Subgroups — #36453076

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 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	#26020412	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
4	#26018136	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	1	no	—
7	C₂×#15667946	—	2	1	no	—
8	C₂×#15667945	—	2	2	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×C₄×C₈×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	#15667946	—	4	2	no	—
23	#15667946	—	4	2	no	—
24	#15667945	—	4	4	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	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	—	2	1	no	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₉	—	2	1	no	—
35	C₄⋊C₈×C₈₉	Q₈	1	2	yes	—
36	C₄×C₈×C₈₉	—	4	2	no	—
37	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₉	Q₈	1	2	yes	—
39	#95×C₈₉	—	2	1	no	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	C₄⋊Dic₈₉	—	8	2	no	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	1	no	—
55	C₂×C₈×C₈₉	—	2	1	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	4	1	no	—
58	C₂×C₈×C₈₉	—	4	1	no	—
59	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	D₈	1	2	yes	—
62	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₄⋊C₄×C₈₉	—	2	1	no	—
65	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
66	C₄⋊C₄×C₈₉	Q₁₆	1	2	yes	—
67	C₄²×C₈₉	—	4	1	no	—
68	C₄²×C₈₉	—	4	1	no	—
69	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₈₉	#41	1	1	yes	characteristic
73	C₂³×C₈₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	4	1	no	—
79	C₂×C₄×C₈₉	—	4	1	no	—
80	C₂×C₄×C₈₉	—	4	1	no	—
81	C₂×C₄×C₈₉	—	4	1	no	—
82	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
93	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
94	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
96	C₂×Dic₈₉	—	8	2	no	—
97	C₂×Dic₈₉	—	8	2	no	—
98	C₂×Dic₈₉	—	8	2	no	—
99	C₈×C₈₉	—	2	2	no	—
100	C₈×C₈₉	—	4	2	no	—
101	C₈×C₈₉	#107	1	4	yes	complemented
102	#10623494	—	4	4	no	—
103	C₂²×C₈₉	#389	1	1	yes	characteristic
104	C₂²×C₈₉	#459	1	1	yes	characteristic
105	C₂²×C₈₉	#462	1	1	yes	characteristic
106	C₂²×C₈₉	#466	1	1	yes	characteristic
107	C₂²×C₈₉	#467	1	1	yes	characteristic
108	C₂²×C₈₉	#472	1	1	yes	characteristic
109	C₂²×C₈₉	#473	1	1	yes	characteristic
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	—	2	1	no	—
112	C₄×C₈₉	—	4	2	no	—
113	C₄×C₈₉	—	4	2	no	—
114	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
115	C₄×C₈₉	#428	1	1	yes	characteristic
116	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
117	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₈₉	#474	1	1	yes	characteristic
119	C₄×C₈₉	#477	1	1	yes	characteristic
120	C₄×C₈₉	#499	1	1	yes	characteristic
121	C₄×C₈₉	#500	1	1	yes	characteristic
122	Dic₈₉	—	16	2	no	—
123	#9614	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	#1667	1	1	yes	characteristic
125	C₂×C₈₉	#1669	1	1	yes	characteristic
126	C₂×C₈₉	#1674	1	1	yes	characteristic
127	C₂×C₈₉	#1676	1	1	yes	characteristic
128	C₂×C₈₉	#1942	1	1	yes	characteristic
129	C₂×C₈₉	#1947	1	1	yes	characteristic
130	C₂×C₈₉	#1952	1	1	yes	characteristic
131	#1947	—	89	2	no	—
132	#1845	—	89	1	no	—
133	C₈×C₄⋊C₄	D₈₉	1	1	yes	characteristic
134	C₂×#335	—	89	1	no	—
135	#1296	—	89	2	no	—
136	C₈₉	#9614	1	1	yes	characteristic, complemented, Sylow
137	C₂×#107	—	178	1	no	—
138	C₂×#107	—	178	1	no	—
139	C₂×#107	—	178	2	no	—
140	C₂×C₄⋊C₈	—	89	2	no	—
141	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
144	#384	—	89	2	no	—
145	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
146	#340	—	178	2	no	—
147	#336	—	2	1	no	—
