Subgroups — C₂²×C₃₇×#476 #22681727

Order 9472.

2790 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	C₂²×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#476	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#106×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#102×C₃₇	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₃₇	—	2	1	no	—
8	C₂³×D₄×C₃₇	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₃₇	—	2	1	no	—
10	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₃₇	—	2	6	no	—
12	C₂×C₄⋊Q₈×C₃₇	C₂²	1	12	yes	complemented
13	C₂×C₄×D₄×C₃₇	—	2	6	no	—
14	C₂×C₄×D₄×C₃₇	C₂²	1	12	yes	—
15	C₂²×C₄⋊C₄×C₃₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₂²⋊C₄×C₃₇	—	2	1	no	—
19	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#476	C₂²	1	64	yes	direct product factor
21	C₂×#106×C₃₇	C₂²	1	24	yes	complemented
22	C₂×C₄⋊C₈×C₃₇	C₂²	1	12	yes	complemented
23	C₂×#102×C₃₇	C₂²	1	24	yes	—
24	C₂⁵×C₃₇	—	2	1	no	—
25	C₂²×Q₈×C₃₇	—	2	12	no	—
26	C₂²×Q₈×C₃₇	—	4	8	no	—
27	C₂²×D₄×C₃₇	—	2	3	no	—
28	C₂²×D₄×C₃₇	—	2	4	no	—
29	C₂²×D₄×C₃₇	Q₈	1	8	yes	complemented
30	C₂²×D₄×C₃₇	C₂³	1	6	yes	—
31	C₂³×C₄×C₃₇	—	2	1	no	—
32	C₂³×C₄×C₃₇	—	2	2	no	—
33	C₂³×C₄×C₃₇	—	2	12	no	—
34	C₂³×C₄×C₃₇	—	4	8	no	—
35	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	—	2	8	no	—
39	C₂²×C₈×C₃₇	C₂³	1	12	yes	—
40	C₄⋊Q₈×C₃₇	—	2	24	no	—
41	C₄⋊Q₈×C₃₇	C₂³	1	16	yes	complemented
42	C₄×D₄×C₃₇	—	2	24	no	—
43	C₄×D₄×C₃₇	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₃₇	—	2	6	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	12	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	3	no	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	12	no	—
50	C₂×C₄²×C₃₇	D₄	1	6	yes	—
51	C₂×C₄²×C₃₇	C₂³	1	6	yes	—
52	#106×C₃₇	C₂³	1	32	yes	complemented
53	C₄⋊C₈×C₃₇	C₂³	1	16	yes	complemented
54	#102×C₃₇	C₂³	1	32	yes	—
55	C₂⁴×C₃₇	—	2	8	no	—
56	C₂⁴×C₃₇	—	2	6	no	—
57	C₂⁴×C₃₇	—	4	4	no	—
58	C₂⁴×C₃₇	—	4	4	no	—
59	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₃₇	—	2	16	no	—
61	C₂×Q₈×C₃₇	—	4	48	no	—
62	C₂×D₄×C₃₇	—	2	24	no	—
63	C₂×D₄×C₃₇	—	2	6	no	—
64	C₂×D₄×C₃₇	—	2	12	no	—
65	C₂×D₄×C₃₇	C₂×Q₈	1	24	yes	complemented
66	C₂×D₄×C₃₇	C₂⁴	1	4	yes	—
67	C₂²×C₄×C₃₇	—	2	4	no	—
68	C₂²×C₄×C₃₇	—	2	6	no	—
69	C₂²×C₄×C₃₇	—	2	12	no	—
70	C₂²×C₄×C₃₇	—	2	16	no	—
71	C₂²×C₄×C₃₇	—	4	4	no	—
72	C₂²×C₄×C₃₇	—	4	8	no	—
73	C₂²×C₄×C₃₇	—	4	48	no	—
74	C₂²×C₄×C₃₇	SD₁₆	1	8	yes	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₃₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₃₇	#41	1	4	yes	—
80	C₂²×C₄×C₃₇	C₂⁴	1	3	yes	—
81	C₂×C₈×C₃₇	—	2	24	no	—
82	C₂×C₈×C₃₇	C₂⁴	1	8	yes	—
83	C₄⋊C₄×C₃₇	—	2	16	no	—
84	C₄⋊C₄×C₃₇	—	2	6	no	—
