Subgroups — #36452980

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020970	C₂	1	2	yes	maximal
2	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
4	#26018092	C₂	1	2	yes	maximal
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667944	C₄	1	1	yes	characteristic
7	C₂×#15667944	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×#15667941	—	2	2	no	—
10	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	#19151361	C₄	1	2	yes	—
12	#19151352	—	2	2	no	—
13	#19151351	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#322×C₈₉	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₈₉	—	2	2	no	—
22	#15667944	—	2	2	no	—
23	#15667944	C₈	1	4	yes	complemented
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	#15667941	—	4	4	no	—
27	#15667919	—	2	2	no	—
28	C₂²×C₈×C₈₉	—	2	1	no	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
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	1	no	—
35	C₄×C₈×C₈₉	—	2	1	no	—
36	C₄×C₈×C₈₉	—	2	1	no	—
37	C₄×C₈×C₈₉	D₄	1	2	yes	—
38	C₄×C₈×C₈₉	Q₈	1	2	yes	—
39	C₄×C₈×C₈₉	Q₈	1	2	yes	—
40	C₂²×Dic₈₉	—	2	2	no	—
41	C₂²×Dic₈₉	—	2	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₂×#10623494	—	4	2	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	—	2	2	no	—
48	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
49	C₄×Dic₈₉	—	4	2	no	—
50	C₄×Dic₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	C₂²×C₄	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	1	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	—	2	2	no	—
60	C₂×C₈×C₈₉	—	2	2	no	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
66	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₈₉	#41	1	1	yes	characteristic
70	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
82	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₈₉	#137	1	2	yes	—
84	C₂×Dic₈₉	—	2	4	no	—
85	C₂×Dic₈₉	—	4	2	no	—
86	C₂×Dic₈₉	—	4	2	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	C₂×Dic₈₉	—	4	2	no	—
90	C₈×C₈₉	—	2	2	no	—
91	C₈×C₈₉	—	4	2	no	—
92	C₈×C₈₉	—	4	1	no	—
93	C₈×C₈₉	—	4	1	no	—
94	C₈×C₈₉	C₄⋊C₈	1	4	yes	complemented
95	#10623494	—	8	2	no	—
96	C₂²×C₈₉	#389	1	1	yes	characteristic
97	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
98	C₂²×C₈₉	#436	1	1	yes	characteristic
99	C₂²×C₈₉	#440	1	2	yes	—
100	C₂²×C₈₉	#444	1	2	yes	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₈₉	#424	1	1	yes	characteristic
105	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
106	C₄×C₈₉	#446	1	1	yes	characteristic
107	C₄×C₈₉	#469	1	1	yes	characteristic
108	C₄×C₈₉	#470	1	1	yes	characteristic
109	C₄×C₈₉	#501	1	2	yes	—
110	Dic₈₉	—	4	4	no	—
111	Dic₈₉	—	8	2	no	—
112	#9596	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	SD₁₆⋊C₈	1	2	yes	—
114	C₂×C₈₉	#1592	1	2	yes	—
115	C₂×C₈₉	#1939	1	1	yes	characteristic
116	C₂×C₈₉	#1944	1	1	yes	characteristic
117	C₂×C₈₉	#1946	1	1	yes	characteristic
118	#1939	—	89	2	no	—
119	C₄×#106	—	89	1	no	—
120	C₂×#335	—	89	1	no	—
121	C₂×#322	D₈₉	1	1	yes	characteristic
122	#1279	—	89	2	no	—
123	C₈₉	#9596	1	1	yes	characteristic, complemented, Sylow
124	C₂×#106	—	89	1	no	—
125	C₂×#106	—	89	1	no	—
126	C₂×C₄⋊C₈	—	89	2	no	—
127	C₂×C₄⋊C₈	—	178	2	no	—
128	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
131	C₄×C₄⋊C₄	—	89	2	no	—
132	#340	—	89	2	no	—
133	#336	—	178	2	no	—
134	#335	—	178	2	no	—
