Subgroups — #32155354

Order 18176.

448 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601232	C₂	1	2	yes	maximal
2	#22590619	C₂	1	2	yes	maximal
3	#22590615	C₂	1	1	yes	maximal, characteristic
4	#22590505	C₂	1	4	yes	maximal
5	#22590483	C₂	1	2	yes	maximal
6	C₄×Dic₅₆₈	C₂	1	2	yes	maximal
7	C₄×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#17986727	—	2	2	no	—
10	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
11	#17986404	C₂²	1	2	yes	—
12	#17986401	C₂²	1	4	yes	—
13	#17986307	—	2	4	no	—
14	#17986285	—	2	2	no	—
15	C₄×Dic₂₈₄	C₂²	1	2	yes	—
16	C₄×Dic₂₈₄	C₂²	1	4	yes	—
17	#17985827	C₂²	1	4	yes	—
18	#17985825	C₂²	1	2	yes	—
19	#17985825	C₂²	1	2	yes	—
20	#17985825	C₂²	1	1	yes	characteristic
21	#17985804	C₂²	1	4	yes	—
22	#17985804	C₂²	1	2	yes	—
23	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
26	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
28	C₂×Dic₂₈₄	C₂³	1	2	yes	—
29	#14556128	—	2	2	no	—
30	#14556126	—	4	2	no	—
31	#14556120	—	2	4	no	—
32	#14556118	—	4	4	no	—
33	C₄×Dic₁₄₂	—	2	4	no	—
34	C₄×Dic₁₄₂	—	2	2	no	—
35	#14556093	—	4	2	no	—
36	Dic₅₆₈	Q₈	1	4	yes	complemented
37	#14556062	—	2	2	no	—
38	#14556062	C₂³	1	4	yes	—
39	#14556062	C₂³	1	2	yes	—
40	#14556060	—	2	4	no	—
41	#14556053	—	2	2	no	—
42	#14556053	—	2	4	no	—
43	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
46	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
48	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	C₂³	1	2	yes	—
50	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₄₂	—	2	2	no	—
52	C₂×Dic₁₄₂	—	4	2	no	—
53	Dic₂₈₄	—	2	2	no	—
54	Dic₂₈₄	C₂×Q₈	1	4	yes	—
55	C₄⋊Dic₇₁	—	2	2	no	—
56	C₄⋊Dic₇₁	—	2	4	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	Dic₇₁⋊C₄	—	4	4	no	—
59	Dic₇₁⋊C₄	—	4	2	no	—
60	Dic₇₁⋊C₄	—	4	4	no	—
61	Dic₇₁⋊C₄	—	4	4	no	—
62	C₄×Dic₇₁	—	4	4	no	—
63	C₄×Dic₇₁	—	4	2	no	—
64	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
69	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
70	C₄²×C₇₁	C₂×D₄	1	2	yes	—
71	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₇₁	—	2	1	no	—
73	C₁₆×C₇₁	—	2	2	no	—
74	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
75	Q₈×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₇₁	—	4	2	no	—
82	C₂×Dic₇₁	—	4	2	no	—
83	C₂×Dic₇₁	—	4	4	no	—
84	Dic₁₄₂	—	2	4	no	—
85	Dic₁₄₂	—	4	2	no	—
86	Dic₁₄₂	—	8	4	no	—
87	C₈×C₇₁	—	2	2	no	—
88	C₈×C₇₁	—	2	1	no	—
89	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	—	2	2	no	—
94	C₄×C₇₁	—	2	1	no	—
95	C₄×C₇₁	#508	1	2	yes	—
96	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
97	C₄×C₇₁	#578	1	1	yes	characteristic
98	Dic₇₁	—	4	4	no	—
99	Dic₇₁	—	8	4	no	—
100	Dic₇₁	—	8	2	no	—
101	#33950	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	Q₈×D₈	1	1	yes	characteristic
103	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
104	C₂×C₇₁	#3417	1	1	yes	characteristic
105	#3383	—	71	2	no	—
106	#2253	—	71	2	no	—
107	#2252	—	71	1	no	—
108	#2229	—	71	4	no	—
109	#2225	—	71	2	no	—
110	Q₈×C₁₆	D₇₁	1	1	yes	characteristic
111	C₄×Q₃₂	—	71	1	no	—
112	C₄×Q₃₂	—	71	2	no	—
113	C₇₁	#33950	1	1	yes	characteristic, complemented, Sylow
114	#557	—	142	2	no	—
115	C₂×Q₃₂	—	71	1	no	—
116	#500	—	71	2	no	—
117	#499	—	71	4	no	—
118	#479	—	142	4	no	—
