Subgroups — #36448096

Order 22784.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020369	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
3	#26018086	C₂	1	4	yes	maximal
4	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×#15667923	C₄	1	2	yes	—
7	C₂×#15667923	C₄	1	1	yes	characteristic
8	C₂×#15667923	C₄	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	#19151353	—	2	2	no	—
11	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	C₄³×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×#10623494	—	2	2	no	—
15	C₂²×#10623494	C₂×C₄	1	2	yes	—
16	C₂²×#10623494	C₂×C₄	1	2	yes	—
17	C₄⋊#10623494	—	2	4	no	—
18	#15667923	—	2	4	no	—
19	#15667923	—	2	2	no	—
20	#15667923	C₈	1	4	yes	complemented
21	C₄×#10623494	—	2	2	no	—
22	C₄×#10623494	—	2	2	no	—
23	C₂²×C₈×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	2	yes	—
27	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	—	2	4	no	—
30	C₄⋊C₈×C₈₉	—	4	4	no	—
31	C₂×#10623494	—	2	2	no	—
32	C₂×#10623494	—	2	2	no	—
33	C₂×#10623494	—	2	2	no	—
34	C₂×#10623494	—	2	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	C₂×C₈	1	4	yes	—
39	C₂×#10623494	M₁₆	1	4	yes	—
40	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₈₉	—	4	2	no	—
46	C₂×C₈×C₈₉	—	4	4	no	—
47	C₄²×C₈₉	—	2	2	no	—
48	C₄²×C₈₉	—	2	2	no	—
49	C₄²×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	—	4	2	no	—
51	C₄²×C₈₉	C₂×C₈	1	2	yes	—
52	C₄²×C₈₉	M₁₆	1	2	yes	—
53	C₄²×C₈₉	#41	1	2	yes	—
54	C₄²×C₈₉	#41	1	2	yes	—
55	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₈₉	#117	1	2	yes	—
64	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#125	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₈₉	#137	1	2	yes	—
72	C₈×C₈₉	—	8	2	no	—
73	#10623494	—	2	4	no	—
74	#10623494	—	2	4	no	—
75	#10623494	—	8	2	no	—
76	C₂²×C₈₉	#382	1	1	yes	characteristic
77	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₈₉	#423	1	1	yes	characteristic
79	C₂²×C₈₉	#428	1	2	yes	—
80	C₂²×C₈₉	#430	1	2	yes	—
81	C₄×C₈₉	—	2	4	no	—
82	C₄×C₈₉	—	2	2	no	—
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	#431	1	2	yes	—
85	C₄×C₈₉	#432	1	2	yes	—
86	C₄×C₈₉	#489	1	2	yes	—
87	C₄×C₈₉	#490	1	1	yes	characteristic
88	C₄×C₈₉	#491	1	1	yes	characteristic
89	#8786	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	#1565	1	2	yes	—
91	C₂×C₈₉	#1569	1	2	yes	—
92	C₂×C₈₉	#1831	1	1	yes	characteristic
93	C₂×C₈₉	#1836	1	1	yes	characteristic
94	C₂×C₈₉	#1838	1	1	yes	characteristic
95	#1842	D₈₉	1	1	yes	characteristic
96	#1831	—	89	1	no	—
97	C₄×#97	—	89	1	no	—
98	#1277	—	89	4	no	—
99	C₈₉	#8786	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	—	89	2	no	—
102	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
103	C₂×#97	—	89	1	no	—
104	C₂×#97	—	89	2	no	—
105	C₂×#97	—	89	1	no	—
106	C₂×C₄×C₈	—	89	2	no	—
107	C₄³	C₂×D₈₉	1	1	yes	characteristic
108	#336	—	178	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	89	2	no	—
111	C₂²×C₈	—	89	2	no	—
112	C₂²×C₈	—	178	2	no	—
113	C₂×C₄²	Dic₁₇₈	1	2	yes	—
114	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
117	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	4	4	no	—
120	C₄⋊C₈	—	178	4	no	—
121	#97	—	89	4	no	—
122	#97	—	178	4	no	—
123	#97	—	178	2	no	—
124	C₄×C₈	—	178	2	no	—
125	C₄×C₈	—	178	2	no	—
126	C₂²×C₄	#13083553	1	2	yes	—
127	C₂²×C₄	#13083553	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	89	4	no	—
134	C₂×C₈	—	89	4	no	—
135	C₂×C₈	—	356	2	no	—
136	C₂×C₈	—	356	2	no	—
137	C₂×C₈	—	356	2	no	—
138	C₂×C₈	—	178	2	no	—
139	C₂×C₈	—	178	2	no	—
140	C₂×C₈	—	178	2	no	—
141	C₂×C₈	—	178	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₈×D₈₉	1	2	yes	—
147	C₄²	#13083557	1	2	yes	—
148	C₄²	#13083573	1	2	yes	—
149	C₄²	#13083575	1	2	yes	—
150	C₂³	C₂×#13083553	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#15667941	1	4	yes	—
158	C₂×C₄	#15667970	1	2	yes	—
159	C₂×C₄	#15667999	1	1	yes	characteristic
160	C₂×C₄	#15668001	1	1	yes	characteristic
161	C₂×C₄	#15668003	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
164	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
165	C₂×C₄	#15668038	1	2	yes	—
166	C₂×C₄	#15668043	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	178	4	no	—
169	C₈	—	178	4	no	—
170	C₈	—	712	2	no	—
171	C₂²	#19151463	1	1	yes	characteristic
172	C₂²	C₂×#15667941	1	1	yes	characteristic
173	C₂²	#19151621	1	1	yes	characteristic
174	C₂²	#19151638	1	2	yes	—
175	C₂²	#19151646	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	#19151650	1	2	yes	—
180	C₄	#19151654	1	2	yes	—
181	C₄	#19151960	1	2	yes	—
182	C₄	#19151962	1	1	yes	characteristic
183	C₄	#19151966	1	1	yes	characteristic
184	C₂	#26019002	1	2	yes	—
185	C₂	#26019017	1	2	yes	—
186	C₂	#26020368	1	1	yes	characteristic
187	C₂	#26020387	1	1	yes	characteristic
188	C₂	#26020392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#8786
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
20. #15667923	C₈	semidirect
99. C₈₉	#8786	semidirect

Maximal subgroups

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

subgroup	index
#26020369	2
C₄×#15667923	2
#26018086	2
C₈₉×#1842	2
#8786	89