Subgroups — #36474046

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151426	C₂	1	2	yes	maximal
3	#26020985	C₂	1	1	yes	maximal, characteristic
4	#26018344	C₂	1	2	yes	maximal
5	C₈₉×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	—	2	1	no	—
7	#19152003	—	2	4	no	—
8	C₂×#15667946	—	2	1	no	—
9	C₂×#15667946	C₂²	1	1	yes	characteristic
10	C₂×#15667946	C₂²	1	2	yes	—
11	C₂×#15667946	C₂²	1	1	yes	characteristic
12	#19151496	—	2	1	no	—
13	#19151426	—	2	4	no	—
14	#19151361	—	2	1	no	—
15	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	—	4	1	no	—
18	#15668012	—	2	4	no	—
19	#15668012	—	4	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	1	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×C₄⋊Dic₈₉	—	2	1	no	—
23	C₂×C₄⋊Dic₈₉	—	4	1	no	—
24	C₂×#13083553	—	4	1	no	—
25	#15667946	—	2	4	no	—
26	#15667946	—	2	2	no	—
27	#15667946	—	2	2	no	—
28	#15667946	—	2	1	no	—
29	#15667946	D₄	1	4	yes	—
30	#15667946	Q₈	1	2	yes	complemented
31	#15667919	—	4	1	no	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
38	C₄×C₈×C₈₉	—	2	2	no	—
39	C₂×Dic₁₇₈	—	4	4	no	—
40	C₂×Dic₁₇₈	—	8	2	no	—
41	C₂×Dic₁₇₈	—	8	2	no	—
42	C₂²×Dic₈₉	—	4	2	no	—
43	C₂²×Dic₈₉	—	4	1	no	—
44	C₂²×Dic₈₉	—	4	1	no	—
45	C₄⋊Dic₈₉	—	2	4	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	—	4	4	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	C₄⋊Dic₈₉	—	4	1	no	—
50	C₄⋊Dic₈₉	—	4	1	no	—
51	C₄⋊Dic₈₉	—	4	1	no	—
52	#13083553	—	8	2	no	—
53	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	2	1	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
59	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
63	C₄²×C₈₉	—	2	1	no	—
64	C₄²×C₈₉	—	2	1	no	—
65	C₁₆×C₈₉	—	2	4	no	—
66	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
72	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
73	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
79	C₂×Dic₈₉	—	4	4	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	—	8	1	no	—
82	C₂×Dic₈₉	—	8	2	no	—
83	C₂×Dic₈₉	—	8	1	no	—
84	C₂×Dic₈₉	—	8	2	no	—
85	C₂×Dic₈₉	—	8	1	no	—
86	C₂×Dic₈₉	—	8	1	no	—
87	C₂×Dic₈₉	—	8	1	no	—
88	Dic₁₇₈	—	8	8	no	—
89	C₈×C₈₉	—	2	2	no	—
90	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
92	C₂²×C₈₉	#447	1	1	yes	characteristic
93	C₂²×C₈₉	#474	1	1	yes	characteristic
94	C₂²×C₈₉	#500	1	1	yes	characteristic
95	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₈₉	#509	1	1	yes	characteristic
98	C₂²×C₈₉	#510	1	1	yes	characteristic
99	C₄×C₈₉	—	2	2	no	—
100	C₄×C₈₉	—	2	2	no	—
101	C₄×C₈₉	#395	1	1	yes	characteristic
102	C₄×C₈₉	#451	1	1	yes	characteristic
103	C₄×C₈₉	#477	1	1	yes	characteristic
104	C₄×C₈₉	#501	1	1	yes	characteristic
105	Dic₈₉	—	8	4	no	—
106	Dic₈₉	—	8	2	no	—
107	Dic₈₉	—	16	1	no	—
108	#13452	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	#2057	1	1	yes	characteristic
110	C₂×C₈₉	#2185	1	1	yes	characteristic
111	C₂×C₈₉	#2187	1	1	yes	characteristic
112	C₂×C₈₉	#2225	1	1	yes	characteristic
113	C₂×C₈₉	#2226	1	1	yes	characteristic
114	C₂×C₈₉	#2254	1	1	yes	characteristic
115	C₂×C₈₉	#2255	1	1	yes	characteristic
116	C₂×#500	—	89	1	no	—
117	C₂×#367	—	89	2	no	—
118	#1942	—	89	1	no	—
119	#1380	—	89	2	no	—
120	#1374	D₈₉	1	1	yes	characteristic
121	C₈₉	#13452	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄⋊Q₈	—	178	1	no	—
123	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
124	#500	—	178	4	no	—