85	C₄⋊C₄×C₃₇	—	2	12	no	—
86	C₄⋊C₄×C₃₇	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₃₇	C₂⁴	1	4	yes	—
88	C₂²⋊C₄×C₃₇	—	2	16	no	—
89	C₄²×C₃₇	C₂×D₄	1	12	yes	—
90	C₄²×C₃₇	C₂⁴	1	4	yes	—
91	C₂³×C₃₇	—	2	24	no	—
92	C₂³×C₃₇	—	2	4	no	—
93	C₂³×C₃₇	—	4	12	no	—
94	C₂³×C₃₇	—	4	12	no	—
95	C₂³×C₃₇	—	4	32	no	—
96	C₂³×C₃₇	C₂²⋊Q₈	1	4	yes	—
97	C₂³×C₃₇	C₂×SD₁₆	1	4	yes	—
98	C₂³×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
99	C₂³×C₃₇	C₂²×D₄	1	3	yes	—
100	Q₈×C₃₇	—	4	32	no	—
101	D₄×C₃₇	—	2	24	no	—
102	D₄×C₃₇	—	2	4	no	—
103	D₄×C₃₇	C₂²×Q₈	1	8	yes	complemented
104	C₂×C₄×C₃₇	—	2	12	no	—
105	C₂×C₄×C₃₇	—	2	4	no	—
106	C₂×C₄×C₃₇	—	2	8	no	—
107	C₂×C₄×C₃₇	—	4	12	no	—
108	C₂×C₄×C₃₇	—	4	24	no	—
109	C₂×C₄×C₃₇	—	4	32	no	—
110	C₂×C₄×C₃₇	C₂²⋊Q₈	1	12	yes	—
111	C₂×C₄×C₃₇	C₂×SD₁₆	1	24	yes	—
112	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
113	C₂×C₄×C₃₇	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
115	C₂×C₄×C₃₇	C₂²×Q₈	1	6	yes	—
116	C₂×C₄×C₃₇	C₂×#41	1	6	yes	—
117	C₂×C₄×C₃₇	C₂⁵	1	1	yes	characteristic
118	C₈×C₃₇	—	2	8	no	—
119	C₂²×#476	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₃₇	—	2	8	no	—
121	C₂²×C₃₇	—	4	4	no	—
122	C₂²×C₃₇	—	4	4	no	—
123	C₂²×C₃₇	—	4	48	no	—
124	C₂²×C₃₇	#476	1	16	yes	direct product factor
125	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	6	yes	—
126	C₂²×C₃₇	C₂²×SD₁₆	1	6	yes	—
127	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
128	C₂²×C₃₇	C₂³×D₄	1	1	yes	characteristic
129	C₄×C₃₇	—	2	4	no	—
130	C₄×C₃₇	—	4	4	no	—
131	C₄×C₃₇	—	4	8	no	—
132	C₄×C₃₇	C₂×C₂²⋊Q₈	1	6	yes	—
133	C₄×C₃₇	C₂²×SD₁₆	1	8	yes	—
134	C₄×C₃₇	C₂³×Q₈	1	1	yes	characteristic
135	C₄×C₃₇	C₂²×#41	1	1	yes	characteristic
136	C₂²×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic, complemented
137	C₂²×C₄×D₄	C₂×C₃₇	1	1	yes	characteristic
138	C₂×#476	C₂×C₃₇	1	24	yes	direct product factor
139	C₂²×#106	C₂×C₃₇	1	2	yes	complemented
140	C₂²×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
141	C₂²×#102	C₂×C₃₇	1	2	yes	—
142	C₂×C₃₇	—	4	8	no	—
143	C₂×C₃₇	C₂×#476	1	12	yes	direct product factor
144	C₂×C₃₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₃₇	C₂³×SD₁₆	1	1	yes	characteristic
146	C₂×C₃₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂³×Q₈	—	2	1	no	—
148	C₂³×D₄	C₂²×C₃₇	1	1	yes	characteristic
149	C₂⁴×C₄	—	2	1	no	—
150	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
151	C₂×C₄⋊Q₈	—	2	6	no	—
152	C₂×C₄⋊Q₈	C₂²×C₃₇	1	12	yes	complemented
153	C₂×C₄×D₄	—	2	6	no	—
154	C₂×C₄×D₄	C₂²×C₃₇	1	12	yes	—
155	C₂²×C₄⋊C₄	—	2	1	no	—
156	C₂²×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
157	C₂²×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	—	2	1	no	—
