Subgroups — #32016404

Order 18176.

822 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₇₁	C₂	1	2	yes	maximal
2	#22589099	C₂	1	6	yes	maximal
3	#22588468	C₂	1	3	yes	maximal
4	C₄×#14556062	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₇₁	—	2	2	no	—
7	C₂×#14556062	C₂²	1	6	yes	—
8	#17985898	—	2	6	no	—
9	#17985869	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₇₁	C₂²	1	6	yes	—
11	#17985762	C₂²	1	3	yes	—
12	#17985762	C₂²	1	6	yes	—
13	#17985762	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₇₁	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
18	Q₈×Dic₇₁	—	4	8	no	—
19	C₂×C₄⋊Dic₇₁	—	2	6	no	—
20	C₂×C₄⋊Dic₇₁	—	2	6	no	—
21	C₂×C₄⋊Dic₇₁	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₇₁	C₂³	1	2	yes	—
23	C₂×C₄×Dic₇₁	—	2	6	no	—
24	#14556062	—	2	12	no	—
25	C₇₁⋊#95	—	2	6	no	—
26	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
30	C₄×Q₈×C₇₁	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
33	C₄⋊C₈×C₇₁	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	12	yes	—
35	C₂²×Dic₇₁	—	2	2	no	—
36	C₂²×Dic₇₁	—	2	6	no	—
37	C₄⋊Dic₇₁	—	2	2	no	—
38	C₄⋊Dic₇₁	—	2	6	no	—
39	C₄⋊Dic₇₁	—	2	6	no	—
40	C₄⋊Dic₇₁	—	4	12	no	—
41	C₄⋊Dic₇₁	—	8	6	no	—
42	C₄⋊Dic₇₁	#41	1	4	yes	—
43	C₄×Dic₇₁	—	4	12	no	—
44	C₂×Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
45	C₂×Q₈×C₇₁	D₈	1	4	yes	—
46	C₂×Q₈×C₇₁	Q₁₆	1	4	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₇₁	—	2	6	no	—
51	C₂×C₈×C₇₁	—	2	2	no	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
54	C₄⋊C₄×C₇₁	C₄⋊C₄	1	12	yes	—
55	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
56	C₄²×C₇₁	C₂×Q₈	1	6	yes	—
57	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₇₁	#107	1	16	yes	complemented
59	C₂×C₄×C₇₁	—	2	6	no	—
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₇₁	C₂×D₈	1	6	yes	—
63	C₂×C₄×C₇₁	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
67	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
68	C₂×Dic₇₁	—	2	4	no	—
69	C₂×Dic₇₁	—	4	2	no	—
70	C₂×Dic₇₁	—	4	2	no	—
71	C₂×Dic₇₁	—	4	6	no	—
72	C₂×Dic₇₁	—	4	6	no	—
73	C₂×Dic₇₁	—	4	6	no	—
74	C₈×C₇₁	—	2	6	no	—
75	C₈×C₇₁	—	2	2	no	—
76	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₇₁	#548	1	2	yes	—
78	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
79	C₂²×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
80	C₂²×C₇₁	#577	1	2	yes	—
81	C₄×C₇₁	—	2	6	no	—
82	C₄×C₇₁	C₂×#107	1	12	yes	—
83	C₄×C₇₁	C₄×#41	1	1	yes	characteristic
84	C₄×C₇₁	#519	1	1	yes	characteristic
85	C₄×C₇₁	#557	1	2	yes	—
86	Dic₇₁	—	4	4	no	—
87	Dic₇₁	—	8	6	no	—
88	#22740	—	71	1	no	maximal, Sylow
89	C₂×C₇₁	#2351	1	1	yes	characteristic
90	C₂×C₇₁	C₂²×#107	1	1	yes	characteristic
91	C₂×C₇₁	#2912	1	1	yes	characteristic
92	C₂×C₇₁	#3358	1	2	yes	—
93	C₂×C₇₁	#3363	1	2	yes	—
94	C₂×C₈×Q₈	D₇₁	1	1	yes	characteristic
95	Q₈×C₄⋊C₄	—	71	2	no	—
96	#1947	—	71	6	no	—
97	#1834	—	71	3	no	—
98	C₄×#107	—	71	3	no	—
99	C₇₁	#22740	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×Q₈	—	142	2	no	—
101	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
