Subgroups — #36447369

Order 22784.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020248	C₂	1	2	yes	maximal
2	#26019011	C₂	1	1	yes	maximal, characteristic
3	#26019008	C₂	1	1	yes	maximal, characteristic
4	#26018127	C₂	1	2	yes	maximal
5	C₂×#446×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151622	C₂²	1	2	yes	—
7	#19151622	C₂²	1	2	yes	—
8	C₂×#15667931	—	2	2	no	—
9	#19151367	—	2	2	no	—
10	#19151351	—	2	1	no	—
11	#19151351	—	2	1	no	—
12	#19151349	—	2	1	no	—
13	#19151349	—	2	1	no	—
14	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	#446×C₈₉	—	2	1	no	—
16	#446×C₈₉	—	2	1	no	—
17	#446×C₈₉	—	2	1	no	—
18	#446×C₈₉	—	2	1	no	—
19	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
20	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#97×C₈₉	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₄⋊#10623494	—	2	2	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	C₄⋊#10623494	—	2	2	no	—
29	#15667931	—	4	4	no	—
30	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₈₉	—	2	1	no	—
35	C₄×Q₈×C₈₉	—	2	1	no	—
36	C₄×Q₈×C₈₉	—	2	1	no	—
37	C₄×Q₈×C₈₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
40	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₈₉	—	2	2	no	—
43	C₄⋊C₈×C₈₉	—	2	2	no	—
44	C₄⋊C₈×C₈₉	—	2	1	no	—
45	C₄⋊C₈×C₈₉	—	2	1	no	—
46	#97×C₈₉	—	2	2	no	—
47	#97×C₈₉	—	2	2	no	—
48	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	#13083558	—	4	2	no	—
53	#13083558	—	4	2	no	—
54	#13083558	—	4	2	no	—
55	#13083558	—	4	2	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×Q₈×C₈₉	—	2	1	no	—
59	C₂×Q₈×C₈₉	—	2	1	no	—
60	C₂×Q₈×C₈₉	—	2	2	no	—
61	C₂×Q₈×C₈₉	—	2	2	no	—
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	4	1	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₈₉	—	2	1	no	—
76	C₄⋊C₄×C₈₉	—	2	1	no	—
77	C₄⋊C₄×C₈₉	—	2	2	no	—
78	C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	—	2	2	no	—
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	#41	1	1	yes	characteristic
84	C₄²×C₈₉	#41	1	1	yes	characteristic
85	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Q₈×C₈₉	—	4	2	no	—
87	Q₈×C₈₉	—	4	1	no	—
88	Q₈×C₈₉	—	4	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	4	no	—
94	C₂×C₄×C₈₉	—	4	1	no	—
95	C₂×C₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₈₉	#137	1	2	yes	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	—	2	1	no	—
106	C₈×C₈₉	—	2	1	no	—
107	C₈×C₈₉	—	4	2	no	—
108	C₈×C₈₉	#101	1	2	yes	complemented
109	#10623494	—	8	2	no	—
110	#10623494	—	8	2	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	#380	1	1	yes	characteristic
114	C₂²×C₈₉	#428	1	2	yes	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	4	2	no	—
119	C₄×C₈₉	—	4	2	no	—
120	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
121	C₄×C₈₉	#413	1	1	yes	characteristic
122	C₄×C₈₉	#468	1	2	yes	—
123	C₄×C₈₉	#469	1	1	yes	characteristic
124	C₄×C₈₉	#470	1	1	yes	characteristic
125	#8619	—	89	1	no	maximal, Sylow
126	C₂×C₈₉	—	2	2	no	—
127	C₂×C₈₉	#1801	1	1	yes	characteristic
128	C₂×C₈₉	#1809	1	1	yes	characteristic
129	C₂×C₈₉	#1813	1	1	yes	characteristic
130	C₂×#446	D₈₉	1	1	yes	characteristic
131	#1801	—	89	2	no	—
132	#1567	—	89	1	no	—
133	#1566	—	89	1	no	—
134	#1292	—	89	2	no	—
135	C₈₉	#8619	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
137	#446	—	2	1	no	—
138	#446	—	2	1	no	—
139	#446	—	2	1	no	—
140	#446	—	2	1	no	—
141	#423	—	89	2	no	—
142	#423	—	89	2	no	—
143	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
144	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
145	C₂×#101	—	178	2	no	—
146	C₂×#97	Dic₈₉	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	#343	—	178	2	no	—
149	#335	—	178	1	no	—
150	#335	—	178	1	no	—
151	#334	—	178	1	no	—
152	#334	—	178	1	no	—
153	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
154	C₂×M₁₆	—	178	2	no	—
155	C₂×M₁₆	—	178	2	no	—
156	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
158	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
164	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
166	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	178	2	no	—
172	C₄⋊C₈	—	178	2	no	—
173	C₄⋊C₈	—	178	2	no	—
174	C₄⋊C₈	—	178	2	no	—
175	#101	—	356	4	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
191	M₁₆	—	356	2	no	—
192	M₁₆	—	356	2	no	—
193	M₁₆	—	356	2	no	—
194	M₁₆	—	356	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	356	2	no	—
200	C₂×C₈	—	356	2	no	—
201	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
213	C₄²	#13083574	1	1	yes	characteristic
214	C₄²	#13083574	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
227	C₂×C₄	#15667971	1	1	yes	characteristic
228	C₂×C₄	#15667986	1	1	yes	characteristic
229	C₂×C₄	#15667986	1	1	yes	characteristic
230	C₂×C₄	#15667991	1	1	yes	characteristic
231	C₂×C₄	#15667991	1	1	yes	characteristic
232	C₂×C₄	#15668040	1	2	yes	—
233	C₂×C₄	#15668045	1	2	yes	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	712	2	no	—
239	C₈	—	712	2	no	—
240	C₈	#15667931	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#19151458	1	1	yes	characteristic
244	C₂²	#19151640	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂×#15667931	1	1	yes	characteristic
251	C₄	#19151582	1	1	yes	characteristic
252	C₄	#19151844	1	2	yes	—
253	C₄	#19151851	1	1	yes	characteristic
254	C₄	#19151856	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#26020248	1	1	yes	characteristic
257	C₂	#26020285	1	1	yes	characteristic
258	C₂	#26020302	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151458
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#446×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#446	D₈₉
hypocenter	C₈₉	#8619
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
108. C₈×C₈₉	#101	semidirect
135. C₈₉	#8619	semidirect
240. C₈	#15667931	semidirect

Maximal subgroups

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

subgroup	index
#26020248	2
#26019011	2
#26019008	2
#26018127	2
C₂×#446×C₈₉	2
#8619	89