Subgroups — #36452976

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020971	C₂	1	2	yes	maximal
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
4	#26018093	C₂	1	2	yes	maximal
5	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	—	2	2	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151353	—	2	2	no	—
11	#19151349	—	2	2	no	—
12	#19151319	—	2	2	no	—
13	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
14	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
17	#340×C₈₉	C₄	1	2	yes	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂²×#10623494	—	2	1	no	—
20	C₂²×#10623494	D₄	1	2	yes	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	4	4	no	—
24	C₄×#10623494	—	2	1	no	—
25	C₄×#10623494	—	2	1	no	—
26	C₄×#10623494	—	2	1	no	—
27	C₄×#10623494	Q₈	1	2	yes	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₉	—	2	2	no	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	#107×C₈₉	—	2	2	no	—
35	#107×C₈₉	C₈	1	4	yes	complemented
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₄×C₈×C₈₉	Q₈	1	2	yes	—
38	#95×C₈₉	—	2	2	no	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₂×#10623494	—	2	2	no	—
42	C₂×#10623494	—	2	2	no	—
43	C₂×#10623494	—	4	1	no	—
44	C₂×#10623494	—	4	1	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₉	—	2	1	no	—
54	C₂×C₈×C₈₉	—	2	1	no	—
55	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	2	2	no	—
60	C₄⋊C₄×C₈₉	—	2	2	no	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
63	C₄²×C₈₉	—	4	2	no	—
64	C₄²×C₈₉	—	4	2	no	—
65	C₄²×C₈₉	C₂×D₄	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	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	—	4	2	no	—
78	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
88	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₈₉	#137	1	2	yes	—
90	C₈×C₈₉	—	2	2	no	—
91	C₈×C₈₉	C₄⋊C₈	1	4	yes	complemented
92	#10623494	—	4	2	no	—
93	#10623494	—	4	2	no	—
94	#10623494	—	8	2	no	—
95	C₂²×C₈₉	#389	1	1	yes	characteristic
96	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
97	C₂²×C₈₉	#436	1	1	yes	characteristic
98	C₂²×C₈₉	#441	1	1	yes	characteristic
99	C₂²×C₈₉	#442	1	1	yes	characteristic
100	C₂²×C₈₉	#444	1	1	yes	characteristic
101	C₂²×C₈₉	#444	1	1	yes	characteristic
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	—	4	4	no	—
105	C₄×C₈₉	—	8	2	no	—
106	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₈₉	#424	1	1	yes	characteristic
108	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
109	C₄×C₈₉	#446	1	1	yes	characteristic
110	C₄×C₈₉	#468	1	2	yes	—
111	C₄×C₈₉	#501	1	2	yes	—
112	#9595	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	#1579	1	1	yes	characteristic
114	C₂×C₈₉	#1580	1	1	yes	characteristic
115	C₂×C₈₉	#1593	1	1	yes	characteristic
116	C₂×C₈₉	#1594	1	1	yes	characteristic
117	C₂×C₈₉	#1939	1	1	yes	characteristic
118	C₂×C₈₉	#1943	1	1	yes	characteristic
119	C₂×C₈₉	#1946	1	1	yes	characteristic
120	#1939	—	89	2	no	—
121	C₄×#107	D₈₉	1	1	yes	characteristic
122	C₂×#334	—	89	1	no	—
123	C₂×#322	—	89	1	no	—
124	#1279	—	89	2	no	—
125	C₈₉	#9595	1	1	yes	characteristic, complemented, Sylow
126	C₂×#107	Dic₈₉	1	1	yes	characteristic
127	C₂×#107	Dic₈₉	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	89	2	no	—
129	C₂×C₄⋊C₈	—	178	2	no	—
130	C₂×C₄×C₈	—	89	1	no	—
131	C₂×C₄×C₈	—	89	1	no	—
132	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
