Subgroups — #36470867

Order 22784.

342 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	C₂×#19151425	C₂	1	2	yes	maximal
2	C₂×#19151423	C₂	1	2	yes	maximal
3	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
4	#26020416	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂²×#13083549	C₂²	1	1	yes	characteristic
8	C₂×#15667946	C₂²	1	2	yes	—
9	C₂×#15667944	—	2	1	no	—
10	#19151464	—	2	1	no	—
11	#19151425	—	2	4	no	—
12	#19151423	—	2	4	no	—
13	C₄×#13083549	C₄	1	4	yes	complemented
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	C₂×#106×C₈₉	—	2	1	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#382×C₈₉	—	2	1	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	C₂×#13083549	C₂×C₄	1	2	yes	—
21	C₂×#13083549	D₄	1	2	yes	—
22	C₂×#13083549	D₄	1	2	yes	—
23	C₂×#13083549	Q₈	1	2	yes	—
24	C₂×#13083549	Q₈	1	2	yes	—
25	#15667946	—	2	2	no	—
26	#15667946	—	2	2	no	—
27	#15667944	—	4	2	no	—
28	#15667919	—	4	2	no	—
29	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	#107×C₈₉	—	2	2	no	—
34	#107×C₈₉	—	2	2	no	—
35	#106×C₈₉	—	4	2	no	—
36	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₈₉	Q₈	1	2	yes	—
38	#95×C₈₉	—	4	2	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	#13083549	C₄⋊C₄	1	4	yes	—
43	#13083549	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₈₉	—	4	2	no	—
45	C₂²×C₄×C₈₉	D₈	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₈₉	D₈	1	2	yes	—
49	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₄²×C₈₉	Q₁₆	1	2	yes	—
60	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	8	2	no	—
63	C₂×C₄×C₈₉	—	8	4	no	—
64	C₂×C₄×C₈₉	#107	1	4	yes	—
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₈₉	—	8	2	no	—
72	C₂×Dic₈₉	—	8	4	no	—
73	C₈×C₈₉	#107	1	4	yes	—
74	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₈₉	#493	1	1	yes	characteristic
80	C₂²×C₈₉	#500	1	1	yes	characteristic
81	C₂²×C₈₉	#508	1	4	yes	—
82	C₄×C₈₉	—	2	4	no	—
83	C₄×C₈₉	—	16	2	no	—
84	C₄×C₈₉	#384	1	1	yes	characteristic
85	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
86	C₄×C₈₉	#494	1	1	yes	characteristic
87	C₄×C₈₉	#500	1	1	yes	characteristic
88	Dic₈₉	—	16	2	no	—
89	#12907	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	#1845	1	1	yes	characteristic
91	C₂×C₈₉	#2158	1	2	yes	—
92	C₂×C₈₉	#2249	1	2	yes	—
93	C₂×C₈₉	#2253	1	2	yes	—
94	C₂×#367	—	89	2	no	—
95	C₂×#366	—	89	2	no	—
96	C₂×C₄×C₁₆	—	89	1	no	—
97	#1846	—	89	1	no	—
98	#1846	D₈₉	1	1	yes	characteristic
99	C₈₉	#12907	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	89	1	no	—
101	C₂²×C₁₆	—	89	1	no	—
102	C₂×#107	—	89	2	no	—
103	C₂×#107	C₂×D₈₉	1	2	yes	—
104	C₂×#106	—	2	1	no	—
105	C₂×#106	—	178	1	no	—
106	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
107	#382	—	2	1	no	—
108	#382	—	178	1	no	—
109	#367	—	178	4	no	—
110	#366	—	178	4	no	—
111	C₄×C₁₆	—	89	4	no	—
112	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	178	2	no	—
116	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
117	C₂×C₁₆	—	89	2	no	—
118	C₂×C₁₆	—	89	2	no	—
119	C₂×C₁₆	—	89	2	no	—
120	C₂×C₁₆	—	89	2	no	—
121	C₂×C₁₆	—	89	2	no	—
122	C₂×C₁₆	—	89	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	178	2	no	—
126	#107	—	178	2	no	—
127	#106	—	4	2	no	—
128	#106	—	356	2	no	—
129	C₄×C₈	Dic₁₇₈	1	2	yes	—
130	C₄×C₈	C₄×D₈₉	1	2	yes	—
131	#95	—	4	2	no	—
132	#95	—	356	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	356	2	no	—
135	C₂²×C₄	#13083559	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂×C₈	#13083553	1	2	yes	—
138	C₂×C₈	#13083559	1	2	yes	—
139	C₂×C₈	#13083564	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
142	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	356	2	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄²	#13083553	1	2	yes	—
151	C₄²	#13083564	1	2	yes	—
152	C₁₆	—	89	4	no	—
153	C₁₆	—	89	4	no	—
154	C₂³	C₂×#13083559	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	712	2	no	—
159	C₂×C₄	—	712	4	no	—
160	C₂×C₄	#15667945	1	4	yes	—
161	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
162	C₂×C₄	#15668008	1	1	yes	characteristic
163	C₂×C₄	#15668010	1	1	yes	characteristic
164	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
165	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
166	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
167	C₈	#15667945	1	4	yes	—
168	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
169	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
170	C₈	#15668011	1	1	yes	characteristic
171	C₈	#15668011	1	1	yes	characteristic
172	C₂²	C₂×#15667945	1	1	yes	characteristic
173	C₂²	#19151974	1	1	yes	characteristic
174	C₂²	#19152001	1	1	yes	characteristic
175	C₂²	#19152028	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	1424	2	no	—
179	C₄	#19151467	1	1	yes	characteristic
180	C₄	C₂×#15667945	1	1	yes	characteristic
181	C₄	#19151977	1	1	yes	characteristic
182	C₄	#19152001	1	1	yes	characteristic
183	C₂	#26020411	1	1	yes	characteristic
184	C₂	#26022034	1	2	yes	—
185	C₂	#26022511	1	2	yes	—
186	C₂	#26022528	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1846	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1846	D₈₉
hypocenter	C₈₉	#12907
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
13. C₄×#13083549	C₄	semidirect
99. C₈₉	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151425	2
C₂×#19151423	2
C₂×C₄×#13083549	2
#26020416	2
C₈₉×#1846	2
#12907	89