Subgroups — #36709928

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	#26022539	C₂	1	4	yes	maximal, complemented
3	#26022535	C₂	1	4	yes	maximal, complemented
4	#26022034	C₂	1	4	yes	maximal
5	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#3160	C₂	1	1	yes	maximal, characteristic
7	#19152155	—	2	1	no	—
8	C₂²×#13083549	C₂²	1	2	yes	—
9	#19152003	C₂²	1	2	yes	—
10	#19152000	—	2	1	no	—
11	#19152000	C₂²	1	2	yes	—
12	#19151996	—	2	1	no	—
13	#19151636	—	2	1	no	—
14	#19151636	C₂²	1	2	yes	—
15	#19151425	C₂²	1	8	yes	—
16	#19151423	C₂²	1	8	yes	—
17	#19151374	C₂²	1	2	yes	—
18	#19151374	C₂²	1	2	yes	—
19	#533×C₈₉	—	2	1	no	—
20	#500×C₈₉	C₂²	1	2	yes	—
21	#499×C₈₉	—	2	1	no	—
22	#499×C₈₉	C₂²	1	2	yes	—
23	#498×C₈₉	—	2	1	no	—
24	#427×C₈₉	—	2	1	no	—
25	#427×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	#15668012	—	2	2	no	—
28	#15668004	—	2	2	no	—
29	#15667991	—	4	2	no	—
30	C₄×Dic₁₇₈	—	4	2	no	—
31	#15667971	—	2	2	no	—
32	C₂×#13083549	Q₈	1	8	yes	complemented
33	C₂×#13083549	C₂³	1	4	yes	—
34	#15667946	—	2	2	no	—
35	#15667946	C₂³	1	4	yes	—
36	#15667944	—	2	2	no	—
37	#15667944	—	2	2	no	—
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₈₉	—	2	2	no	—
41	#125×C₈₉	—	2	2	no	—
42	C₂²⋊Q₈×C₈₉	—	4	2	no	—
43	C₄×Q₈×C₈₉	—	4	2	no	—
44	#117×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	#107×C₈₉	—	2	2	no	—
47	#107×C₈₉	C₂³	1	4	yes	—
48	#106×C₈₉	—	2	2	no	—
49	#106×C₈₉	—	2	2	no	—
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₇₈	—	4	2	no	—
55	C₄⋊Dic₈₉	—	2	4	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	#13083553	—	4	4	no	—
58	C₂²⋊Dic₈₉	—	4	2	no	—
59	C₄×Dic₈₉	—	4	2	no	—
60	#13083549	C₂×Q₈	1	8	yes	—
61	C₂×Q₈×C₈₉	—	4	2	no	—
62	C₂²×C₄×C₈₉	D₈	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₄⋊C₄×C₈₉	—	4	4	no	—
72	C₄⋊C₄×C₈₉	—	4	2	no	—
73	C₂²⋊C₄×C₈₉	—	4	2	no	—
74	C₄²×C₈₉	—	4	2	no	—
75	C₄²×C₈₉	D₈	1	2	yes	—
76	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₈₉	—	8	4	no	—
80	C₂×C₄×C₈₉	—	4	2	no	—
81	C₂×C₄×C₈₉	—	4	4	no	—
82	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₈₉	—	4	4	no	—
89	Dic₁₇₈	—	8	4	no	—
90	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
91	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
92	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
93	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
94	C₂²×C₈₉	—	2	2	no	—
95	C₂²×C₈₉	#500	1	2	yes	—
96	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	—	8	4	no	—
100	C₄×C₈₉	#500	1	2	yes	—
101	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₈₉	—	8	4	no	—
104	#33728	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	—	2	2	no	—
106	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
107	C₂×C₈₉	#3414	1	2	yes	—
108	#3160	—	89	1	no	—
109	#3160	D₈₉	1	1	yes	characteristic
110	#2255	—	89	4	no	—
111	#2255	—	89	4	no	—
112	#2158	—	89	4	no	—
113	C₂×#346	—	89	1	no	—
114	C₈₉	#33728	1	1	yes	characteristic, complemented, Sylow
115	#533	—	2	1	no	—
116	#533	—	178	1	no	—
117	C₂²×C₁₆	—	89	2	no	—
118	#500	—	89	2	no	—
119	#500	C₂×D₈₉	1	2	yes	—
120	#499	—	2	1	no	—
121	#499	—	89	2	no	—
122	#499	—	178	1	no	—
123	#499	C₂×D₈₉	1	2	yes	—
124	#498	—	2	1	no	—
125	#498	—	178	1	no	—
126	#427	—	2	1	no	—
127	#427	—	89	2	no	—
128	#427	—	178	1	no	—
129	#427	C₂×D₈₉	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
131	#367	—	89	8	no	—
132	#366	—	89	8	no	—
133	#346	—	89	2	no	—
134	#346	—	89	2	no	—
135	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	178	2	no	—
139	#125	—	2	2	no	—
140	#125	—	178	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	356	2	no	—
143	C₄×Q₈	—	4	2	no	—
144	C₄×Q₈	—	356	2	no	—
145	#117	—	2	2	no	—
146	#117	—	178	2	no	—
147	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
148	C₂×C₁₆	—	89	8	no	—
149	C₂×C₁₆	—	89	4	no	—
150	#107	—	2	2	no	—
151	#107	—	89	4	no	—
152	#107	—	178	2	no	—
153	#107	C₂²×D₈₉	1	4	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	—	178	2	no	—
157	#106	—	178	2	no	—
158	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	356	2	no	—
164	C₂²×C₄	#13083559	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
166	C₂×C₈	#13083564	1	4	yes	—
167	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₈₉	1	4	yes	—
171	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	356	4	no	—
176	C₄⋊C₄	—	356	2	no	—
177	C₄⋊C₄	—	178	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	356	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	356	2	no	—
182	C₄²	#13083559	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
185	C₁₆	—	89	8	no	—
186	C₂³	C₂×#13083559	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	712	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	356	4	no	—
192	C₂×C₄	#15668010	1	2	yes	—
193	C₂×C₄	C₂×#13083559	1	2	yes	—
194	C₂×C₄	C₂×#13083559	1	2	yes	—
195	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
196	C₂×C₄	C₂×#13083564	1	2	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
198	C₈	#15668010	1	2	yes	—
199	C₈	C₂×#13083564	1	2	yes	—
200	C₈	C₂×#13083564	1	2	yes	—
201	C₈	C₂×Q₈×D₈₉	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	#19152001	1	2	yes	—
204	C₂²	C₂²×#13083559	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	8	4	no	—
208	C₄	—	712	4	no	—
209	C₄	#19152001	1	2	yes	—
210	C₄	C₂×#15668010	1	1	yes	characteristic
211	C₄	C₂²×#13083564	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₂×#19152001	1	1	yes	characteristic
214	C₂	#26033386	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3160	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#3160	D₈₉
hypocenter	C₈₉	#33728
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #26022539	C₂	semidirect
3. #26022535	C₂	semidirect
32. C₂×#13083549	Q₈	semidirect
114. C₈₉	#33728	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26022539	2
#26022535	2
#26022034	2
C₂×#19151374	2
C₈₉×#3160	2
#33728	89