Subgroups — #36472469

Order 22784.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151373	C₂	1	1	yes	maximal, characteristic
3	#26020986	C₂	1	2	yes	maximal
4	#26018343	C₂	1	2	yes	maximal
5	C₂×#367×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
9	#19151497	—	2	2	no	—
10	#19151424	—	2	2	no	—
11	#19151373	—	2	2	no	—
12	#19151360	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	#367×C₈₉	C₄	1	4	yes	complemented
16	C₂²×#10623494	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	4	2	no	—
19	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₂×#13083549	Q₈	1	2	yes	—
23	#15667946	—	2	2	no	—
24	#15667946	—	2	2	no	—
25	C₈×Dic₈₉	—	2	1	no	—
26	C₈×Dic₈₉	—	2	1	no	—
27	#15667919	—	4	2	no	—
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
33	#107×C₈₉	—	2	2	no	—
34	#107×C₈₉	C₂×C₄	1	4	yes	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
37	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
38	C₂×#10623494	—	2	1	no	—
39	C₂×#10623494	—	2	1	no	—
40	C₂×#10623494	Q₁₆	1	2	yes	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₄⋊Dic₈₉	—	8	4	no	—
44	C₄×Dic₈₉	—	2	1	no	—
45	C₄×Dic₈₉	—	2	1	no	—
46	#13083549	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	4	2	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	2	4	no	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	4	4	no	—
60	C₂×C₄×C₈₉	—	8	2	no	—
61	C₂×C₄×C₈₉	—	8	2	no	—
62	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
68	C₂×Dic₈₉	—	2	1	no	—
69	C₂×Dic₈₉	—	2	1	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	—	2	2	no	—
72	C₂×Dic₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	8	2	no	—
74	C₂×Dic₈₉	—	8	2	no	—
75	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₈₉	#125	1	1	yes	characteristic
79	#10623494	—	2	2	no	—
80	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₈₉	#466	1	1	yes	characteristic
82	C₂²×C₈₉	#500	1	1	yes	characteristic
83	C₂²×C₈₉	#509	1	2	yes	—
84	C₂²×C₈₉	#510	1	2	yes	—
85	C₄×C₈₉	—	8	4	no	—
86	C₄×C₈₉	#389	1	1	yes	characteristic
87	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₈₉	#467	1	1	yes	characteristic
89	C₄×C₈₉	#499	1	1	yes	characteristic
90	Dic₈₉	—	4	1	no	—
91	Dic₈₉	—	4	1	no	—
92	Dic₈₉	—	16	2	no	—
93	#13167	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	#1942	1	1	yes	characteristic
95	C₂×C₈₉	#2175	1	2	yes	—
96	C₂×C₈₉	#2218	1	1	yes	characteristic
97	C₂×C₈₉	#2219	1	1	yes	characteristic
98	C₂×C₈₉	#2255	1	2	yes	—
99	C₂×#367	D₈₉	1	1	yes	characteristic
100	C₂×#366	—	89	1	no	—
101	C₂×#346	—	89	1	no	—
102	#1942	—	89	2	no	—
103	#1380	—	89	2	no	—
104	C₈₉	#13167	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	89	1	no	—
106	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
107	C₂×#107	—	89	2	no	—
108	C₂×#107	C₂×D₈₉	1	2	yes	—
109	C₂×C₄×C₈	—	89	1	no	—
110	#389	—	178	2	no	—
111	#367	Dic₈₉	1	4	yes	complemented
112	#366	—	178	2	no	—
113	#346	—	178	2	no	—
114	#340	—	178	2	no	—
115	C₂²×C₈	—	89	1	no	—
116	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	356	2	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄²	—	89	1	no	—
121	C₂×C₁₆	—	89	2	no	—
122	C₂×C₁₆	—	178	1	no	—
123	C₂×C₁₆	—	178	1	no	—
124	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
125	C₂×C₁₆	D₃₅₆	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	178	2	no	—
129	#107	—	178	2	no	—
130	#107	C₂×Dic₈₉	1	4	yes	—
131	C₄×C₈	—	178	1	no	—
132	C₄×C₈	—	178	1	no	—
133	#95	—	356	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	89	1	no	—
136	C₂²×C₄	—	89	1	no	—
137	C₂²×C₄	—	356	2	no	—
138	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₈	—	89	2	no	—
140	C₂×C₈	—	178	1	no	—
141	C₂×C₈	—	178	1	no	—
142	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
143	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
146	C₂×C₈	#13083573	1	1	yes	characteristic
147	C₂×C₈	#13083575	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	356	2	no	—
151	C₄⋊C₄	—	356	2	no	—
152	C₄⋊C₄	—	712	4	no	—
153	C₄²	—	178	1	no	—
154	C₄²	—	178	1	no	—
155	C₁₆	—	178	2	no	—
156	C₁₆	C₄⋊Dic₈₉	1	4	yes	complemented
157	C₂³	D₈×D₈₉	1	1	yes	characteristic
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	178	2	no	—
162	C₂×C₄	—	178	2	no	—
163	C₂×C₄	—	178	1	no	—
164	C₂×C₄	—	178	1	no	—
165	C₂×C₄	—	712	2	no	—
166	C₂×C₄	—	712	2	no	—
167	C₂×C₄	—	712	2	no	—
168	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
169	C₂×C₄	#15667987	1	1	yes	characteristic
170	C₂×C₄	#15668011	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
172	C₂×C₄	#15668031	1	1	yes	characteristic
173	C₂×C₄	#15668035	1	1	yes	characteristic
174	C₈	—	178	2	no	—
175	C₈	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
176	C₈	Q₈×Dic₈₉	1	1	yes	characteristic
177	C₈	#15667992	1	1	yes	characteristic
178	C₈	#15668003	1	1	yes	characteristic
179	C₂²	D₈×Dic₈₉	1	1	yes	characteristic
180	C₂²	#19151835	1	1	yes	characteristic
181	C₂²	#19152002	1	1	yes	characteristic
182	C₂²	#19152035	1	2	yes	—
183	C₂²	#19152037	1	2	yes	—
184	C₄	—	8	4	no	—
185	C₄	—	356	1	no	—
186	C₄	—	356	1	no	—
187	C₄	—	1424	2	no	—
188	C₄	#19151497	1	1	yes	characteristic
189	C₄	Q₁₆×Dic₈₉	1	1	yes	characteristic
190	C₄	#19151840	1	1	yes	characteristic
191	C₄	#19151999	1	1	yes	characteristic
192	C₂	#26020986	1	1	yes	characteristic
193	C₂	#26022117	1	2	yes	—
194	C₂	#26022356	1	1	yes	characteristic
195	C₂	#26022361	1	1	yes	characteristic
196	C₂	#26022537	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#367×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#367	D₈₉
hypocenter	C₈₉	#13167
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. #367×C₈₉	C₄	semidirect
57. C₁₆×C₈₉	C₄⋊C₄	semidirect
104. C₈₉	#13167	semidirect
111. #367	Dic₈₉	semidirect
156. C₁₆	C₄⋊Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151424	2
C₂×#19151373	2
#26020986	2
#26018343	2
C₂×#367×C₈₉	2
#13167	89