Subgroups — #36709938

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030792	C₂	1	2	yes	maximal
2	#26022537	C₂	1	4	yes	maximal, complemented
3	#26022529	C₂	1	2	yes	maximal, complemented
4	#26022526	C₂	1	2	yes	maximal, complemented
5	#26022041	C₂	1	2	yes	maximal
6	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
7	#26021848	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic
9	#19152226	—	2	2	no	—
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	#19152005	—	2	4	no	—
12	#19152002	C₂²	1	4	yes	—
13	#19151999	C₂²	1	4	yes	—
14	#19151641	—	2	2	no	—
15	C₂×#15667946	C₂²	1	2	yes	—
16	#19151553	C₂²	1	1	yes	characteristic
17	#19151425	C₂²	1	4	yes	—
18	#19151424	C₂²	1	2	yes	—
19	#19151424	C₂²	1	2	yes	—
20	#19151423	C₂²	1	4	yes	—
21	#19151373	C₂²	1	2	yes	—
22	C₁₆×Dic₈₉	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	#427×C₈₉	C₂²	1	2	yes	—
25	#367×C₈₉	C₂²	1	2	yes	—
26	#366×C₈₉	C₂²	1	1	yes	characteristic
27	#366×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#13083558	D₄	1	1	yes	characteristic
29	#15668012	—	4	2	no	—
30	#15668011	—	2	4	no	—
31	#15668004	—	4	2	no	—
32	#15668003	—	2	4	no	—
33	#15667989	—	4	4	no	—
34	#15667971	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	#15667952	Q₈	1	4	yes	complemented
37	C₂×#13083549	C₂³	1	2	yes	—
38	#15667946	—	2	2	no	—
39	#15667946	C₂³	1	4	yes	—
40	#15667945	—	2	4	no	—
41	#15667943	—	2	4	no	—
42	#15667922	D₄	1	2	yes	—
43	C₈×Dic₈₉	C₂³	1	2	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	#117×C₈₉	—	2	2	no	—
46	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	#107×C₈₉	C₂³	1	2	yes	—
50	#107×C₈₉	C₂³	1	2	yes	—
51	#106×C₈₉	—	2	2	no	—
52	C₂×Dic₁₇₈	—	4	4	no	—
53	C₂²×Dic₈₉	—	4	2	no	—
54	#13083558	Q₁₆	1	4	yes	—
55	C₂×#10623494	C₂×D₄	1	2	yes	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	—	4	4	no	—
59	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
60	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
61	#13083553	—	4	4	no	—
62	#13083553	—	4	4	no	—
63	C₂²⋊Dic₈₉	D₈	1	2	yes	—
64	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
65	#13083549	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₈₉	—	2	2	no	—
72	C₄⋊C₄×C₈₉	—	2	2	no	—
73	C₂²⋊C₄×C₈₉	—	4	2	no	—
74	C₄²×C₈₉	—	4	2	no	—
75	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
76	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
77	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	4	2	no	—
79	C₂×C₄×C₈₉	—	4	2	no	—
80	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	—	16	2	no	—
86	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
87	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
88	Dic₁₇₈	—	16	4	no	—
89	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
90	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
92	#10623494	C₂×Q₁₆	1	4	yes	—
93	C₂²×C₈₉	—	2	1	no	—
94	C₂²×C₈₉	—	2	1	no	—
95	C₂²×C₈₉	#500	1	2	yes	—
96	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₈₉	—	8	2	no	—
98	C₄×C₈₉	—	8	2	no	—
99	C₄×C₈₉	#500	1	2	yes	—
100	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₈₉	—	2	2	no	—
103	Dic₈₉	—	2	2	no	—
104	Dic₈₉	—	8	4	no	—
105	#33729	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	—	2	2	no	—
107	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
108	C₂×C₈₉	#3415	1	1	yes	characteristic
109	C₂×C₈₉	#3417	1	1	yes	characteristic
