Subgroups — #36459346

Order 22784.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021016	C₂	1	2	yes	maximal
2	#26020970	C₂	1	1	yes	maximal, characteristic
3	#26020412	C₂	1	1	yes	maximal, characteristic
4	#26018136	C₂	1	2	yes	maximal
5	C₈₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	1	no	—
7	C₂×#15667946	—	2	1	no	—
8	C₂×#15667944	—	2	2	no	—
9	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₂×#15667941	—	2	1	no	—
11	#19151468	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
13	#19151361	—	2	2	no	—
14	#19151352	—	2	1	no	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	1	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₉	—	2	1	no	—
24	#15667946	—	4	2	no	—
25	#15667946	—	4	2	no	—
26	#15667944	—	4	4	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	#15667941	—	4	2	no	—
29	#15667919	—	2	1	no	—
30	C₂²×C₈×C₈₉	—	2	2	no	—
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₄⋊C₈×C₈₉	Q₈	1	4	yes	—
36	C₄×C₈×C₈₉	—	2	2	no	—
37	C₂²×Dic₈₉	—	2	1	no	—
38	C₂²×Dic₈₉	—	2	1	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₂×#10623494	—	4	1	no	—
41	C₂×#10623494	—	4	1	no	—
42	C₂×#10623494	—	4	1	no	—
43	C₄⋊Dic₈₉	—	2	1	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	C₄⋊Dic₈₉	—	8	2	no	—
50	C₄⋊Dic₈₉	D₈	1	2	yes	—
51	C₄×Dic₈₉	—	4	1	no	—
52	C₄×Dic₈₉	—	4	1	no	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	2	1	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	—	2	1	no	—
59	C₂×C₈×C₈₉	—	2	1	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
64	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
65	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
66	C₄²×C₈₉	#41	1	2	yes	—
67	C₂³×C₈₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#125	1	2	yes	—
73	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
78	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#137	1	2	yes	—
80	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
81	C₂×Dic₈₉	—	2	2	no	—
82	C₂×Dic₈₉	—	4	1	no	—
83	C₂×Dic₈₉	—	4	1	no	—
84	C₂×Dic₈₉	—	4	1	no	—
85	C₂×Dic₈₉	—	4	1	no	—
86	C₂×Dic₈₉	—	4	1	no	—
87	C₂×Dic₈₉	—	8	2	no	—
88	C₂×Dic₈₉	—	8	2	no	—
89	C₂×Dic₈₉	—	8	2	no	—
90	C₈×C₈₉	—	2	2	no	—
91	C₈×C₈₉	—	2	2	no	—
92	C₈×C₈₉	—	4	4	no	—
93	#10623494	—	8	1	no	—
94	C₂²×C₈₉	#398	1	1	yes	characteristic
95	C₂²×C₈₉	#460	1	1	yes	characteristic
96	C₂²×C₈₉	#461	1	1	yes	characteristic
97	C₂²×C₈₉	#466	1	1	yes	characteristic
98	C₂²×C₈₉	#470	1	1	yes	characteristic
99	C₂²×C₈₉	#472	1	1	yes	characteristic
100	C₂²×C₈₉	#473	1	1	yes	characteristic
101	C₄×C₈₉	—	2	2	no	—
102	C₄×C₈₉	#474	1	2	yes	—
103	C₄×C₈₉	#479	1	2	yes	—
104	C₄×C₈₉	#481	1	1	yes	characteristic
105	C₄×C₈₉	#483	1	1	yes	characteristic
106	C₄×C₈₉	#498	1	1	yes	characteristic
107	C₄×C₈₉	#501	1	1	yes	characteristic
108	Dic₈₉	—	4	2	no	—
109	Dic₈₉	—	8	1	no	—
110	Dic₈₉	—	16	2	no	—
111	#10651	—	89	1	no	maximal, Sylow
112	C₂×C₈₉	#1668	1	1	yes	characteristic
113	C₂×C₈₉	#1673	1	1	yes	characteristic
114	C₂×C₈₉	#1690	1	1	yes	characteristic
115	C₂×C₈₉	#1695	1	1	yes	characteristic
116	C₂×C₈₉	#2084	1	1	yes	characteristic
117	C₂×C₈₉	#2086	1	1	yes	characteristic
118	C₂×C₈₉	#2089	1	1	yes	characteristic
119	#1948	—	89	2	no	—
120	#1939	—	89	1	no	—
121	#1845	—	89	1	no	—
122	#1296	—	89	2	no	—
123	#1277	D₈₉	1	1	yes	characteristic
124	C₈₉	#10651	1	1	yes	characteristic, complemented, Sylow
125	C₂×#107	—	178	1	no	—
126	C₂×#107	—	178	1	no	—
127	C₂×#106	—	178	2	no	—
128	C₂×C₄⋊C₈	—	89	1	no	—
