Subgroups — C₂²×#17985800 #32150324

Order 18176.

1470 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985800	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₁	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	—	2	4	no	—
7	C₂²×C₇₁⋊C₁₆	C₂²	1	6	yes	complemented
8	C₂×#14556062	—	2	3	no	—
9	C₂×#14556062	C₂²	1	6	yes	complemented
10	#17985800	C₂²	1	16	yes	direct product factor
11	C₂³×D₄×C₇₁	—	2	1	no	—
12	C₂²×D₈×C₇₁	—	2	3	no	—
13	C₂²×D₈×C₇₁	—	2	4	no	—
14	C₂²×D₈×C₇₁	C₄	1	8	yes	complemented
15	C₂²×D₈×C₇₁	C₂²	1	6	yes	—
16	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂³×Dic₇₁	—	4	1	no	—
18	C₂×C₄⋊Dic₇₁	—	2	6	no	—
19	C₂×C₄⋊Dic₇₁	—	4	3	no	—
20	C₂×C₇₁⋊C₁₆	—	2	12	no	—
21	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	complemented
22	#14556062	—	2	6	no	—
23	#14556062	C₂³	1	4	yes	complemented
24	C₂⁵×C₇₁	—	4	1	no	—
25	C₂²×D₄×C₇₁	—	2	8	no	—
26	C₂²×D₄×C₇₁	—	2	6	no	—
27	C₂²×D₄×C₇₁	—	4	3	no	—
28	C₂²×D₄×C₇₁	—	4	4	no	—
29	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂×D₈×C₇₁	—	2	24	no	—
31	C₂×D₈×C₇₁	—	2	6	no	—
32	C₂×D₈×C₇₁	—	2	12	no	—
33	C₂×D₈×C₇₁	C₂×C₄	1	24	yes	complemented
34	C₂×D₈×C₇₁	C₂³	1	4	yes	—
35	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₇₁	D₄	1	3	yes	—
37	C₂²×C₈×C₇₁	D₄	1	4	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
39	C₂²×Dic₇₁	—	4	6	no	—
40	C₂²×Dic₇₁	—	8	4	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	4	6	no	—
43	C₇₁⋊C₁₆	—	2	4	no	—
44	C₂⁴×C₇₁	—	4	8	no	—
45	C₂⁴×C₇₁	—	4	6	no	—
46	C₂⁴×C₇₁	—	8	4	no	—
47	C₂⁴×C₇₁	—	8	4	no	—
48	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
49	C₂×D₄×C₇₁	—	2	24	no	—
50	C₂×D₄×C₇₁	—	2	4	no	—
51	C₂×D₄×C₇₁	—	4	24	no	—
52	C₂×D₄×C₇₁	—	4	6	no	—
53	C₂×D₄×C₇₁	—	4	12	no	—
54	C₂²×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₇₁	D₈	1	3	yes	—
56	C₂²×C₄×C₇₁	SD₁₆	1	4	yes	—
57	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
58	D₈×C₇₁	—	2	24	no	—
59	D₈×C₇₁	—	2	4	no	—
60	D₈×C₇₁	C₂²×C₄	1	8	yes	complemented
61	C₂×C₈×C₇₁	C₂²⋊C₄	1	12	yes	—
62	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
63	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
64	C₂×C₈×C₇₁	C₂×D₄	1	6	yes	—
65	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
66	C₂³×C₇₁	—	4	4	no	—
67	C₂³×C₇₁	—	4	24	no	—
68	C₂³×C₇₁	—	8	32	no	—
69	C₂³×C₇₁	—	8	12	no	—
70	C₂³×C₇₁	—	8	12	no	—
71	C₂³×C₇₁	#102	1	4	yes	—
72	C₂³×C₇₁	D₁₆	1	4	yes	—
73	C₂³×C₇₁	SD₃₂	1	4	yes	—
74	C₂³×C₇₁	C₂×D₈	1	3	yes	—
75	D₄×C₇₁	—	2	8	no	—
76	D₄×C₇₁	—	4	4	no	—
77	D₄×C₇₁	—	4	24	no	—
78	C₂×C₄×C₇₁	#102	1	12	yes	—
79	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₇₁	C₂×SD₁₆	1	6	yes	—
