Subgroups — C₂²×#17985803 #32150328

Order 18176.

1214 conjugacy classes of subgroups.

The subgroup lattice has 196 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
2	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985803	C₂	1	12	yes	maximal, direct product factor
4	C₂³×Q₁₆×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	—
8	C₂×#14556062	—	2	3	no	—
9	C₂×#14556062	C₂²	1	6	yes	—
10	#17985803	C₂²	1	16	yes	direct product factor
11	C₂³×Q₈×C₇₁	—	2	1	no	—
12	C₂²×Q₁₆×C₇₁	—	2	3	no	—
13	C₂²×Q₁₆×C₇₁	—	2	4	no	—
14	C₂²×Q₁₆×C₇₁	C₄	1	8	yes	complemented
15	C₂²×Q₁₆×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	—
22	#14556062	—	2	6	no	—
23	#14556062	C₂³	1	4	yes	—
24	C₂²×Q₈×C₇₁	—	2	8	no	—
25	C₂²×Q₈×C₇₁	—	2	6	no	—
26	C₂²×Q₈×C₇₁	—	4	3	no	—
27	C₂²×Q₈×C₇₁	—	4	4	no	—
28	C₂³×C₄×C₇₁	—	4	1	no	—
29	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂×Q₁₆×C₇₁	—	2	24	no	—
31	C₂×Q₁₆×C₇₁	—	2	6	no	—
32	C₂×Q₁₆×C₇₁	—	2	12	no	—
33	C₂×Q₁₆×C₇₁	C₂×C₄	1	24	yes	complemented
34	C₂×Q₁₆×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	D₈	1	1	yes	characteristic
45	C₂×Q₈×C₇₁	—	2	24	no	—
46	C₂×Q₈×C₇₁	—	2	4	no	—
47	C₂×Q₈×C₇₁	—	4	24	no	—
48	C₂×Q₈×C₇₁	—	4	6	no	—
49	C₂×Q₈×C₇₁	—	4	12	no	—
50	C₂²×C₄×C₇₁	—	4	8	no	—
51	C₂²×C₄×C₇₁	—	4	6	no	—
52	C₂²×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₇₁	D₈	1	3	yes	—
54	C₂²×C₄×C₇₁	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
56	Q₁₆×C₇₁	—	2	24	no	—
57	Q₁₆×C₇₁	—	2	4	no	—
58	Q₁₆×C₇₁	C₂²×C₄	1	8	yes	complemented
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	12	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
62	C₂×C₈×C₇₁	C₂×D₄	1	6	yes	—
63	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
64	C₂³×C₇₁	#102	1	4	yes	—
65	C₂³×C₇₁	SD₃₂	1	4	yes	—
66	C₂³×C₇₁	Q₃₂	1	4	yes	—
67	C₂³×C₇₁	C₂×D₈	1	3	yes	—
68	Q₈×C₇₁	—	2	8	no	—
69	Q₈×C₇₁	—	4	4	no	—
70	Q₈×C₇₁	—	4	24	no	—
71	C₂×C₄×C₇₁	—	4	4	no	—
72	C₂×C₄×C₇₁	—	4	24	no	—
73	C₂×C₄×C₇₁	#102	1	12	yes	—
74	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
75	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
76	C₂×C₄×C₇₁	C₂×SD₁₆	1	6	yes	—
77	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₇₁	—	4	4	no	—
79	C₂×Dic₇₁	—	8	12	no	—
80	C₈×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
81	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₇₁	#358	1	16	yes	complemented
84	C₂²×C₇₁	C₂×#102	1	6	yes	—
85	C₂²×C₇₁	C₂×SD₃₂	1	6	yes	—
86	C₂²×C₇₁	C₂×Q₃₂	1	6	yes	—
87	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₇₁	—	4	8	no	—
89	C₄×C₇₁	C₂×#102	1	6	yes	—
90	C₄×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
92	Dic₇₁	—	8	4	no	—
93	C₂²×#358	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	C₂×#358	1	12	yes	complemented
95	C₂×C₇₁	C₂²×#102	1	1	yes	characteristic
96	C₂×C₇₁	C₂²×SD₃₂	1	1	yes	characteristic
97	C₂×C₇₁	C₂²×Q₃₂	1	1	yes	characteristic
98	C₂³×Q₁₆	D₇₁	1	1	yes	characteristic
99	C₂³×C₁₆	—	71	1	no	—
100	C₂²×#107	—	71	1	no	—
101	C₂×#358	—	71	12	no	—
102	C₇₁	C₂²×#358	1	1	yes	characteristic, complemented, Sylow
103	C₂³×Q₈	—	2	1	no	—
104	C₂²×Q₁₆	—	2	3	no	—
105	C₂²×Q₁₆	—	2	4	no	—
106	C₂²×Q₁₆	Dic₇₁	1	8	yes	complemented
107	C₂²×Q₁₆	C₂×D₇₁	1	6	yes	—
108	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
109	C₂²×C₄⋊C₄	—	142	1	no	—
110	C₂²×C₁₆	—	71	6	no	—
111	C₂²×C₁₆	—	142	4	no	—
112	C₂×#107	—	71	6	no	—
113	C₂×#107	—	142	3	no	—
114	#358	—	71	16	no	—
115	C₂²×Q₈	—	2	8	no	—
116	C₂²×Q₈	—	2	6	no	—
117	C₂²×Q₈	—	4	3	no	—