159	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
160	#476	C₂²×C₃₇	1	64	yes	direct product factor
161	C₂×#106	C₂²×C₃₇	1	24	yes	complemented
162	C₂×C₄⋊C₈	C₂²×C₃₇	1	12	yes	complemented
163	C₂×#102	C₂²×C₃₇	1	24	yes	—
164	C₃₇	C₂²×#476	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁵	—	2	1	no	—
166	C₂²×Q₈	—	2	12	no	—
167	C₂²×Q₈	—	4	8	no	—
168	C₂²×D₄	—	2	3	no	—
169	C₂²×D₄	—	2	4	no	—
170	C₂²×D₄	Q₈×C₃₇	1	8	yes	complemented
171	C₂²×D₄	C₂³×C₃₇	1	6	yes	—
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	2	12	no	—
175	C₂³×C₄	—	4	8	no	—
176	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	8	no	—
180	C₂²×C₈	C₂³×C₃₇	1	12	yes	—
181	C₄⋊Q₈	—	2	24	no	—
182	C₄⋊Q₈	C₂³×C₃₇	1	16	yes	complemented
183	C₄×D₄	—	2	24	no	—
184	C₄×D₄	C₂³×C₃₇	1	16	yes	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	2	3	no	—
187	C₂×C₄⋊C₄	—	2	6	no	—
188	C₂×C₄⋊C₄	C₂³×C₃₇	1	6	yes	—
189	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
190	C₂×C₂²⋊C₄	—	2	12	no	—
191	C₂×C₄²	D₄×C₃₇	1	6	yes	—
192	C₂×C₄²	C₂³×C₃₇	1	6	yes	—
193	#106	C₂³×C₃₇	1	32	yes	complemented
194	C₄⋊C₈	C₂³×C₃₇	1	16	yes	complemented
195	#102	C₂³×C₃₇	1	32	yes	—
196	C₂⁴	—	2	8	no	—
197	C₂⁴	—	2	6	no	—
198	C₂⁴	—	4	4	no	—
199	C₂⁴	—	4	4	no	—
200	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
201	C₂×Q₈	—	2	16	no	—
202	C₂×Q₈	—	4	48	no	—
203	C₂×D₄	—	2	24	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	—	2	12	no	—
206	C₂×D₄	C₂×Q₈×C₃₇	1	24	yes	complemented
207	C₂×D₄	C₂⁴×C₃₇	1	4	yes	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	12	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	8	no	—
214	C₂²×C₄	—	4	48	no	—
215	C₂²×C₄	SD₁₆×C₃₇	1	8	yes	—
216	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
217	C₂²×C₄	C₂×D₄×C₃₇	1	3	yes	—
218	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
219	C₂²×C₄	C₂×Q₈×C₃₇	1	4	yes	—
220	C₂²×C₄	#41×C₃₇	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₃₇	1	3	yes	—
222	C₂×C₈	—	2	24	no	—
223	C₂×C₈	C₂⁴×C₃₇	1	8	yes	—
224	C₄⋊C₄	—	2	16	no	—
225	C₄⋊C₄	—	2	6	no	—
226	C₄⋊C₄	—	2	12	no	—
227	C₄⋊C₄	C₂⁴×C₃₇	1	8	yes	—
228	C₄⋊C₄	C₂⁴×C₃₇	1	4	yes	—
229	C₂²⋊C₄	—	2	16	no	—
230	C₄²	C₂×D₄×C₃₇	1	12	yes	—
231	C₄²	C₂⁴×C₃₇	1	4	yes	—
232	C₂³	—	2	24	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	4	12	no	—
235	C₂³	—	4	12	no	—
236	C₂³	—	4	32	no	—
237	C₂³	C₂²⋊Q₈×C₃₇	1	4	yes	—
238	C₂³	C₂×SD₁₆×C₃₇	1	4	yes	—
239	C₂³	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
240	C₂³	C₂²×D₄×C₃₇	1	3	yes	—
241	Q₈	—	4	32	no	—
242	D₄	—	2	24	no	—
243	D₄	—	2	4	no	—
244	D₄	C₂²×Q₈×C₃₇	1	8	yes	complemented
245	C₂×C₄	—	2	12	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	12	no	—