148	#335	—	89	4	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	178	2	no	—
151	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	178	2	no	—
154	C₂×C₄⋊C₄	—	178	2	no	—
155	C₂×C₄⋊C₄	—	178	2	no	—
156	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
157	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
160	#107	—	356	2	no	—
161	#107	—	356	2	no	—
162	#107	—	356	4	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	89	4	no	—
165	C₄⋊C₈	—	178	2	no	—
166	C₄⋊C₈	Dic₁₇₈	1	2	yes	—
167	C₄×C₈	—	4	2	no	—
168	C₄×C₈	Dic₁₇₈	1	2	yes	—
169	C₄×C₈	C₄×D₈₉	1	2	yes	—
170	#95	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	356	2	no	—
174	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
176	C₂²×C₄	#13083574	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	356	2	no	—
183	C₂×C₈	—	356	2	no	—
184	C₂×C₈	—	178	4	no	—
185	C₂×C₈	#13083553	1	2	yes	—
186	C₂×C₈	#13083553	1	2	yes	—
187	C₂×C₈	#13083559	1	2	yes	—
188	C₂×C₈	#13083564	1	2	yes	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	356	2	no	—
192	C₄⋊C₄	—	356	2	no	—
193	C₄⋊C₄	—	356	2	no	—
194	C₄⋊C₄	—	356	2	no	—
195	C₄⋊C₄	—	712	2	no	—
196	C₄⋊C₄	D₇₁₂	1	2	yes	—
197	C₄⋊C₄	Dic₃₅₆	1	2	yes	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
201	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
202	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
203	C₄²	#13083574	1	1	yes	characteristic
204	C₂³	#15667986	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	712	2	no	—
214	C₂×C₄	—	712	2	no	—
215	C₂×C₄	—	712	2	no	—
216	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
217	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
218	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
219	C₂×C₄	#15667991	1	1	yes	characteristic
220	C₂×C₄	#15668004	1	1	yes	characteristic
221	C₂×C₄	#15668012	1	1	yes	characteristic
222	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
223	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
224	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
226	C₂×C₄	#15668034	1	1	yes	characteristic
227	C₂×C₄	#15668034	1	1	yes	characteristic
228	C₂×C₄	#15668037	1	1	yes	characteristic
229	C₂×C₄	#15668042	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	356	4	no	—
233	C₈	#15667945	1	4	yes	complemented
234	C₂²	#19151496	1	1	yes	characteristic
235	C₂²	#19151793	1	1	yes	characteristic
236	C₂²	#19151808	1	1	yes	characteristic
237	C₂²	#19151834	1	1	yes	characteristic
238	C₂²	#19151839	1	1	yes	characteristic
239	C₂²	#19151868	1	1	yes	characteristic
240	C₂²	#19151873	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	1424	2	no	—
246	C₄	C₂×#15667945	1	1	yes	characteristic
247	C₄	#19151641	1	1	yes	characteristic
248	C₄	C₄×D₇₁₂	1	1	yes	characteristic
249	C₄	C₄×Dic₃₅₆	1	1	yes	characteristic
250	C₄	#19151878	1	1	yes	characteristic
251	C₄	#19151893	1	1	yes	characteristic
252	C₄	#19152000	1	1	yes	characteristic
253	C₄	#19152003	1	1	yes	characteristic
254	C₂	#26019607	1	1	yes	characteristic
255	C₂	#26019617	1	1	yes	characteristic
256	C₂	#26019646	1	1	yes	characteristic
257	C₂	#26019656	1	1	yes	characteristic
258	C₂	#26020985	1	1	yes	characteristic
259	C₂	#26021010	1	1	yes	characteristic
260	C₂	#26021037	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈×C₄⋊C₄×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#9614
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #19151351	C₄	semidirect
101. C₈×C₈₉	#107	semidirect
136. C₈₉	#9614	semidirect
233. 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
#26020412	2
C₂×#19151351	2
#26018136	2
C₈×C₄⋊C₄×C₈₉	2
#9614	89