125	C₂×#107	—	89	2	no	—
126	C₂×#107	—	89	1	no	—
127	C₂×#107	—	89	1	no	—
128	C₂×#107	—	178	1	no	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	#389	—	178	1	no	—
131	#367	—	178	4	no	—
132	#340	—	178	1	no	—
133	C₂²×Q₈	—	356	1	no	—
134	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	356	2	no	—
137	C₄⋊Q₈	—	178	4	no	—
138	C₂×C₄⋊C₄	—	356	1	no	—
139	C₂×C₄⋊C₄	—	356	1	no	—
140	C₂×C₄⋊C₄	—	178	1	no	—
141	C₂×C₄⋊C₄	—	178	2	no	—
142	C₂×C₄⋊C₄	—	178	1	no	—
143	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
147	#107	—	89	4	no	—
148	#107	—	89	2	no	—
149	#107	—	178	2	no	—
150	#107	—	178	4	no	—
151	#107	—	178	2	no	—
152	#107	—	178	1	no	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	356	1	no	—
155	C₂×Q₈	—	356	4	no	—
156	C₂×Q₈	—	712	2	no	—
157	C₂×Q₈	—	712	2	no	—
158	C₂²×C₄	—	356	1	no	—
159	C₂²×C₄	—	356	2	no	—
160	C₂²×C₄	—	356	1	no	—
161	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
162	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	Dic₃₅₆	1	2	yes	—
167	C₂×C₈	D₄×D₈₉	1	2	yes	—
168	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
170	C₂×C₈	#13083573	1	1	yes	characteristic
171	C₄⋊C₄	—	356	4	no	—
172	C₄⋊C₄	—	356	2	no	—
173	C₄⋊C₄	—	356	1	no	—
174	C₄⋊C₄	—	356	1	no	—
175	C₄⋊C₄	—	356	1	no	—
176	C₄⋊C₄	—	178	4	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄⋊C₄	—	712	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	2	4	no	—
182	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
183	Q₈	—	712	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	356	2	no	—
189	C₂×C₄	—	356	4	no	—
190	C₂×C₄	—	712	1	no	—
191	C₂×C₄	—	712	1	no	—
192	C₂×C₄	—	712	1	no	—
193	C₂×C₄	—	712	1	no	—
194	C₂×C₄	—	712	2	no	—
195	C₂×C₄	—	712	1	no	—
196	C₂×C₄	—	712	2	no	—
197	C₂×C₄	#15667957	1	2	yes	—
198	C₂×C₄	#15667957	1	2	yes	—
199	C₂×C₄	#15667982	1	1	yes	characteristic
200	C₂×C₄	#15667992	1	1	yes	characteristic
201	C₂×C₄	#15668012	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
203	C₂×C₄	#15668040	1	1	yes	characteristic
204	C₂×C₄	#15668045	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#15667989	1	2	yes	—
207	C₈	#15668011	1	2	yes	—
208	C₂²	#19151734	1	1	yes	characteristic
209	C₂²	#19151879	1	1	yes	characteristic
210	C₂²	#19152003	1	1	yes	characteristic
211	C₂²	C₂×#15667957	1	1	yes	characteristic
212	C₂²	C₂×#15667957	1	1	yes	characteristic
213	C₂²	#19152036	1	1	yes	characteristic
214	C₂²	#19152041	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	1424	1	no	—
218	C₄	—	712	2	no	—
219	C₄	—	712	4	no	—
220	C₄	#19151519	1	1	yes	characteristic
221	C₄	#19151756	1	1	yes	characteristic
222	C₄	#19151894	1	1	yes	characteristic
223	C₄	#19152007	1	1	yes	characteristic
224	C₂	#26021581	1	1	yes	characteristic
225	C₂	#26022171	1	1	yes	characteristic
226	C₂	#26022183	1	1	yes	characteristic
227	C₂	#26022395	1	1	yes	characteristic
228	C₂	#26022400	1	1	yes	characteristic
229	C₂	#26022534	1	1	yes	characteristic
230	C₂	#26022538	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1374	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#1374	D₈₉
hypocenter	C₈₉	#13452
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
30. #15667946	Q₈	semidirect
37. C₂×C₁₆×C₈₉	Q₈	semidirect
121. C₈₉	#13452	semidirect
146. 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
C₂×#19152003	2
C₂×#19151426	2
#26020985	2
#26018344	2
C₈₉×#1374	2
#13452	89