Subgroups — #36453093

Order 22784.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021010	C₂	1	2	yes	maximal
2	#26020409	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
4	#26018136	C₂	1	2	yes	maximal
5	C₈₉×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667945	—	2	2	no	—
7	C₂×#15667944	—	2	1	no	—
8	C₂×#15667944	—	2	1	no	—
9	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
10	#19151468	C₂²	1	2	yes	—
11	#19151360	—	2	2	no	—
12	#19151351	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#97×C₈₉	—	2	1	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	#336×C₈₉	—	2	1	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	#15667945	—	4	4	no	—
23	#15667944	—	4	2	no	—
24	#15667944	—	4	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₄⋊#10623494	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₈₉	—	2	1	no	—
28	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	—	2	1	no	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	—	2	1	no	—
34	C₄⋊C₈×C₈₉	Q₈	1	2	yes	—
35	#97×C₈₉	—	4	2	no	—
36	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₈₉	Q₈	1	2	yes	—
38	#95×C₈₉	—	2	1	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₂×#10623494	—	2	4	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	8	2	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	1	no	—
53	C₂×C₈×C₈₉	—	2	1	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	4	1	no	—
56	C₂×C₈×C₈₉	—	4	1	no	—
57	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₈₉	D₈	1	2	yes	—
60	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₄²×C₈₉	—	4	1	no	—
64	C₄²×C₈₉	—	4	1	no	—
65	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₈₉	#41	1	1	yes	characteristic
69	C₂³×C₈₉	#121	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#137	1	2	yes	—
86	C₂×Dic₈₉	—	8	2	no	—
87	C₂×Dic₈₉	—	8	2	no	—
88	C₂×Dic₈₉	—	8	2	no	—
89	C₈×C₈₉	—	2	2	no	—
90	C₈×C₈₉	—	4	2	no	—
91	C₈×C₈₉	#107	1	4	yes	complemented
92	#10623494	—	4	4	no	—
93	C₂²×C₈₉	#389	1	1	yes	characteristic
94	C₂²×C₈₉	#459	1	1	yes	characteristic
95	C₂²×C₈₉	#462	1	1	yes	characteristic
96	C₂²×C₈₉	#468	1	2	yes	—
97	C₂²×C₈₉	#471	1	2	yes	—
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	8	2	no	—
101	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
102	C₄×C₈₉	#428	1	1	yes	characteristic
103	C₄×C₈₉	#441	1	1	yes	characteristic
104	C₄×C₈₉	#442	1	1	yes	characteristic
105	C₄×C₈₉	#474	1	1	yes	characteristic
106	C₄×C₈₉	#477	1	1	yes	characteristic
107	C₄×C₈₉	#501	1	2	yes	—
108	Dic₈₉	—	16	2	no	—
109	#9617	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	#1685	1	2	yes	—
111	C₂×C₈₉	#1692	1	2	yes	—
112	C₂×C₈₉	#1943	1	1	yes	characteristic
113	C₂×C₈₉	#1947	1	1	yes	characteristic
114	C₂×C₈₉	#1951	1	1	yes	characteristic
115	#1947	—	89	2	no	—
116	#1844	—	89	1	no	—
117	#1770	D₈₉	1	1	yes	characteristic
118	C₂×#335	—	89	1	no	—
119	#1296	—	89	2	no	—
120	C₈₉	#9617	1	1	yes	characteristic, complemented, Sylow
121	C₂×#107	—	178	2	no	—
122	C₂×#106	—	178	1	no	—
123	C₂×#106	—	178	1	no	—
124	C₂×C₄⋊C₈	—	89	2	no	—
125	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₂×#97	—	2	1	no	—
127	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
128	#384	—	89	2	no	—
129	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
130	#340	—	178	2	no	—
131	#336	—	2	1	no	—
132	#335	—	89	4	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	178	2	no	—
135	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	178	2	no	—
138	C₂×C₄⋊C₄	—	178	2	no	—
139	C₂×C₄⋊C₄	—	178	2	no	—
140	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
143	#107	—	356	4	no	—
144	#106	—	356	2	no	—
145	#106	—	356	2	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	89	4	no	—
148	C₄⋊C₈	—	178	2	no	—
149	C₄⋊C₈	Dic₁₇₈	1	2	yes	—
150	#97	—	4	2	no	—
151	C₄×C₈	Dic₁₇₈	1	2	yes	—
152	C₄×C₈	C₄×D₈₉	1	2	yes	—
153	#95	—	2	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	356	2	no	—
156	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
158	C₂²×C₄	#13083574	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	356	2	no	—
165	C₂×C₈	—	356	2	no	—
166	C₂×C₈	—	178	4	no	—
167	C₂×C₈	#13083553	1	2	yes	—
168	C₂×C₈	#13083553	1	2	yes	—
169	C₂×C₈	#13083559	1	2	yes	—
170	C₂×C₈	#13083564	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	356	2	no	—
174	C₄⋊C₄	—	356	2	no	—
175	C₄⋊C₄	—	356	2	no	—
176	C₄⋊C₄	—	356	2	no	—
177	C₄⋊C₄	—	712	2	no	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
181	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
182	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
183	C₄²	#13083574	1	1	yes	characteristic
184	C₂³	#15667986	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	712	2	no	—
190	C₂×C₄	—	712	2	no	—
191	C₂×C₄	—	712	2	no	—
192	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
193	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
194	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
195	C₂×C₄	#15667991	1	1	yes	characteristic
196	C₂×C₄	#15668004	1	1	yes	characteristic
197	C₂×C₄	#15668012	1	1	yes	characteristic
198	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
199	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
200	C₂×C₄	#15668037	1	1	yes	characteristic
201	C₂×C₄	#15668040	1	2	yes	—
202	C₂×C₄	#15668042	1	1	yes	characteristic
203	C₂×C₄	#15668045	1	2	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	356	4	no	—
207	C₈	#15667945	1	4	yes	complemented
208	C₂²	#19151496	1	1	yes	characteristic
209	C₂²	#19151793	1	1	yes	characteristic
210	C₂²	#19151808	1	1	yes	characteristic
211	C₂²	#19151844	1	2	yes	—
212	C₂²	#19151861	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	—	1424	2	no	—
217	C₄	C₂×#15667945	1	1	yes	characteristic
218	C₄	#19151641	1	1	yes	characteristic
219	C₄	#19151706	1	1	yes	characteristic
220	C₄	#19151711	1	1	yes	characteristic
221	C₄	#19151878	1	1	yes	characteristic
222	C₄	#19151893	1	1	yes	characteristic
223	C₄	#19152007	1	2	yes	—
224	C₂	#26019717	1	2	yes	—
225	C₂	#26019758	1	2	yes	—
226	C₂	#26020990	1	1	yes	characteristic
227	C₂	#26021010	1	1	yes	characteristic
228	C₂	#26021032	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1770	C₂
socle, prime radical	C₂³×C₈₉	#121
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1770	D₈₉
hypocenter	C₈₉	#9617
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
12. #19151351	C₄	semidirect
91. C₈×C₈₉	#107	semidirect
120. C₈₉	#9617	semidirect
207. C₈	#15667945	semidirect

Maximal subgroups

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

subgroup	index
#26021010	2
#26020409	2
C₂×#19151351	2
#26018136	2
C₈₉×#1770	2
#9617	89