Subgroups — #36573380

Order 22784.

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⋊Dic₈₉	C₂	1	2	yes	maximal
2	#26021011	C₂	1	6	yes	maximal
3	#26020380	C₂	1	3	yes	maximal
4	C₄×#15667946	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₈₉	—	2	2	no	—
7	C₂×#15667946	C₂²	1	6	yes	—
8	#19151497	—	2	6	no	—
9	#19151468	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₈₉	C₂²	1	6	yes	—
11	#19151361	C₂²	1	3	yes	—
12	#19151361	C₂²	1	6	yes	—
13	#19151361	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₈₉	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
18	Q₈×Dic₈₉	—	4	8	no	—
19	C₂×C₄⋊Dic₈₉	—	2	6	no	—
20	C₂×C₄⋊Dic₈₉	—	2	6	no	—
21	C₂×C₄⋊Dic₈₉	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
23	C₂×C₄×Dic₈₉	—	2	6	no	—
24	#15667946	—	2	12	no	—
25	#15667919	—	2	6	no	—
26	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
29	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₈₉	D₄	1	3	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
33	C₄⋊C₈×C₈₉	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₈₉	C₂×C₄	1	12	yes	—
35	C₂²×Dic₈₉	—	2	2	no	—
36	C₂²×Dic₈₉	—	2	6	no	—
37	C₄⋊Dic₈₉	—	2	2	no	—
38	C₄⋊Dic₈₉	—	2	6	no	—
39	C₄⋊Dic₈₉	—	2	6	no	—
40	C₄⋊Dic₈₉	—	4	12	no	—
41	C₄⋊Dic₈₉	—	8	6	no	—
42	C₄⋊Dic₈₉	#41	1	4	yes	—
43	C₄×Dic₈₉	—	4	12	no	—
44	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
45	C₂×Q₈×C₈₉	D₈	1	4	yes	—
46	C₂×Q₈×C₈₉	Q₁₆	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₈₉	—	2	6	no	—
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
54	C₄⋊C₄×C₈₉	C₄⋊C₄	1	12	yes	—
55	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
56	C₄²×C₈₉	C₂×Q₈	1	6	yes	—
57	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₈₉	#107	1	16	yes	complemented
59	C₂×C₄×C₈₉	—	2	6	no	—
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
63	C₂×C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
67	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
68	C₂×Dic₈₉	—	2	4	no	—
69	C₂×Dic₈₉	—	4	2	no	—
70	C₂×Dic₈₉	—	4	2	no	—
71	C₂×Dic₈₉	—	4	6	no	—
72	C₂×Dic₈₉	—	4	6	no	—
73	C₂×Dic₈₉	—	4	6	no	—
74	C₈×C₈₉	—	2	6	no	—
75	C₈×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈₉	#548	1	2	yes	—
78	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
79	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
80	C₂²×C₈₉	#577	1	2	yes	—
81	C₄×C₈₉	—	2	6	no	—
82	C₄×C₈₉	C₂×#107	1	12	yes	—
83	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
84	C₄×C₈₉	#519	1	1	yes	characteristic
85	C₄×C₈₉	#557	1	2	yes	—
86	Dic₈₉	—	4	4	no	—
87	Dic₈₉	—	8	6	no	—
88	#22740	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	#2351	1	1	yes	characteristic
90	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
91	C₂×C₈₉	#2912	1	1	yes	characteristic
92	C₂×C₈₉	#3358	1	2	yes	—
93	C₂×C₈₉	#3363	1	2	yes	—
94	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
95	Q₈×C₄⋊C₄	—	89	2	no	—
96	#1947	—	89	6	no	—
97	#1834	—	89	3	no	—
98	C₄×#107	—	89	3	no	—
99	C₈₉	#22740	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×Q₈	—	178	2	no	—