110	#3164	—	89	2	no	—
111	#2255	—	89	4	no	—
112	#2253	—	89	2	no	—
113	#2252	—	89	2	no	—
114	#2159	—	89	2	no	—
115	#2158	D₈₉	1	1	yes	characteristic
116	C₂×#366	—	89	1	no	—
117	#2108	—	89	1	no	—
118	C₈₉	#33729	1	1	yes	characteristic, complemented, Sylow
119	#544	—	178	2	no	—
120	C₂×M₃₂	—	89	1	no	—
121	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
122	#501	—	178	4	no	—
123	#500	—	89	4	no	—
124	#499	—	89	4	no	—
125	#428	—	178	2	no	—
126	#427	C₂×D₈₉	1	2	yes	—
127	C₂×#107	—	89	2	no	—
128	#405	—	89	1	no	—
129	#367	—	89	4	no	—
130	#367	C₂×D₈₉	1	2	yes	—
131	#366	—	89	2	no	—
132	#366	—	89	2	no	—
133	#366	—	89	4	no	—
134	#366	C₂×D₈₉	1	1	yes	characteristic
135	#366	C₂×D₈₉	1	1	yes	characteristic
136	#346	—	89	2	no	—
137	C₄×C₁₆	—	89	2	no	—
138	C₂×M₁₆	—	89	1	no	—
139	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
140	C₄⋊Q₈	—	356	2	no	—
141	C₄⋊Q₈	—	178	4	no	—
142	#125	—	356	2	no	—
143	#125	—	178	4	no	—
144	C₂²⋊Q₈	—	356	4	no	—
145	#117	—	2	2	no	—
146	#117	—	89	1	no	—
147	C₂×C₄⋊C₄	—	178	2	no	—
148	M₃₂	—	89	4	no	—
149	C₂×C₁₆	—	89	2	no	—
150	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
153	#107	—	89	4	no	—
154	#107	—	178	4	no	—
155	#107	—	178	2	no	—
156	#107	C₂²×D₈₉	1	2	yes	—
157	#107	C₂²×D₈₉	1	2	yes	—
158	#106	—	2	2	no	—
159	#106	—	178	4	no	—
160	#97	—	89	2	no	—
161	C₄×C₈	—	89	2	no	—
162	C₂×Q₈	—	356	4	no	—
163	C₂²×C₄	—	356	2	no	—
164	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
165	M₁₆	—	89	4	no	—
166	C₂×C₈	—	89	2	no	—
167	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₈₉	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
170	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	89	1	no	—
174	C₄⋊C₄	—	89	1	no	—
175	C₄⋊C₄	—	356	4	no	—
176	C₄⋊C₄	—	356	4	no	—
177	C₄⋊C₄	—	356	4	no	—
178	C₄⋊C₄	—	356	2	no	—
179	C₄⋊C₄	—	178	4	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	89	2	no	—
182	C₄²	—	4	2	no	—
183	C₄²	—	89	2	no	—
184	C₁₆	—	89	4	no	—
185	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
186	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
187	C₂³	D₈×D₈₉	1	1	yes	characteristic
188	Q₈	—	1424	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	89	2	no	—
192	C₂×C₄	—	89	2	no	—
193	C₂×C₄	—	356	4	no	—
194	C₂×C₄	—	1424	2	no	—
195	C₂×C₄	#15668011	1	2	yes	—
196	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
199	C₈	—	89	4	no	—
200	C₈	#15668011	1	2	yes	—
201	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#19152002	1	2	yes	—
206	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	—	178	2	no	—
210	C₄	—	178	2	no	—
211	C₄	—	712	4	no	—
212	C₄	#19152002	1	2	yes	—
213	C₄	C₂×#15668011	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#19152002	1	1	yes	characteristic
217	C₂	#26033390	1	1	yes	characteristic
218	C₂	#26033402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2158	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#2158	D₈₉
hypocenter	C₈₉	#33729
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #26022537	C₂	semidirect
3. #26022529	C₂	semidirect
4. #26022526	C₂	semidirect
36. #15667952	Q₈	semidirect
46. C₂×C₁₆×C₈₉	Q₈	semidirect
118. C₈₉	#33729	semidirect
150. 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
#26030792	2
#26022537	2
#26022529	2
#26022526	2
#26022041	2
C₂×#19151424	2
#26021848	2
C₈₉×#2158	2
#33729	89