249	C₂×C₄	—	4	24	no	—
250	C₂×C₄	—	4	32	no	—
251	C₂×C₄	C₂²⋊Q₈×C₃₇	1	12	yes	—
252	C₂×C₄	C₂×SD₁₆×C₃₇	1	24	yes	—
253	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
254	C₂×C₄	C₂²×D₄×C₃₇	1	3	yes	—
255	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
256	C₂×C₄	C₂²×Q₈×C₃₇	1	6	yes	—
257	C₂×C₄	C₂×#41×C₃₇	1	6	yes	—
258	C₂×C₄	C₂⁵×C₃₇	1	1	yes	characteristic
259	C₈	—	2	8	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	48	no	—
264	C₂²	C₃₇×#476	1	16	yes	direct product factor
265	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	6	yes	—
266	C₂²	C₂²×SD₁₆×C₃₇	1	6	yes	—
267	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	6	yes	—
268	C₂²	C₂³×D₄×C₃₇	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	8	no	—
272	C₄	C₂×C₂²⋊Q₈×C₃₇	1	6	yes	—
273	C₄	C₂²×SD₁₆×C₃₇	1	8	yes	—
274	C₄	C₂³×Q₈×C₃₇	1	1	yes	characteristic
275	C₄	C₂²×#41×C₃₇	1	1	yes	characteristic
276	C₂	—	4	8	no	—
277	C₂	C₂×C₃₇×#476	1	12	yes	direct product factor
278	C₂	C₂²×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
279	C₂	C₂³×SD₁₆×C₃₇	1	1	yes	characteristic
280	C₂	C₂²×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₇
prime radical	C₂³×D₄×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ C₂²×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₃₇	C₂	semidirect
3. C₂×C₃₇×#476	C₂	direct (N × H)
4. C₂²×#106×C₃₇	C₂	semidirect
5. C₂²×C₄⋊C₈×C₃₇	C₂	semidirect
12. C₂×C₄⋊Q₈×C₃₇	C₂²	semidirect
20. C₃₇×#476	C₂²	direct (N × H)
21. C₂×#106×C₃₇	C₂²	semidirect
22. C₂×C₄⋊C₈×C₃₇	C₂²	semidirect
29. C₂²×D₄×C₃₇	Q₈	semidirect
41. C₄⋊Q₈×C₃₇	C₂³	semidirect
52. #106×C₃₇	C₂³	semidirect
53. C₄⋊C₈×C₃₇	C₂³	semidirect
65. C₂×D₄×C₃₇	C₂×Q₈	semidirect
103. D₄×C₃₇	C₂²×Q₈	semidirect
119. C₂²×#476	C₃₇	direct (N × H)
124. C₂²×C₃₇	#476	direct (N × H)
136. C₂²×C₄⋊Q₈	C₂×C₃₇	semidirect
138. C₂×#476	C₂×C₃₇	direct (N × H)
139. C₂²×#106	C₂×C₃₇	semidirect
140. C₂²×C₄⋊C₈	C₂×C₃₇	semidirect
143. C₂×C₃₇	C₂×#476	direct (N × H)
152. C₂×C₄⋊Q₈	C₂²×C₃₇	semidirect
160. #476	C₂²×C₃₇	direct (N × H)
161. C₂×#106	C₂²×C₃₇	semidirect
162. C₂×C₄⋊C₈	C₂²×C₃₇	semidirect
164. C₃₇	C₂²×#476	direct (N × H)
170. C₂²×D₄	Q₈×C₃₇	semidirect
182. C₄⋊Q₈	C₂³×C₃₇	semidirect
193. #106	C₂³×C₃₇	semidirect
194. C₄⋊C₈	C₂³×C₃₇	semidirect
206. C₂×D₄	C₂×Q₈×C₃₇	semidirect
244. D₄	C₂²×Q₈×C₃₇	semidirect
264. C₂²	C₃₇×#476	direct (N × H)
277. C₂	C₂×C₃₇×#476	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₃₇	2
C₂²×C₄×D₄×C₃₇	2
C₂×C₃₇×#476	2
C₂²×#106×C₃₇	2
C₂²×C₄⋊C₈×C₃₇	2
C₂²×#102×C₃₇	2
C₂²×#476	37

Subgroups — C22×C37×#476 #22681727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#476 #22681727