102	C₈×Q₈	Dic₇₁	1	8	yes	complemented
103	C₂×#107	—	71	6	no	—
104	C₂×C₄⋊C₈	C₂×D₇₁	1	3	yes	—
105	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
106	#389	—	142	6	no	—
107	#384	—	71	6	no	—
108	C₄×C₄⋊C₄	—	71	6	no	—
109	#340	—	71	3	no	—
110	#340	—	71	6	no	—
111	#340	—	71	1	no	—
112	C₂²×Q₈	D₂₈₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
115	C₄×Q₈	—	284	8	no	—
116	C₄×Q₈	Dic₁₄₂	1	4	yes	—
117	C₄×Q₈	C₂×Dic₇₁	1	4	yes	—
118	C₂×C₄⋊C₄	—	71	2	no	—
119	C₂×C₄⋊C₄	—	71	6	no	—
120	C₂×C₄⋊C₄	—	142	6	no	—
121	C₂×C₄⋊C₄	—	142	6	no	—
122	C₂×C₄⋊C₄	D₂₈₄	1	3	yes	—
123	C₂×C₄²	—	142	6	no	—
124	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
125	#107	—	142	12	no	—
126	C₄⋊C₈	C₂×Dic₇₁	1	12	yes	—
127	C₄×C₈	C₂×Dic₇₁	1	12	yes	—
128	#95	—	142	6	no	—
129	C₂×Q₈	C₄⋊Dic₇₁	1	4	yes	—
130	C₂×Q₈	D₅₆₈	1	4	yes	—
131	C₂×Q₈	Dic₂₈₄	1	4	yes	—
132	C₂²×C₄	—	142	6	no	—
133	C₂²×C₄	—	142	2	no	—
134	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
135	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
136	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
140	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
141	C₄⋊C₄	—	284	12	no	—
142	C₄⋊C₄	—	568	6	no	—
143	C₄⋊C₄	—	71	4	no	—
144	C₄⋊C₄	—	142	6	no	—
145	C₄⋊C₄	—	142	6	no	—
146	C₄⋊C₄	—	142	2	no	—
147	C₄⋊C₄	C₄⋊Dic₇₁	1	12	yes	—
148	C₄²	—	284	12	no	—
149	C₄²	C₂²×Dic₇₁	1	6	yes	—
150	C₄²	C₂×Dic₁₄₂	1	6	yes	—
151	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
152	Q₈	#14556062	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	284	2	no	—
155	C₂×C₄	—	284	2	no	—
156	C₂×C₄	—	284	6	no	—
157	C₂×C₄	—	284	6	no	—
158	C₂×C₄	—	284	6	no	—
159	C₂×C₄	—	142	4	no	—
160	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
162	C₂×C₄	C₂×D₅₆₈	1	6	yes	—
163	C₂×C₄	C₂×Dic₂₈₄	1	6	yes	—
164	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
166	C₂×C₄	#41×D₇₁	1	2	yes	—
167	C₂×C₄	#14556176	1	2	yes	—
168	C₈	—	2	6	no	—
169	C₈	—	2	2	no	—
170	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
171	C₂²	#17986661	1	2	yes	—
172	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
173	C₂²	C₂²×Dic₂₈₄	1	1	yes	characteristic
174	C₂²	#17986811	1	2	yes	—
175	C₄	—	2	6	no	—
176	C₄	—	284	4	no	—
177	C₄	—	568	6	no	—
178	C₄	C₂×#14556062	1	12	yes	—
179	C₄	#41×Dic₇₁	1	1	yes	characteristic
180	C₄	#17986477	1	1	yes	characteristic
181	C₄	#17986725	1	2	yes	—
182	C₂	#22591261	1	1	yes	characteristic
183	C₂	C₂²×#14556062	1	1	yes	characteristic
184	C₂	#22596935	1	1	yes	characteristic
185	C₂	#22600976	1	2	yes	—
186	C₂	#22601019	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×C₈×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×D₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₁
hypocenter	C₇₁	#22740
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₇₁	C₄	semidirect
58. Q₈×C₇₁	#107	semidirect
99. C₇₁	#22740	semidirect
102. C₈×Q₈	Dic₇₁	semidirect
152. Q₈	#14556062	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₇₁	2
#22589099	2
#22588468	2
C₄×#14556062	2
C₂×C₈×Q₈×C₇₁	2
#22740	71