101	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
102	C₈×Q₈	Dic₈₉	1	8	yes	complemented
103	C₂×#107	—	89	6	no	—
104	C₂×C₄⋊C₈	C₂×D₈₉	1	3	yes	—
105	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
106	#389	—	178	6	no	—
107	#384	—	89	6	no	—
108	C₄×C₄⋊C₄	—	89	6	no	—
109	#340	—	89	3	no	—
110	#340	—	89	6	no	—
111	#340	—	89	1	no	—
112	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
115	C₄×Q₈	—	356	8	no	—
116	C₄×Q₈	Dic₁₇₈	1	4	yes	—
117	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
118	C₂×C₄⋊C₄	—	89	2	no	—
119	C₂×C₄⋊C₄	—	89	6	no	—
120	C₂×C₄⋊C₄	—	178	6	no	—
121	C₂×C₄⋊C₄	—	178	6	no	—
122	C₂×C₄⋊C₄	D₃₅₆	1	3	yes	—
123	C₂×C₄²	—	178	6	no	—
124	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
125	#107	—	178	12	no	—
126	C₄⋊C₈	C₂×Dic₈₉	1	12	yes	—
127	C₄×C₈	C₂×Dic₈₉	1	12	yes	—
128	#95	—	178	6	no	—
129	C₂×Q₈	C₄⋊Dic₈₉	1	4	yes	—
130	C₂×Q₈	D₇₁₂	1	4	yes	—
131	C₂×Q₈	Dic₃₅₆	1	4	yes	—
132	C₂²×C₄	—	178	6	no	—
133	C₂²×C₄	—	178	2	no	—
134	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
135	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
136	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
140	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
141	C₄⋊C₄	—	89	4	no	—
142	C₄⋊C₄	—	356	12	no	—
143	C₄⋊C₄	—	178	6	no	—
144	C₄⋊C₄	—	178	6	no	—
145	C₄⋊C₄	—	178	2	no	—
146	C₄⋊C₄	—	712	6	no	—
147	C₄⋊C₄	C₄⋊Dic₈₉	1	12	yes	—
148	C₄²	—	356	12	no	—
149	C₄²	C₂²×Dic₈₉	1	6	yes	—
150	C₄²	C₂×Dic₁₇₈	1	6	yes	—
151	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
152	Q₈	#15667946	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	356	2	no	—
155	C₂×C₄	—	356	2	no	—
156	C₂×C₄	—	356	6	no	—
157	C₂×C₄	—	356	6	no	—
158	C₂×C₄	—	356	6	no	—
159	C₂×C₄	—	178	4	no	—
160	C₂×C₄	C₂×C₄⋊Dic₈₉	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₈₉	1	6	yes	—
162	C₂×C₄	C₂×D₇₁₂	1	6	yes	—
163	C₂×C₄	C₂×Dic₃₅₆	1	6	yes	—
164	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
166	C₂×C₄	#41×D₈₉	1	2	yes	—
167	C₂×C₄	#15668060	1	2	yes	—
168	C₈	—	2	6	no	—
169	C₈	—	2	2	no	—
170	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
171	C₂²	#19152260	1	2	yes	—
172	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
173	C₂²	C₂²×Dic₃₅₆	1	1	yes	characteristic
174	C₂²	#19152410	1	2	yes	—
175	C₄	—	2	6	no	—
176	C₄	—	356	4	no	—
177	C₄	—	712	6	no	—
178	C₄	C₂×#15667946	1	12	yes	—
179	C₄	#41×Dic₈₉	1	1	yes	characteristic
180	C₄	#19152076	1	1	yes	characteristic
181	C₄	#19152324	1	2	yes	—
182	C₂	#26023173	1	1	yes	characteristic
183	C₂	C₂²×#15667946	1	1	yes	characteristic
184	C₂	#26028847	1	1	yes	characteristic
185	C₂	#26032888	1	2	yes	—
186	C₂	#26032931	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#22740
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
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×C₈₉	C₄	semidirect
58. Q₈×C₈₉	#107	semidirect
99. C₈₉	#22740	semidirect
102. C₈×Q₈	Dic₈₉	semidirect
152. Q₈	#15667946	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₈₉	2
#26021011	2
#26020380	2
C₄×#15667946	2
C₂×C₈×Q₈×C₈₉	2
#22740	89