Subgroups — #36709945

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030791	C₂	1	2	yes	maximal
2	#26022538	C₂	1	4	yes	maximal, complemented
3	#26022534	C₂	1	2	yes	maximal, complemented
4	#26022531	C₂	1	2	yes	maximal, complemented
5	#26022040	C₂	1	2	yes	maximal
6	#26022036	C₂	1	1	yes	maximal, characteristic
7	C₂×#19151426	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic
9	#19152226	—	2	2	no	—
10	#19152007	—	2	4	no	—
11	#19152003	C₂²	1	4	yes	—
12	#19152000	C₂²	1	4	yes	—
13	#19151640	—	2	2	no	—
14	#19151636	C₂²	1	2	yes	—
15	C₂×#15667946	C₂²	1	2	yes	—
16	#19151426	C₂²	1	1	yes	characteristic
17	#19151426	C₂²	1	2	yes	—
18	#19151426	C₂²	1	1	yes	characteristic
19	#19151426	C₂²	1	4	yes	—
20	#19151426	C₂²	1	2	yes	—
21	#19151424	C₂²	1	4	yes	—
22	#19151424	C₂²	1	2	yes	—
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	#405×C₈₉	C₂²	1	1	yes	characteristic
26	#346×C₈₉	C₂²	1	2	yes	—
27	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
28	#15668012	—	2	4	no	—
29	#15668011	—	4	2	no	—
30	#15668004	—	2	4	no	—
31	#15668003	—	4	2	no	—
32	#15667989	—	4	4	no	—
33	#15667971	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	—	2	2	no	—
35	#15667946	—	2	2	no	—
36	#15667946	—	2	4	no	—
37	#15667946	C₂³	1	4	yes	—
38	#15667946	C₂³	1	2	yes	—
39	#15667946	C₂³	1	2	yes	—
40	#15667944	—	2	2	no	—
41	#15667944	—	2	4	no	—
42	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	#117×C₈₉	D₄	1	1	yes	characteristic
45	M₃₂×C₈₉	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
49	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
50	#97×C₈₉	D₄	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂×Dic₁₇₈	—	4	4	no	—
53	C₂²×Dic₈₉	—	4	2	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	4	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	#13083553	—	4	4	no	—
60	#13083553	—	4	4	no	—
61	C₂²⋊Dic₈₉	—	4	2	no	—
62	C₄×Dic₈₉	—	4	2	no	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₈₉	Q₁₆	1	4	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
71	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
72	C₄²×C₈₉	C₂×D₄	1	2	yes	—
73	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
74	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
75	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
76	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	—	16	2	no	—
86	Dic₁₇₈	—	16	4	no	—
87	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
88	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
89	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₈₉	—	2	2	no	—
92	C₂²×C₈₉	#500	1	2	yes	—
93	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	#500	1	2	yes	—
96	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
98	Dic₈₉	—	8	2	no	—
99	Dic₈₉	—	8	4	no	—
100	Dic₈₉	—	8	2	no	—
101	#33730	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	—	2	2	no	—
103	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
104	C₂×C₈₉	#3416	1	2	yes	—
105	#3164	—	89	2	no	—
106	#2255	—	89	4	no	—
107	#2254	—	89	2	no	—
108	#2253	—	89	2	no	—
109	#2159	—	89	2	no	—
110	#2158	—	89	1	no	—
111	C₂×#367	—	89	1	no	—
112	#2108	D₈₉	1	1	yes	characteristic
113	C₈₉	#33730	1	1	yes	characteristic, complemented, Sylow
114	#544	—	178	2	no	—
115	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
116	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
117	#501	—	178	4	no	—
118	#500	—	89	4	no	—
119	#499	—	89	4	no	—
120	#428	—	178	2	no	—
121	#427	—	89	2	no	—
122	C₂×#107	—	89	2	no	—
123	#405	C₂×D₈₉	1	1	yes	characteristic
124	#367	—	89	2	no	—
125	#367	—	89	2	no	—
126	#367	—	89	1	no	—
127	#367	—	89	1	no	—
128	#367	—	89	4	no	—
129	#366	—	89	2	no	—
130	#366	—	89	4	no	—
131	#346	C₂×D₈₉	1	2	yes	—
132	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
133	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	356	2	no	—
136	C₄⋊Q₈	—	178	4	no	—
137	#125	—	356	2	no	—
138	#125	—	178	4	no	—
139	C₂²⋊Q₈	—	356	4	no	—
140	#117	—	178	2	no	—
141	#117	D₃₅₆	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	178	2	no	—
143	M₃₂	Dic₁₇₈	1	4	yes	complemented
144	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
146	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
147	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
148	#107	—	89	2	no	—
149	#107	—	89	4	no	—
150	#107	—	89	2	no	—
151	#107	—	178	2	no	—
152	#107	—	178	4	no	—
153	#106	—	178	2	no	—
154	#106	—	178	4	no	—
155	#97	D₃₅₆	1	2	yes	—
156	C₄×C₈	C₂²×D₈₉	1	2	yes	—
157	C₂×Q₈	—	356	4	no	—
158	C₂²×C₄	—	356	2	no	—
159	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
160	M₁₆	Dic₃₅₆	1	4	yes	—
161	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
162	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
165	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
166	C₄⋊C₄	—	356	2	no	—
167	C₄⋊C₄	—	356	4	no	—
168	C₄⋊C₄	—	356	4	no	—
169	C₄⋊C₄	—	356	4	no	—
170	C₄⋊C₄	—	178	2	no	—
171	C₄⋊C₄	—	178	4	no	—
172	C₄⋊C₄	—	178	2	no	—
173	C₄⋊C₄	D₇₁₂	1	2	yes	—
174	C₂²⋊C₄	—	356	2	no	—
175	C₂²⋊C₄	D₇₁₂	1	2	yes	—
176	C₄²	—	356	2	no	—
177	C₄²	C₂×D₃₅₆	1	2	yes	—
178	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
179	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
180	C₁₆	C₂×Dic₁₇₈	1	4	yes	—
181	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
182	Q₈	—	1424	4	no	—
183	C₂×C₄	—	356	2	no	—
184	C₂×C₄	—	356	2	no	—
185	C₂×C₄	—	356	4	no	—
186	C₂×C₄	—	1424	2	no	—
187	C₂×C₄	#15668012	1	2	yes	—
188	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
189	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
191	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
192	C₈	#15668012	1	2	yes	—
193	C₈	C₂×Dic₃₅₆	1	4	yes	—
194	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#19152003	1	2	yes	—
198	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	712	4	no	—
201	C₄	—	712	2	no	—
202	C₄	—	712	2	no	—
203	C₄	#19152003	1	2	yes	—
204	C₄	C₂×#15668012	1	1	yes	characteristic
205	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#19152003	1	1	yes	characteristic
208	C₂	#26033400	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#33730
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #26022538	C₂	semidirect
3. #26022534	C₂	semidirect
4. #26022531	C₂	semidirect
45. M₃₂×C₈₉	Q₈	semidirect
46. C₂×C₁₆×C₈₉	Q₈	semidirect
113. C₈₉	#33730	semidirect
143. M₃₂	Dic₁₇₈	semidirect
144. C₂×C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030791	2
#26022538	2
#26022534	2
#26022531	2
#26022040	2
#26022036	2
C₂×#19151426	2
C₈₉×#2108	2
#33730	89