135	#322	—	2	2	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	178	2	no	—
138	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
139	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
140	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
141	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	89	2	no	—
143	C₂×C₄⋊C₄	—	89	2	no	—
144	C₂×C₄²	—	178	2	no	—
145	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
146	#106	—	89	4	no	—
147	#106	—	178	2	no	—
148	C₄⋊C₈	—	356	4	no	—
149	C₄⋊C₈	—	178	2	no	—
150	C₄⋊C₈	—	178	2	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	Dic₁₇₈	1	2	yes	—
155	C₄×C₈	Dic₁₇₈	1	2	yes	—
156	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
157	#95	—	178	2	no	—
158	C₂²×C₄	—	178	2	no	—
159	C₂²×C₄	—	178	2	no	—
160	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
162	C₂²×C₄	#13083574	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	356	2	no	—
172	C₂×C₈	—	356	2	no	—
173	C₂×C₈	—	356	2	no	—
174	C₂×C₈	#13083553	1	2	yes	—
175	C₂×C₈	#13083553	1	2	yes	—
176	C₂×C₈	C₈×D₈₉	1	2	yes	—
177	C₂×C₈	#13083557	1	2	yes	—
178	C₄⋊C₄	—	89	4	no	—
179	C₄⋊C₄	—	178	2	no	—
180	C₄⋊C₄	—	178	2	no	—
181	C₄⋊C₄	—	178	2	no	—
182	C₄²	—	356	2	no	—
183	C₄²	—	356	2	no	—
184	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
185	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
186	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
187	C₄²	#13083574	1	1	yes	characteristic
188	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	356	2	no	—
191	C₂×C₄	—	356	2	no	—
192	C₂×C₄	—	356	2	no	—
193	C₂×C₄	—	356	2	no	—
194	C₂×C₄	—	356	2	no	—
195	C₂×C₄	—	178	4	no	—
196	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
197	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
198	C₂×C₄	#15667986	1	1	yes	characteristic
199	C₂×C₄	#15667991	1	1	yes	characteristic
200	C₂×C₄	#15668004	1	1	yes	characteristic
201	C₂×C₄	#15668012	1	1	yes	characteristic
202	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
204	C₂×C₄	#15668018	1	1	yes	characteristic
205	C₂×C₄	#15668018	1	1	yes	characteristic
206	C₂×C₄	#15668040	1	2	yes	—
207	C₂×C₄	#15668045	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	712	2	no	—
213	C₈	#15667941	1	4	yes	complemented
214	C₂²	#19151496	1	1	yes	characteristic
215	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
216	C₂²	#19151677	1	1	yes	characteristic
217	C₂²	#19151700	1	2	yes	—
218	C₂²	#19151721	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	356	4	no	—
222	C₄	—	712	2	no	—
223	C₄	C₂×#15667941	1	1	yes	characteristic
224	C₄	#19151624	1	1	yes	characteristic
225	C₄	C₈×Dic₁₇₈	1	1	yes	characteristic
226	C₄	#19151728	1	1	yes	characteristic
227	C₄	#19151851	1	1	yes	characteristic
228	C₄	#19151856	1	1	yes	characteristic
229	C₄	#19152007	1	2	yes	—
230	C₂	#26019066	1	2	yes	—
231	C₂	#26019156	1	2	yes	—
232	C₂	#26020970	1	1	yes	characteristic
233	C₂	#26020995	1	1	yes	characteristic
234	C₂	#26021005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9596
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
23. #15667944	C₈	semidirect
94. C₈×C₈₉	C₄⋊C₈	semidirect
123. C₈₉	#9596	semidirect
213. C₈	#15667941	semidirect

Maximal subgroups

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

subgroup	index
#26020970	2
C₄×#15667944	2
C₂×#19151351	2
#26018092	2
C₂×#322×C₈₉	2
#9596	89