119	#475	—	142	2	no	—
120	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
121	C₄×Q₁₆	—	71	2	no	—
122	C₄×Q₁₆	—	71	4	no	—
123	#367	—	71	4	no	—
124	#366	—	71	2	no	—
125	#366	—	71	1	no	—
126	#366	—	71	2	no	—
127	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
128	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
129	#358	—	71	4	no	—
130	#358	—	71	2	no	—
131	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
132	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
133	C₂×Q₁₆	—	71	2	no	—
134	C₄⋊Q₈	—	284	2	no	—
135	C₄⋊Q₈	—	142	2	no	—
136	#125	—	284	4	no	—
137	#125	—	142	4	no	—
138	C₄×Q₈	—	284	2	no	—
139	C₄×Q₈	—	142	2	no	—
140	C₄×Q₈	—	142	4	no	—
141	C₄×Q₈	D₂₈₄	1	1	yes	characteristic
142	Q₃₂	—	71	4	no	—
143	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
144	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
145	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
146	#107	—	71	4	no	—
147	#107	—	71	2	no	—
148	#107	—	142	2	no	—
149	#106	—	142	4	no	—
150	C₄⋊C₈	D₂₈₄	1	1	yes	characteristic
151	C₄⋊C₈	D₂₈₄	1	2	yes	—
152	#103	—	142	4	no	—
153	#103	—	142	2	no	—
154	C₄×C₈	C₂²×D₇₁	1	2	yes	—
155	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
156	C₂×Q₈	—	284	2	no	—
157	C₂×Q₈	—	142	2	no	—
158	C₂×Q₈	D₅₆₈	1	1	yes	characteristic
159	Q₁₆	—	71	4	no	—
160	Q₁₆	—	142	2	no	—
161	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
162	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
163	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
164	C₄⋊C₄	—	284	4	no	—
165	C₄⋊C₄	—	284	2	no	—
166	C₄⋊C₄	—	284	4	no	—
167	C₄⋊C₄	—	284	4	no	—
168	C₄⋊C₄	—	284	2	no	—
169	C₄⋊C₄	—	142	2	no	—
170	C₄⋊C₄	—	142	4	no	—
171	C₄⋊C₄	D₅₆₈	1	1	yes	characteristic
172	C₄⋊C₄	D₅₆₈	1	2	yes	—
173	C₄²	—	284	4	no	—
174	C₄²	—	284	2	no	—
175	C₄²	C₂×D₂₈₄	1	2	yes	—
176	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
177	C₁₆	—	2	1	no	—
178	C₁₆	—	2	2	no	—
179	C₁₆	Q₈×D₇₁	1	2	yes	—
180	Q₈	—	2	2	no	—
181	Q₈	—	284	2	no	—
182	Q₈	—	568	4	no	—
183	Q₈	—	142	4	no	—
184	C₂×C₄	—	284	2	no	—
185	C₂×C₄	—	284	2	no	—
186	C₂×C₄	—	284	4	no	—
187	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
188	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
189	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
190	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	1	no	—
194	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
195	C₈	#14556175	1	1	yes	characteristic
196	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	284	4	no	—
201	C₄	—	568	2	no	—
202	C₄	—	568	4	no	—
203	C₄	#17986431	1	2	yes	—
204	C₄	Q₈×D₂₈₄	1	1	yes	characteristic
205	C₄	#17986816	1	1	yes	characteristic
206	C₂	Q₈×D₅₆₈	1	1	yes	characteristic
207	C₂	C₂×#17986431	1	1	yes	characteristic
208	C₂	#22601493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	Q₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂²×D₈
hypercenter	Q₈×C₁₆	D₇₁
hypocenter	C₇₁	#33950
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
36. Dic₅₆₈	Q₈	semidirect
113. C₇₁	#33950	semidirect

Maximal subgroups

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

subgroup	index
#22601232	2
#22590619	2
#22590615	2
#22590505	2
#22590483	2
C₄×Dic₅₆₈	2
Q₈×C₁₆×C₇₁	2
#33950	71