82	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₇₁	—	4	4	no	—
84	C₂×Dic₇₁	—	8	12	no	—
85	C₈×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
86	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₇₁	—	4	8	no	—
89	C₂²×C₇₁	—	8	4	no	—
90	C₂²×C₇₁	—	8	4	no	—
91	C₂²×C₇₁	—	8	48	no	—
92	C₂²×C₇₁	#357	1	16	yes	complemented
93	C₂²×C₇₁	C₂×#102	1	6	yes	—
94	C₂²×C₇₁	C₂×D₁₆	1	6	yes	—
95	C₂²×C₇₁	C₂×SD₃₂	1	6	yes	—
96	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₇₁	C₂×#102	1	6	yes	—
98	C₄×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
100	Dic₇₁	—	8	4	no	—
101	C₂²×#357	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	—	8	8	no	—
103	C₂×C₇₁	C₂×#357	1	12	yes	complemented
104	C₂×C₇₁	C₂²×#102	1	1	yes	characteristic
105	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
106	C₂×C₇₁	C₂²×SD₃₂	1	1	yes	characteristic
107	C₂³×D₈	D₇₁	1	1	yes	characteristic
108	C₂³×C₁₆	—	71	1	no	—
109	C₂²×#107	—	71	1	no	—
110	C₂×#357	—	71	12	no	—
111	C₇₁	C₂²×#357	1	1	yes	characteristic, complemented, Sylow
112	C₂³×D₄	—	2	1	no	—
113	C₂²×D₈	—	2	3	no	—
114	C₂²×D₈	—	2	4	no	—
115	C₂²×D₈	Dic₇₁	1	8	yes	complemented
116	C₂²×D₈	C₂×D₇₁	1	6	yes	—
117	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
118	C₂²×C₄⋊C₄	—	142	1	no	—
119	C₂²×C₁₆	—	71	6	no	—
120	C₂²×C₁₆	—	142	4	no	—
121	C₂×#107	—	71	6	no	—
122	C₂×#107	—	142	3	no	—
123	#357	—	71	16	no	—
124	C₂⁵	—	4	1	no	—
125	C₂²×D₄	—	2	8	no	—
126	C₂²×D₄	—	2	6	no	—
127	C₂²×D₄	—	4	3	no	—
128	C₂²×D₄	—	4	4	no	—
129	C₂³×C₄	—	284	1	no	—
130	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
131	C₂×D₈	—	2	24	no	—
132	C₂×D₈	—	2	12	no	—
133	C₂×D₈	—	2	6	no	—
134	C₂×D₈	C₂×Dic₇₁	1	24	yes	complemented
135	C₂×D₈	C₂²×D₇₁	1	4	yes	—
136	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
137	C₂²×C₈	C₂²⋊D₇₁	1	3	yes	—
138	C₂²×C₈	C₂²⋊D₇₁	1	4	yes	—
139	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
140	C₂×C₄⋊C₄	—	284	3	no	—
141	C₂×C₄⋊C₄	—	142	6	no	—
142	C₂×C₁₆	—	71	4	no	—
143	C₂×C₁₆	—	142	12	no	—
144	#107	—	71	4	no	—
145	#107	—	142	6	no	—
146	C₂⁴	—	4	8	no	—
147	C₂⁴	—	4	6	no	—
148	C₂⁴	—	8	4	no	—
149	C₂⁴	—	8	4	no	—
150	C₂⁴	#11765850	1	1	yes	characteristic
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	2	24	no	—
153	C₂×D₄	—	4	24	no	—
154	C₂×D₄	—	4	6	no	—
155	C₂×D₄	—	4	12	no	—
156	C₂²×C₄	—	284	6	no	—
157	C₂²×C₄	—	568	4	no	—
158	C₂²×C₄	C₂²⋊Dic₇₁	1	4	yes	—
159	C₂²×C₄	#11765850	1	3	yes	—
160	C₂²×C₄	#11765853	1	4	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
162	D₈	—	2	4	no	—
163	D₈	—	2	24	no	—
164	D₈	C₂²×Dic₇₁	1	8	yes	complemented
165	C₂×C₈	C₂²⋊Dic₇₁	1	12	yes	—
166	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
