Subgroups — #36447356

Order 22784.

486 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	#26020244	C₂	1	2	yes	maximal
2	#26019011	C₂	1	2	yes	maximal, complemented
3	#26018127	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151622	C₂²	1	4	yes	—
6	C₂×#15667929	—	2	2	no	—
7	#19151367	—	2	2	no	—
8	#19151351	—	2	2	no	—
9	#19151351	—	2	2	no	—
10	C₂³×C₈×C₈₉	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₈₉	—	2	2	no	—
13	C₈×D₄×C₈₉	—	2	2	no	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083558	—	2	4	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄⋊#10623494	—	2	4	no	—
20	#15667929	—	4	4	no	—
21	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₈₉	—	2	2	no	—
24	C₂²×C₈×C₈₉	—	2	2	no	—
25	C₂²×C₈×C₈₉	—	4	4	no	—
26	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×D₄×C₈₉	—	2	2	no	—
30	C₄×D₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₈₉	Q₈	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₂²⋊C₈×C₈₉	—	2	4	no	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₄×C₈×C₈₉	D₄	1	2	yes	—
38	#13083558	—	4	4	no	—
39	#13083558	—	4	4	no	—
40	C₂×#10623494	—	4	4	no	—
41	C₂⁴×C₈₉	C₄⋊C₄	1	2	yes	—
42	C₂×D₄×C₈₉	—	2	2	no	—
43	C₂×D₄×C₈₉	—	2	2	no	—
44	C₂×D₄×C₈₉	D₈	1	2	yes	—
45	C₂×D₄×C₈₉	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₈₉	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	4	2	no	—
49	C₂²×C₄×C₈₉	—	4	2	no	—
50	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	4	1	no	—
57	C₂×C₈×C₈₉	—	4	4	no	—
58	C₂×C₈×C₈₉	—	4	4	no	—
59	C₂×C₈×C₈₉	—	4	2	no	—
60	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₂²⋊C₄×C₈₉	—	2	4	no	—
65	C₄²×C₈₉	C₂×D₄	1	2	yes	—
66	C₄²×C₈₉	#41	1	2	yes	—
67	C₂³×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	—	2	2	no	—
69	C₂³×C₈₉	—	4	2	no	—
70	C₂³×C₈₉	—	4	2	no	—
71	C₂³×C₈₉	#107	1	4	yes	—
72	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	D₄×C₈₉	—	2	4	no	—
74	D₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	1	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	4	1	no	—
79	C₂×C₄×C₈₉	—	4	4	no	—
80	C₂×C₄×C₈₉	—	4	2	no	—
81	C₂×C₄×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#121	1	2	yes	—
86	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
87	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#135	1	2	yes	—
90	C₈×C₈₉	—	2	1	no	—
91	C₈×C₈₉	—	2	2	no	—
92	C₈×C₈₉	—	4	2	no	—
93	C₈×C₈₉	#100	1	2	yes	complemented
94	#10623494	—	8	4	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	—	2	4	no	—
98	C₂²×C₈₉	—	4	4	no	—
99	C₂²×C₈₉	—	4	4	no	—
100	C₂²×C₈₉	—	4	2	no	—
101	C₂²×C₈₉	—	4	2	no	—
102	C₂²×C₈₉	#380	1	1	yes	characteristic
103	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
104	C₂²×C₈₉	#427	1	1	yes	characteristic
105	C₄×C₈₉	—	2	1	no	—
106	C₄×C₈₉	—	2	2	no	—
107	C₄×C₈₉	—	4	2	no	—
108	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
109	C₄×C₈₉	#413	1	1	yes	characteristic
110	C₄×C₈₉	#466	1	2	yes	—
111	C₄×C₈₉	#467	1	2	yes	—
112	#8616	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	—	2	1	no	—
114	C₂×C₈₉	—	2	1	no	—
115	C₂×C₈₉	—	4	4	no	—
116	C₂×C₈₉	#1800	1	1	yes	characteristic
117	C₂×C₈₉	#1808	1	1	yes	characteristic
118	C₂×C₈₉	#1813	1	1	yes	characteristic
119	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
120	#1800	—	89	2	no	—
121	#1567	—	89	2	no	—
122	#1292	—	89	2	no	—
123	C₈₉	#8616	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₈	Dic₈₉	1	2	yes	—
125	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
126	C₈×D₄	—	2	2	no	—
127	C₈×D₄	—	2	2	no	—
128	#423	—	89	4	no	—
129	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
130	C₂×#100	—	178	2	no	—
131	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
133	#343	—	178	2	no	—
134	#335	—	178	2	no	—
135	#335	—	178	2	no	—
136	C₂²×D₄	D₃₅₆	1	1	yes	characteristic
137	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
138	C₂×M₁₆	—	178	4	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	4	4	no	—
142	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
144	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	Dic₁₇₈	1	2	yes	—
149	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	178	4	no	—
152	C₄⋊C₈	—	178	4	no	—
153	#100	—	356	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
156	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
157	C₂⁴	C₄⋊Dic₈₉	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	D₇₁₂	1	2	yes	—
161	C₂×D₄	Dic₃₅₆	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
170	M₁₆	—	356	4	no	—
171	M₁₆	—	356	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	356	4	no	—
179	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
180	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
185	C₄²	#13083574	1	2	yes	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	4	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	#15667946	1	4	yes	—
191	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
192	D₄	—	2	4	no	—
193	D₄	—	2	4	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
203	C₂×C₄	#15667971	1	1	yes	characteristic
204	C₂×C₄	#15667986	1	2	yes	—
205	C₂×C₄	#15667991	1	2	yes	—
206	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
207	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
208	C₂×C₄	#15668034	1	2	yes	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	712	4	no	—
213	C₈	#15667929	1	2	yes	complemented
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	#19151458	1	1	yes	characteristic
222	C₂²	C₂×#15667946	1	1	yes	characteristic
223	C₂²	#19151636	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×#15667929	1	1	yes	characteristic
228	C₄	#19151582	1	1	yes	characteristic
229	C₄	#19151834	1	2	yes	—
230	C₄	#19151839	1	2	yes	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	—	4	4	no	—
234	C₂	#26020244	1	1	yes	characteristic
235	C₂	#26020281	1	1	yes	characteristic
236	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×C₈×D₄×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂²×D₄×C₈₉	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	#8616
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #26019011	C₂	semidirect
3. #26018127	C₂	semidirect
93. C₈×C₈₉	#100	semidirect
123. C₈₉	#8616	semidirect
213. C₈	#15667929	semidirect

Maximal subgroups

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

subgroup	index
#26020244	2
#26019011	2
#26018127	2
C₂×C₈×D₄×C₈₉	2
#8616	89