Subgroups — #36472980

Order 22784.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#26020131	C₂	1	1	yes	maximal, characteristic
3	#26018367	C₂	1	2	yes	maximal
4	#26018354	C₂	1	2	yes	maximal
5	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19152155	—	2	1	no	—
7	#19152003	—	2	1	no	—
8	#19152000	—	2	1	no	—
9	#19152000	C₄	1	2	yes	complemented
10	#19151996	C₄	1	2	yes	complemented
11	C₂×#15667948	C₂²	1	1	yes	characteristic
12	#19151636	—	2	1	no	—
13	#19151636	C₂²	1	2	yes	—
14	#19151636	C₂²	1	1	yes	characteristic
15	#19151553	—	2	1	no	—
16	#19151358	—	2	1	no	—
17	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#364×C₈₉	—	2	1	no	—
20	#364×C₈₉	—	2	1	no	—
21	C₂×#13083558	—	2	1	no	—
22	#15668012	—	2	2	no	—
23	#15668004	—	2	2	no	—
24	#15667991	—	4	2	no	—
25	C₄×Dic₁₇₈	—	4	2	no	—
26	#15667971	—	2	2	no	—
27	#15667971	—	2	1	no	—
28	#15667948	—	2	1	no	—
29	#15667948	Q₈	1	2	yes	complemented
30	#15667946	—	2	2	no	—
31	#15667946	—	2	2	no	—
32	#15667946	—	2	1	no	—
33	#15667944	—	2	2	no	—
34	#15667944	C₂×C₄	1	4	yes	—
35	#15667944	Q₈	1	2	yes	complemented
36	#15667922	—	4	1	no	—
37	C₈×Dic₈₉	—	4	1	no	—
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	M₃₂×C₈₉	—	2	2	no	—
42	M₃₂×C₈₉	—	2	2	no	—
43	C₂×C₁₆×C₈₉	—	2	2	no	—
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂×Dic₁₇₈	—	4	2	no	—
49	#13083558	—	2	2	no	—
50	#13083558	—	4	1	no	—
51	C₂×#10623494	—	4	1	no	—
52	C₄⋊Dic₈₉	—	2	4	no	—
53	C₄⋊Dic₈₉	—	2	2	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	#13083553	—	4	4	no	—
56	C₂²⋊Dic₈₉	—	4	1	no	—
57	C₂²⋊Dic₈₉	—	4	2	no	—
58	C₄×Dic₈₉	—	4	1	no	—
59	C₄×Dic₈₉	—	4	2	no	—
60	C₂²×C₄×C₈₉	—	2	1	no	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
67	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
74	C₄²×C₈₉	—	2	1	no	—
75	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₁₆×C₈₉	—	4	2	no	—
78	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×Dic₈₉	—	4	4	no	—
89	C₂×Dic₈₉	—	4	2	no	—
90	Dic₁₇₈	—	8	4	no	—
91	C₈×C₈₉	#103	1	1	yes	characteristic
92	C₈×C₈₉	#103	1	1	yes	characteristic
93	C₈×C₈₉	#103	1	1	yes	characteristic
94	C₈×C₈₉	#103	1	1	yes	characteristic
95	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₈₉	#124	1	1	yes	characteristic
98	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
99	#10623494	—	4	2	no	—
100	C₂²×C₈₉	—	4	1	no	—
101	C₂²×C₈₉	#418	1	1	yes	characteristic
102	C₂²×C₈₉	#476	1	1	yes	characteristic
103	C₂²×C₈₉	#477	1	1	yes	characteristic
104	C₄×C₈₉	—	2	2	no	—
105	C₄×C₈₉	—	4	1	no	—
106	C₄×C₈₉	#391	1	1	yes	characteristic
107	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
108	C₄×C₈₉	#476	1	1	yes	characteristic
109	C₄×C₈₉	#477	1	1	yes	characteristic
110	Dic₈₉	—	8	4	no	—
111	Dic₈₉	—	8	2	no	—
112	#13271	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	—	2	1	no	—
114	C₂×C₈₉	—	2	1	no	—
115	C₂×C₈₉	—	2	1	no	—
116	C₂×C₈₉	#1987	1	1	yes	characteristic
117	#3160	—	89	1	no	—
118	C₂×#364	D₈₉	1	1	yes	characteristic
119	#1772	—	89	1	no	—
120	#1389	—	89	2	no	—
121	#1384	—	89	2	no	—
122	C₈₉	#13271	1	1	yes	characteristic, complemented, Sylow
123	#533	—	178	1	no	—
124	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
125	#500	—	178	1	no	—
126	#499	—	89	2	no	—
127	#499	—	178	1	no	—
128	#498	—	89	2	no	—
129	C₂×#108	—	89	1	no	—
130	#427	—	89	2	no	—
131	#427	—	89	1	no	—
132	#427	—	178	1	no	—
133	#405	—	178	1	no	—