167	C₂×C₈	C₂×C₂²⋊D₇₁	1	6	yes	—
168	C₂×C₈	C₂×C₂²⋊D₇₁	1	3	yes	—
169	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
170	C₄⋊C₄	—	284	6	no	—
171	C₄⋊C₄	—	142	4	no	—
172	C₁₆	—	142	4	no	—
173	C₂³	—	4	24	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	8	32	no	—
176	C₂³	—	8	12	no	—
177	C₂³	—	8	12	no	—
178	C₂³	#14556049	1	4	yes	—
179	C₂³	#14556069	1	4	yes	—
180	C₂³	#14556072	1	4	yes	—
181	C₂³	C₂×#11765850	1	3	yes	—
182	D₄	—	2	8	no	—
183	D₄	—	4	24	no	—
184	D₄	—	4	4	no	—
185	C₂×C₄	—	284	4	no	—
186	C₂×C₄	—	568	12	no	—
187	C₂×C₄	#14556049	1	12	yes	—
188	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	6	yes	—
189	C₂×C₄	C₂×#11765850	1	3	yes	—
190	C₂×C₄	C₂×#11765853	1	6	yes	—
191	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
192	C₈	C₂×C₂²⋊Dic₇₁	1	6	yes	—
193	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
194	C₈	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
195	C₂²	—	4	8	no	—
196	C₂²	—	8	48	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	4	no	—
199	C₂²	#17985800	1	16	yes	direct product factor
200	C₂²	C₂×#14556049	1	6	yes	—
201	C₂²	C₂×#14556069	1	6	yes	—
202	C₂²	C₂×#14556072	1	6	yes	—
203	C₂²	C₂²×#11765850	1	1	yes	characteristic
204	C₄	—	568	4	no	—
205	C₄	C₂×#14556049	1	6	yes	—
206	C₄	C₂²×C₂²⋊Dic₇₁	1	1	yes	characteristic
207	C₄	C₂²×#11765853	1	1	yes	characteristic
208	C₂	—	8	8	no	—
209	C₂	C₂×#17985800	1	12	yes	direct product factor
210	C₂	C₂²×#14556049	1	1	yes	characteristic
211	C₂	C₂²×#14556069	1	1	yes	characteristic
212	C₂	C₂²×#14556072	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11765850
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting, prime radical	C₂³×D₈×C₇₁	C₂
socle	C₂⁴×C₇₁	D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂³×D₈	D₇₁
hypocenter	C₇₁	C₂²×#357
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇₁⋊C₁₆	C₂	semidirect
2. C₂²×#14556062	C₂	semidirect
3. C₂×#17985800	C₂	direct (N × H)
7. C₂²×C₇₁⋊C₁₆	C₂²	semidirect
9. C₂×#14556062	C₂²	semidirect
10. #17985800	C₂²	direct (N × H)
14. C₂²×D₈×C₇₁	C₄	semidirect
21. C₂×C₇₁⋊C₁₆	C₂³	semidirect
23. #14556062	C₂³	semidirect
33. C₂×D₈×C₇₁	C₂×C₄	semidirect
60. D₈×C₇₁	C₂²×C₄	semidirect
92. C₂²×C₇₁	#357	semidirect
103. C₂×C₇₁	C₂×#357	semidirect
111. C₇₁	C₂²×#357	semidirect
115. C₂²×D₈	Dic₇₁	semidirect
134. C₂×D₈	C₂×Dic₇₁	semidirect
164. D₈	C₂²×Dic₇₁	semidirect
199. C₂²	#17985800	direct (N × H)
209. C₂	C₂×#17985800	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₁⋊C₁₆	2
C₂²×#14556062	2
C₂×#17985800	2
C₂³×D₈×C₇₁	2
C₂²×#357	71

Subgroups — C22×#17985800 #32150324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17985800 #32150324