118	C₂²×Q₈	—	4	4	no	—
119	C₂³×C₄	—	4	1	no	—
120	C₂³×C₄	—	284	1	no	—
121	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂×Q₁₆	—	2	24	no	—
123	C₂×Q₁₆	—	2	12	no	—
124	C₂×Q₁₆	—	2	6	no	—
125	C₂×Q₁₆	C₂×Dic₇₁	1	24	yes	complemented
126	C₂×Q₁₆	C₂²×D₇₁	1	4	yes	—
127	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
128	C₂²×C₈	C₂²⋊D₇₁	1	3	yes	—
129	C₂²×C₈	C₂²⋊D₇₁	1	4	yes	—
130	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
131	C₂×C₄⋊C₄	—	284	3	no	—
132	C₂×C₄⋊C₄	—	142	6	no	—
133	C₂×C₁₆	—	71	4	no	—
134	C₂×C₁₆	—	142	12	no	—
135	#107	—	71	4	no	—
136	#107	—	142	6	no	—
137	C₂⁴	#11765850	1	1	yes	characteristic
138	C₂×Q₈	—	2	4	no	—
139	C₂×Q₈	—	2	24	no	—
140	C₂×Q₈	—	4	24	no	—
141	C₂×Q₈	—	4	6	no	—
142	C₂×Q₈	—	4	12	no	—
143	C₂²×C₄	—	4	8	no	—
144	C₂²×C₄	—	4	6	no	—
145	C₂²×C₄	—	284	6	no	—
146	C₂²×C₄	—	568	4	no	—
147	C₂²×C₄	C₂²⋊Dic₇₁	1	4	yes	—
148	C₂²×C₄	#11765850	1	3	yes	—
149	C₂²×C₄	#11765853	1	4	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
151	Q₁₆	—	2	4	no	—
152	Q₁₆	—	2	24	no	—
153	Q₁₆	C₂²×Dic₇₁	1	8	yes	complemented
154	C₂×C₈	C₂²⋊Dic₇₁	1	12	yes	—
155	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
156	C₂×C₈	C₂×C₂²⋊D₇₁	1	6	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₁	1	3	yes	—
158	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
159	C₄⋊C₄	—	284	6	no	—
160	C₄⋊C₄	—	142	4	no	—
161	C₁₆	—	142	4	no	—
162	C₂³	#14556049	1	4	yes	—
163	C₂³	#14556071	1	4	yes	—
164	C₂³	#14556074	1	4	yes	—
165	C₂³	C₂×#11765850	1	3	yes	—
166	Q₈	—	2	8	no	—
167	Q₈	—	4	24	no	—
168	Q₈	—	4	4	no	—
169	C₂×C₄	—	4	24	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	284	4	no	—
172	C₂×C₄	—	568	12	no	—
173	C₂×C₄	#14556049	1	12	yes	—
174	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	6	yes	—
175	C₂×C₄	C₂×#11765850	1	3	yes	—
176	C₂×C₄	C₂×#11765853	1	6	yes	—
177	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
178	C₈	C₂×C₂²⋊Dic₇₁	1	6	yes	—
179	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
180	C₈	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
181	C₂²	#17985803	1	16	yes	direct product factor
182	C₂²	C₂×#14556049	1	6	yes	—
183	C₂²	C₂×#14556071	1	6	yes	—
184	C₂²	C₂×#14556074	1	6	yes	—
185	C₂²	C₂²×#11765850	1	1	yes	characteristic
186	C₄	—	4	8	no	—
187	C₄	—	568	4	no	—
188	C₄	C₂×#14556049	1	6	yes	—
189	C₄	C₂²×C₂²⋊Dic₇₁	1	1	yes	characteristic
190	C₄	C₂²×#11765853	1	1	yes	characteristic
191	C₂	C₂×#17985803	1	12	yes	direct product factor
192	C₂	C₂²×#14556049	1	1	yes	characteristic
193	C₂	C₂²×#14556071	1	1	yes	characteristic
194	C₂	C₂²×#14556074	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11765850
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂³×Q₁₆×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂³×Q₁₆	D₇₁
hypocenter	C₇₁	C₂²×#358
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#17985803	C₂	direct (N × H)
10. #17985803	C₂²	direct (N × H)
14. C₂²×Q₁₆×C₇₁	C₄	semidirect
33. C₂×Q₁₆×C₇₁	C₂×C₄	semidirect
58. Q₁₆×C₇₁	C₂²×C₄	semidirect
83. C₂²×C₇₁	#358	semidirect
94. C₂×C₇₁	C₂×#358	semidirect
102. C₇₁	C₂²×#358	semidirect
106. C₂²×Q₁₆	Dic₇₁	semidirect
125. C₂×Q₁₆	C₂×Dic₇₁	semidirect
153. Q₁₆	C₂²×Dic₇₁	semidirect
181. C₂²	#17985803	direct (N × H)
191. C₂	C₂×#17985803	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×#17985803	2
C₂³×Q₁₆×C₇₁	2
C₂²×#358	71

Subgroups — C22×#17985803 #32150328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17985803 #32150328