Subgroups — #36452981

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	#26020971	C₂	1	2	yes	maximal
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
4	#26018093	C₂	1	2	yes	maximal
5	C₄×#106×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	#19151351	—	2	2	no	—
12	#19151319	—	2	2	no	—
13	C₂×#106×C₈₉	C₄	1	1	yes	characteristic
14	C₂×#106×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	1	yes	characteristic
21	C₂²×#10623494	D₄	1	1	yes	characteristic
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	4	4	no	—
25	C₄×#10623494	—	2	1	no	—
26	C₄×#10623494	—	2	1	no	—
27	C₄×#10623494	—	2	1	no	—
28	C₄×#10623494	Q₈	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₈₉	—	2	2	no	—
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	#106×C₈₉	—	2	2	no	—
36	#106×C₈₉	C₈	1	4	yes	complemented
37	C₄×C₈×C₈₉	D₄	1	2	yes	—
38	C₄×C₈×C₈₉	Q₈	1	2	yes	—
39	#95×C₈₉	—	2	2	no	—
40	C₂×#10623494	—	2	1	no	—
41	C₂×#10623494	—	2	1	no	—
42	C₂×#10623494	—	2	2	no	—
43	C₂×#10623494	—	2	2	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	—	2	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	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
60	C₄⋊C₄×C₈₉	—	2	2	no	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
64	C₄²×C₈₉	—	4	2	no	—
65	C₄²×C₈₉	—	4	2	no	—
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	—	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	1	no	—
94	#10623494	—	4	1	no	—
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	—	4	4	no	—
104	C₄×C₈₉	—	8	2	no	—
105	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₈₉	#424	1	1	yes	characteristic
107	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
108	C₄×C₈₉	#446	1	1	yes	characteristic
109	C₄×C₈₉	#469	1	1	yes	characteristic
110	C₄×C₈₉	#470	1	1	yes	characteristic
111	C₄×C₈₉	#501	1	2	yes	—
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	D₈₉	1	1	yes	characteristic
120	C₂×#335	—	89	1	no	—
121	C₂×#322	—	89	1	no	—
122	#1279	—	89	2	no	—
123	C₈₉	#9596	1	1	yes	characteristic, complemented, Sylow
124	C₂×#106	Dic₈₉	1	1	yes	characteristic
125	C₂×#106	Dic₈₉	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	89	2	no	—
127	C₂×C₄⋊C₈	—	178	2	no	—
128	C₂×C₄×C₈	—	89	1	no	—
129	C₂×C₄×C₈	—	89	1	no	—
130	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
131	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
132	#340	Dic₈₉	1	2	yes	—
133	#336	—	178	2	no	—
134	#335	—	178	2	no	—
135	#322	—	178	2	no	—
136	C₂²×C₈	—	89	1	no	—
137	C₂²×C₈	—	89	1	no	—
138	C₂²×C₈	—	178	2	no	—
139	C₂²×C₈	—	178	1	no	—
140	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
146	#106	—	2	2	no	—
147	#106	#10623494	1	4	yes	complemented
148	C₄⋊C₈	—	356	4	no	—
149	C₄⋊C₈	—	178	2	no	—
150	C₄⋊C₈	—	178	2	no	—
151	C₄×C₈	—	89	2	no	—
152	C₄×C₈	—	178	1	no	—
153	C₄×C₈	—	178	1	no	—
154	C₄×C₈	—	178	1	no	—
155	C₄×C₈	Dic₁₇₈	1	2	yes	—
156	C₄×C₈	D₃₅₆	1	2	yes	—
157	#95	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
162	C₂²×C₄	#13083573	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	356	2	no	—
166	C₂×C₈	—	356	2	no	—
167	C₂×C₈	—	356	2	no	—
168	C₂×C₈	—	356	2	no	—
169	C₂×C₈	—	178	1	no	—
170	C₂×C₈	—	178	1	no	—
171	C₂×C₈	—	178	2	no	—
172	C₂×C₈	—	178	2	no	—
173	C₂×C₈	—	178	2	no	—
174	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
175	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
176	C₂×C₈	C₂×#10623494	1	2	yes	—
177	C₂×C₈	#13083558	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂×#10623494	1	4	yes	—
182	C₄²	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
185	C₄²	Q₈×D₈₉	1	1	yes	characteristic
186	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
187	C₄²	#13083575	1	1	yes	characteristic
188	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
197	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
198	C₂×C₄	#15667987	1	1	yes	characteristic
199	C₂×C₄	#15667992	1	1	yes	characteristic
200	C₂×C₄	#15668003	1	1	yes	characteristic
201	C₂×C₄	#15668011	1	1	yes	characteristic
202	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
203	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
204	C₂×C₄	#15668019	1	1	yes	characteristic
205	C₂×C₄	#15668019	1	1	yes	characteristic
206	C₂×C₄	#15668039	1	2	yes	—
207	C₂×C₄	#15668041	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	356	2	no	—
210	C₈	—	356	1	no	—
211	C₈	—	356	1	no	—
212	C₈	—	712	2	no	—
213	C₈	C₄⋊#10623494	1	4	yes	complemented
214	C₂²	#19151497	1	1	yes	characteristic
215	C₂²	D₄×#10623494	1	1	yes	characteristic
216	C₂²	#19151676	1	1	yes	characteristic
217	C₂²	#19151699	1	2	yes	—
218	C₂²	#19151720	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
224	C₄	#19151625	1	1	yes	characteristic
225	C₄	Q₈×#10623494	1	1	yes	characteristic
226	C₄	#19151729	1	1	yes	characteristic
227	C₄	#19151852	1	1	yes	characteristic
228	C₄	#19151857	1	1	yes	characteristic
229	C₄	#19152005	1	2	yes	—
230	C₂	SD₁₆⋊#10623494	1	2	yes	—
231	C₂	#26019155	1	2	yes	—
232	C₂	#26020971	1	1	yes	characteristic
233	C₂	#26020996	1	1	yes	characteristic
234	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×#106×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#106	D₈₉
hypocenter	C₈₉	#9596
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
36. #106×C₈₉	C₈	semidirect
91. C₈×C₈₉	C₄⋊C₈	semidirect
123. C₈₉	#9596	semidirect
147. #106	#10623494	semidirect
213. 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×#19151351	2
C₂×#19151319	2
#26018093	2
C₄×#106×C₈₉	2
#9596	89