129	C₂×C₄⋊C₈	—	178	1	no	—
130	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
132	#384	—	89	2	no	—
133	C₄×C₄⋊C₄	—	89	1	no	—
134	#340	—	178	2	no	—
135	#336	—	178	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	178	1	no	—
138	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
139	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	89	1	no	—
141	C₂×C₄⋊C₄	—	89	1	no	—
142	C₂×C₄⋊C₄	—	178	2	no	—
143	C₂×C₄⋊C₄	—	178	2	no	—
144	C₂×C₄⋊C₄	—	178	2	no	—
145	C₂×C₄²	—	178	1	no	—
146	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
147	#107	—	356	2	no	—
148	#107	—	356	2	no	—
149	#106	—	356	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	356	2	no	—
152	C₄⋊C₈	—	178	2	no	—
153	C₄⋊C₈	Dic₁₇₈	1	4	yes	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	178	1	no	—
156	C₂²×C₄	—	356	2	no	—
157	C₂²×C₄	—	178	1	no	—
158	C₂²×C₄	—	178	1	no	—
159	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
161	C₂²×C₄	#13083575	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	356	1	no	—
170	C₂×C₈	—	356	1	no	—
171	C₂×C₈	—	356	1	no	—
172	C₂×C₈	#13083563	1	2	yes	—
173	C₂×C₈	#13083563	1	2	yes	—
174	C₄⋊C₄	—	89	2	no	—
175	C₄⋊C₄	—	356	2	no	—
176	C₄⋊C₄	—	356	2	no	—
177	C₄⋊C₄	—	356	2	no	—
178	C₄⋊C₄	—	356	2	no	—
179	C₄⋊C₄	—	178	2	no	—
180	C₄⋊C₄	—	178	1	no	—
181	C₄⋊C₄	—	712	2	no	—
182	C₄²	—	356	1	no	—
183	C₄²	—	356	1	no	—
184	C₄²	C₂×Dic₁₇₈	1	2	yes	—
185	C₄²	#13083573	1	2	yes	—
186	C₂³	#15667987	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	356	1	no	—
190	C₂×C₄	—	356	1	no	—
191	C₂×C₄	—	356	1	no	—
192	C₂×C₄	—	356	1	no	—
193	C₂×C₄	—	356	1	no	—
194	C₂×C₄	—	178	2	no	—
195	C₂×C₄	—	712	2	no	—
196	C₂×C₄	—	712	2	no	—
197	C₂×C₄	—	712	2	no	—
198	C₂×C₄	#15667989	1	2	yes	—
199	C₂×C₄	#15667996	1	1	yes	characteristic
200	C₂×C₄	#15668003	1	2	yes	—
201	C₂×C₄	#15668012	1	1	yes	characteristic
202	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
204	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
205	C₂×C₄	#15668035	1	1	yes	characteristic
206	C₂×C₄	#15668040	1	1	yes	characteristic
207	C₂×C₄	#15668041	1	2	yes	—
208	C₂×C₄	#15668045	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	712	1	no	—
213	C₂²	#19151530	1	1	yes	characteristic
214	C₂²	#19151799	1	1	yes	characteristic
215	C₂²	#19151804	1	1	yes	characteristic
216	C₂²	#19151835	1	1	yes	characteristic
217	C₂²	#19151857	1	1	yes	characteristic
218	C₂²	#19151869	1	1	yes	characteristic
219	C₂²	#19151874	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	356	2	no	—
222	C₄	—	1424	2	no	—
223	C₄	—	712	1	no	—
224	C₄	#19151876	1	2	yes	—
225	C₄	#19151907	1	2	yes	—
226	C₄	#19151917	1	1	yes	characteristic
227	C₄	#19151929	1	1	yes	characteristic
228	C₄	#19151996	1	1	yes	characteristic
229	C₄	#19152007	1	1	yes	characteristic
230	C₂	#26019613	1	1	yes	characteristic
231	C₂	#26019642	1	1	yes	characteristic
232	C₂	#26019747	1	1	yes	characteristic
233	C₂	#26019776	1	1	yes	characteristic
234	C₂	#26021730	1	1	yes	characteristic
235	C₂	#26021742	1	1	yes	characteristic
236	C₂	#26021759	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1277	C₂
socle, prime radical	C₂³×C₈₉	#121
hypercenter	#1277	D₈₉
hypocenter	C₈₉	#10651
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
124. C₈₉	#10651	semidirect

Maximal subgroups

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

subgroup	index
#26021016	2
#26020970	2
#26020412	2
#26018136	2
C₈₉×#1277	2
#10651	89