Subgroups — #36473048

Order 22784.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 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	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
3	#26018359	C₂	1	4	yes	maximal
4	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
5	#19152155	—	2	1	no	—
6	C₂×#15667952	C₂²	1	2	yes	—
7	#19152003	C₄	1	2	yes	complemented
8	#19152000	—	2	1	no	—
9	#19152000	C₄	1	2	yes	complemented
10	#19151996	—	2	1	no	—
11	#19151636	—	2	1	no	—
12	#19151636	C₂²	1	2	yes	—
13	#19151374	—	2	1	no	—
14	C₄×#13083549	—	2	1	no	—
15	C₂×#106×C₈₉	—	2	1	no	—
16	C₂×#106×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#384×C₈₉	—	2	1	no	—
19	#15668012	—	2	2	no	—
20	#15668004	—	2	2	no	—
21	#15667991	—	4	2	no	—
22	C₄×Dic₁₇₈	—	4	2	no	—
23	#15667971	—	2	2	no	—
24	#15667952	—	2	2	no	—
25	#15667952	Q₈	1	4	yes	complemented
26	C₂×#13083549	—	2	2	no	—
27	#15667946	—	2	2	no	—
28	#15667946	C₂×C₄	1	4	yes	—
29	#15667944	—	2	2	no	—
30	#15667944	—	2	2	no	—
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	#106×C₈₉	—	2	4	no	—
37	#106×C₈₉	—	4	2	no	—
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₄⋊Dic₈₉	—	2	4	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	#13083553	—	4	4	no	—
46	C₂²⋊Dic₈₉	—	4	2	no	—
47	C₄×Dic₈₉	—	4	2	no	—
48	#13083549	—	2	4	no	—
49	C₂²×C₄×C₈₉	—	4	2	no	—
50	C₂²×C₄×C₈₉	D₈	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
55	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
56	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₈₉	#41	1	2	yes	—
60	C₄⋊C₄×C₈₉	—	4	2	no	—
61	C₄⋊C₄×C₈₉	—	4	2	no	—
62	C₄⋊C₄×C₈₉	—	8	4	no	—
63	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₈₉	SD₁₆	1	2	yes	—
66	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	4	no	—
68	C₂×C₄×C₈₉	—	8	4	no	—
69	C₂×C₄×C₈₉	—	8	2	no	—
70	C₂×C₄×C₈₉	#102	1	2	yes	—
71	C₂×C₄×C₈₉	#102	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₈₉	—	4	4	no	—
79	Dic₁₇₈	—	8	4	no	—
80	C₈×C₈₉	#103	1	2	yes	—
81	C₈×C₈₉	#103	1	2	yes	—
82	C₈×C₈₉	C₄×Q₈	1	2	yes	—
83	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
86	C₂²×C₈₉	#486	1	1	yes	characteristic
87	C₂²×C₈₉	#500	1	1	yes	characteristic
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	—	16	2	no	—
90	C₄×C₈₉	#360	1	4	yes	—
91	C₄×C₈₉	#391	1	1	yes	characteristic
92	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
93	C₄×C₈₉	#487	1	1	yes	characteristic
94	C₄×C₈₉	#498	1	1	yes	characteristic
95	Dic₈₉	—	8	4	no	—
96	#13287	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	2	1	no	—
99	C₂×C₈₉	#1990	1	1	yes	characteristic
100	C₂×C₈₉	C₂×#360	1	2	yes	—
101	#3160	—	89	1	no	—
102	C₄×M₃₂	—	89	1	no	—
103	#1844	D₈₉	1	1	yes	characteristic
104	#1386	—	89	4	no	—
105	C₈₉	#13287	1	1	yes	characteristic, complemented, Sylow
106	#533	—	178	1	no	—
107	C₂×M₃₂	—	89	2	no	—
108	#500	—	89	2	no	—
109	#499	—	89	2	no	—
110	#499	—	178	1	no	—
111	#498	—	178	1	no	—
112	#427	—	89	2	no	—
113	#427	—	178	1	no	—
114	C₂×#106	—	2	1	no	—
115	C₂×#106	C₂×D₈₉	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	#384	—	2	1	no	—
118	#346	—	178	1	no	—
119	C₄×C₁₆	—	178	1	no	—
120	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	C₄⋊Q₈	—	178	2	no	—
123	#125	—	178	2	no	—
124	C₂²⋊Q₈	—	356	2	no	—
125	C₄×Q₈	—	356	2	no	—
126	#117	—	178	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	2	no	—
129	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
130	M₃₂	—	89	4	no	—
131	M₃₂	—	178	2	no	—
132	C₂×C₁₆	—	178	2	no	—
133	#107	—	89	4	no	—
134	#107	—	178	2	no	—
135	#106	—	2	4	no	—
136	#106	—	4	2	no	—
137	#106	—	178	2	no	—
138	#106	—	178	2	no	—
139	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
140	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
141	C₄×C₈	D₃₅₆	1	1	yes	characteristic
142	C₄×C₈	D₃₅₆	1	1	yes	characteristic
143	C₂×Q₈	—	356	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	#13083559	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₈	#13083551	1	1	yes	characteristic
148	C₂×C₈	#13083551	1	1	yes	characteristic
149	C₂×C₈	#13083562	1	2	yes	—
150	C₂×C₈	#13083564	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
153	C₂×C₈	Q₈×D₈₉	1	2	yes	—
154	C₂×C₈	#13083573	1	2	yes	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	8	4	no	—
158	C₄⋊C₄	—	356	4	no	—
159	C₄⋊C₄	—	356	2	no	—
160	C₄⋊C₄	—	178	4	no	—
161	C₂²⋊C₄	—	356	2	no	—
162	C₄²	—	356	2	no	—
163	C₄²	#13083551	1	1	yes	characteristic
164	C₄²	#13083551	1	1	yes	characteristic
165	C₄²	#13083561	1	2	yes	—
166	C₁₆	—	178	4	no	—
167	C₂³	C₂×#13083559	1	1	yes	characteristic
168	Q₈	—	712	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	356	4	no	—
173	C₂×C₄	#15667932	1	2	yes	—
174	C₂×C₄	#15667932	1	2	yes	—
175	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
176	C₂×C₄	#15667999	1	1	yes	characteristic
177	C₂×C₄	#15668010	1	1	yes	characteristic
178	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
179	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
180	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
181	C₈	#15667935	1	2	yes	—
182	C₈	#15667935	1	2	yes	—
183	C₈	#15667977	1	2	yes	—
184	C₈	#15667992	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#15667932	1	1	yes	characteristic
187	C₂²	#19151944	1	1	yes	characteristic
188	C₂²	#19152001	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	16	2	no	—
191	C₄	—	712	4	no	—
192	C₄	#19151407	1	4	yes	—
193	C₄	#19151503	1	1	yes	characteristic
194	C₄	C₂×#15667935	1	1	yes	characteristic
195	C₄	#19151948	1	1	yes	characteristic
196	C₄	#19151994	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#26021212	1	1	yes	characteristic
200	C₂	C₂×#19151407	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#15667932
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1844	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1844	D₈₉
hypocenter	C₈₉	#13287
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
7. #19152003	C₄	semidirect
9. #19152000	C₄	semidirect
25. #15667952	Q₈	semidirect
105. C₈₉	#13287	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
C₄×#15667952	2
#26018359	2
C₈₉×#1844	2
#13287	89