134	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
135	#364	—	2	1	no	—
136	#364	—	2	1	no	—
137	#339	—	178	1	no	—
138	C₂×M₁₆	—	178	1	no	—
139	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	178	2	no	—
142	#125	—	178	2	no	—
143	C₂²⋊Q₈	—	356	2	no	—
144	C₄×Q₈	—	356	2	no	—
145	#117	—	178	2	no	—
146	#117	—	178	1	no	—
147	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	89	2	no	—
152	#108	—	178	1	no	—
153	#107	—	178	2	no	—
154	#107	—	178	2	no	—
155	#107	—	178	1	no	—
156	#106	—	89	4	no	—
157	#106	—	89	2	no	—
158	#106	—	178	2	no	—
159	#97	—	356	1	no	—
160	C₄×C₈	—	356	1	no	—
161	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
162	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂×Q₈	—	356	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
168	M₁₆	—	356	1	no	—
169	M₁₆	—	178	2	no	—
170	C₂×C₈	—	356	1	no	—
171	C₂×C₈	#13083551	1	1	yes	characteristic
172	C₂×C₈	#13083551	1	1	yes	characteristic
173	C₂×C₈	#13083563	1	1	yes	characteristic
174	C₂×C₈	#13083563	1	1	yes	characteristic
175	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
176	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
179	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
180	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
181	C₂×C₈	#13083573	1	1	yes	characteristic
182	C₂×C₈	#13083573	1	1	yes	characteristic
183	C₄⋊C₄	—	356	4	no	—
184	C₄⋊C₄	—	356	2	no	—
185	C₄⋊C₄	—	178	2	no	—
186	C₄⋊C₄	—	178	4	no	—
187	C₂²⋊C₄	—	356	2	no	—
188	C₂²⋊C₄	—	356	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	356	1	no	—
191	C₄²	—	356	2	no	—
192	C₄²	#13083551	1	1	yes	characteristic
193	C₄²	#13083551	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	#15668040	1	1	yes	characteristic
196	Q₈	—	712	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	356	2	no	—
201	C₂×C₄	—	356	4	no	—
202	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
203	C₂×C₄	#15667992	1	1	yes	characteristic
204	C₂×C₄	#15667992	1	1	yes	characteristic
205	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
207	C₂×C₄	#15668040	1	1	yes	characteristic
208	C₈	—	356	2	no	—
209	C₈	#15667937	1	1	yes	characteristic
210	C₈	#15667937	1	1	yes	characteristic
211	C₈	#15667937	1	1	yes	characteristic
212	C₈	#15667937	1	1	yes	characteristic
213	C₈	#15667977	1	1	yes	characteristic
214	C₈	#15667977	1	1	yes	characteristic
215	C₈	#15667999	1	1	yes	characteristic
216	C₈	#15668010	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#19151604	1	1	yes	characteristic
219	C₂²	#19151889	1	1	yes	characteristic
220	C₂²	#19151894	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	712	4	no	—
224	C₄	—	712	2	no	—
225	C₄	#19151503	1	1	yes	characteristic
226	C₄	C₂×#15667937	1	1	yes	characteristic
227	C₄	#19151889	1	1	yes	characteristic
228	C₄	#19151894	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#26021199	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151889
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂×C₈₉	#1987
prime radical	C₂³×C₈₉	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#13271
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #19152000	C₄	semidirect
10. #19151996	C₄	semidirect
29. #15667948	Q₈	semidirect
35. #15667944	Q₈	semidirect
122. C₈₉	#13271	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26020131	2
#26018367	2
#26018354	2
C₂×#364×C₈₉	2
#13271	89