133	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
134	#340	Dic₈₉	1	2	yes	—
135	#336	—	178	2	no	—
136	#334	—	178	2	no	—
137	#322	—	178	2	no	—
138	C₂²×C₈	—	89	2	no	—
139	C₂²×C₈	—	178	2	no	—
140	C₂²×C₈	—	178	1	no	—
141	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
145	C₂×C₄²	—	2	2	no	—
146	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
147	#107	—	2	2	no	—
148	#107	#10623494	1	4	yes	complemented
149	C₄⋊C₈	—	356	4	no	—
150	C₄⋊C₈	—	178	2	no	—
151	C₄⋊C₈	—	178	2	no	—
152	C₄×C₈	—	89	2	no	—
153	C₄×C₈	—	178	1	no	—
154	C₄×C₈	—	178	1	no	—
155	C₄×C₈	—	178	1	no	—
156	C₄×C₈	Dic₁₇₈	1	2	yes	—
157	C₄×C₈	D₃₅₆	1	2	yes	—
158	#95	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
162	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
163	C₂²×C₄	#13083573	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	356	1	no	—
167	C₂×C₈	—	356	1	no	—
168	C₂×C₈	—	356	2	no	—
169	C₂×C₈	—	356	2	no	—
170	C₂×C₈	—	356	2	no	—
171	C₂×C₈	—	178	2	no	—
172	C₂×C₈	—	178	2	no	—
173	C₂×C₈	—	178	2	no	—
174	C₂×C₈	—	178	2	no	—
175	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
176	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
177	C₂×C₈	C₂×#10623494	1	2	yes	—
178	C₂×C₈	#13083558	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₂×#10623494	1	4	yes	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
186	C₄²	Q₈×D₈₉	1	1	yes	characteristic
187	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
188	C₄²	#13083575	1	1	yes	characteristic
189	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
199	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
200	C₂×C₄	#15667987	1	1	yes	characteristic
201	C₂×C₄	#15667992	1	1	yes	characteristic
202	C₂×C₄	#15668003	1	1	yes	characteristic
203	C₂×C₄	#15668011	1	1	yes	characteristic
204	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
205	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
206	C₂×C₄	#15668019	1	1	yes	characteristic
207	C₂×C₄	#15668019	1	1	yes	characteristic
208	C₂×C₄	#15668036	1	2	yes	—
209	C₂×C₄	#15668044	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	—	356	2	no	—
212	C₈	—	356	2	no	—
213	C₈	—	712	2	no	—
214	C₈	C₄⋊#10623494	1	4	yes	complemented
215	C₂²	#19151497	1	1	yes	characteristic
216	C₂²	D₄×#10623494	1	1	yes	characteristic
217	C₂²	#19151676	1	1	yes	characteristic
218	C₂²	#19151705	1	1	yes	characteristic
219	C₂²	#19151710	1	1	yes	characteristic
220	C₂²	#19151720	1	1	yes	characteristic
221	C₂²	#19151720	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
227	C₄	#19151625	1	1	yes	characteristic
228	C₄	Q₈×#10623494	1	1	yes	characteristic
229	C₄	#19151729	1	1	yes	characteristic
230	C₄	#19151845	1	2	yes	—
231	C₄	#19152006	1	2	yes	—
232	C₂	#26019071	1	1	yes	characteristic
233	C₂	#26019076	1	1	yes	characteristic
234	C₂	#26019161	1	1	yes	characteristic
235	C₂	#26019166	1	1	yes	characteristic
236	C₂	#26020971	1	1	yes	characteristic
237	C₂	#26020991	1	1	yes	characteristic
238	C₂	#26021006	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#9595
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
35. #107×C₈₉	C₈	semidirect
91. C₈×C₈₉	C₄⋊C₈	semidirect
125. C₈₉	#9595	semidirect
148. #107	#10623494	semidirect
214. C₈	C₄⋊#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26020971	2
C₂×#19151349	2
C₂×#19151319	2
#26018093	2
C₄×#107×C₈₉	2
#9595	89