C₄	C₂³×C₄×C₃₇	1	16	yes	—
226	C₂×C₄	C₂³×C₄×C₃₇	1	8	yes	—
227	C₂×C₄	C₂²×D₄×C₃₇	1	8	yes	—
228	C₂×C₄	C₂²×D₄×C₃₇	1	8	yes	—
229	C₂×C₄	C₂²×Q₈×C₃₇	1	16	yes	complemented
230	C₂²	C₃₇×#391	1	64	yes	direct product factor
231	C₂²	C₂²×C₂²⋊C₄×C₃₇	1	12	yes	—
232	C₂²	C₂×C₄×Q₈×C₃₇	1	24	yes	—
233	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	24	yes	—
234	C₂²	C₂×#124×C₃₇	1	12	yes	—
235	C₂²	C₂×C₄⋊Q₈×C₃₇	1	12	yes	—
236	C₂²	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
237	C₂²	C₂³×D₄×C₃₇	1	1	yes	characteristic
238	C₂²	C₂³×D₄×C₃₇	1	1	yes	characteristic
239	C₂²	C₂³×Q₈×C₃₇	1	2	yes	—
240	C₂²	C₂²×#41×C₃₇	1	2	yes	—
241	C₄	—	2	16	no	—
242	C₄	—	2	16	no	—
243	C₄	—	4	8	no	—
244	C₄	C₂²×C₂²⋊C₄×C₃₇	1	16	yes	—
245	C₂	C₂×C₃₇×#391	1	24	yes	direct product factor
246	C₂	C₂³×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
247	C₂	C₂²×C₄×Q₈×C₃₇	1	2	yes	—
248	C₂	C₂²×C₂²⋊Q₈×C₃₇	1	2	yes	—
249	C₂	C₂²×#124×C₃₇	1	1	yes	characteristic
250	C₂	C₂²×C₄⋊Q₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁵×C₃₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₃₇×#391	C₂	direct (N × H)
8. C₂³×Q₈×C₃₇	C₄	semidirect
18. C₃₇×#391	C₂²	direct (N × H)
24. C₂²×Q₈×C₃₇	C₂×C₄	semidirect
36. C₂³×C₄×C₃₇	Q₈	semidirect
55. C₂×Q₈×C₃₇	C₂²×C₄	semidirect
70. C₂²×C₄×C₃₇	C₂×Q₈	semidirect
103. C₂×C₄×C₃₇	C₂²×Q₈	semidirect
104. C₂²×#391	C₃₇	direct (N × H)
105. C₂²×C₃₇	#391	direct (N × H)
123. C₂×#391	C₂×C₃₇	direct (N × H)
125. C₂×C₃₇	C₂×#391	direct (N × H)
133. C₂³×Q₈	C₄×C₃₇	semidirect
143. #391	C₂²×C₃₇	direct (N × H)
145. C₃₇	C₂²×#391	direct (N × H)
150. C₂²×Q₈	C₂×C₄×C₃₇	semidirect
162. C₂³×C₄	Q₈×C₃₇	semidirect
181. C₂×Q₈	C₂²×C₄×C₃₇	semidirect
196. C₂²×C₄	C₂×Q₈×C₃₇	semidirect
229. C₂×C₄	C₂²×Q₈×C₃₇	semidirect
230. C₂²	C₃₇×#391	direct (N × H)
245. C₂	C₂×C₃₇×#391	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₃₇	2
C₂³×C₄⋊C₄×C₃₇	2
C₂³×C₄²×C₃₇	2
C₂×C₃₇×#391	2
C₂²×#95×C₃₇	2
C₂²×#391	37

Subgroups — C22×C37×#391 #22655274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#391 #22655274