Subgroups — #36573441

Order 22784.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029203	C₂	1	2	yes	maximal
2	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
3	#26021026	C₂	1	4	yes	maximal
4	#26021025	C₂	1	2	yes	maximal
5	#26020391	C₂	1	2	yes	maximal
6	#26020157	C₂	1	2	yes	maximal
7	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1837	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#19151728	C₄	1	4	yes	—
11	#19151728	C₄	1	4	yes	—
12	#19151727	—	2	4	no	—
13	#19151725	—	2	2	no	—
14	#19151722	—	2	2	no	—
15	C₂×#15667948	C₂²	1	2	yes	—
16	C₂×#15667947	C₂²	1	4	yes	—
17	#19151621	C₂²	1	4	yes	—
18	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
19	C₂×#15667941	C₂²	1	2	yes	—
20	C₄×#13083558	C₂²	1	2	yes	—
21	M₁₆×Dic₈₉	C₂²	1	4	yes	—
22	C₂×#15667922	C₂²	1	2	yes	—
23	#19151363	C₂²	1	1	yes	characteristic
24	#19151363	C₂²	1	1	yes	characteristic
25	#19151362	C₂²	1	4	yes	—
26	#423×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	#341×C₈₉	C₂²	1	2	yes	—
29	#341×C₈₉	C₂²	1	2	yes	—
30	C₂²×Dic₁₇₈	Q₈	1	1	yes	characteristic
31	C₂×#13083558	C₂³	1	2	yes	—
32	C₂×#13083558	C₂³	1	2	yes	—
33	C₂²×#10623494	C₂³	1	2	yes	—
34	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₇₈	D₄	1	2	yes	—
37	C₄×Dic₁₇₈	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₈₉	Q₈	1	1	yes	characteristic
39	C₂×#13083553	Q₈	1	2	yes	—
40	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
41	#15667948	—	2	4	no	—
42	#15667947	—	2	8	no	—
43	C₄⋊#10623494	C₂×C₄	1	2	yes	—
44	C₄⋊#10623494	C₂×C₄	1	2	yes	—
45	#15667941	—	2	4	no	—
46	#15667941	—	2	4	no	—
47	#15667941	C₂×C₄	1	8	yes	—
48	#15667923	—	2	2	no	—
49	#15667922	—	2	4	no	—
50	#15667922	C₂×C₄	1	8	yes	—
51	C₄×#10623494	—	2	2	no	—
52	C₈×Dic₈₉	—	2	4	no	—
53	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₉	—	2	2	no	—
58	C₄⋊C₈×C₈₉	—	2	2	no	—
59	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	C₂×Dic₁₇₈	—	4	2	no	—
62	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₈₉	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₈₉	C₂×Q₈	1	2	yes	—
66	#13083558	—	2	2	no	—
67	#13083558	—	2	2	no	—
68	#13083558	—	2	2	no	—
69	#13083558	#41	1	4	yes	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	2	4	no	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	C₂²×C₄	1	4	yes	—
74	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
75	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
76	#13083553	C₄⋊C₄	1	4	yes	—
77	#13083553	C₄⋊C₄	1	4	yes	—
78	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
79	C₄×Dic₈₉	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	4	1	no	—
97	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
104	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
109	Dic₁₇₈	—	2	4	no	—
110	Dic₁₇₈	—	4	2	no	—
111	C₈×C₈₉	—	2	2	no	—
112	C₈×C₈₉	—	2	2	no	—
113	C₈×C₈₉	—	4	4	no	—
114	#10623494	—	2	4	no	—
115	#10623494	—	2	4	no	—
116	#10623494	—	4	2	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₈₉	#557	1	2	yes	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	#430	1	2	yes	—
125	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
126	C₄×C₈₉	#519	1	1	yes	characteristic
127	C₄×C₈₉	#548	1	2	yes	—
128	Dic₈₉	—	2	4	no	—
129	Dic₈₉	—	2	4	no	—
130	#22747	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	—	2	2	no	—
132	C₂×C₈₉	#2351	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
134	C₂×C₈₉	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	—	89	2	no	—
136	C₂×#446	—	89	1	no	—
137	#1950	—	89	2	no	—
138	#1950	—	89	4	no	—
139	#1838	—	89	2	no	—
140	#1837	D₈₉	1	1	yes	characteristic
141	#1778	—	89	2	no	—
142	C₄×#108	—	89	1	no	—
143	C₈₉	#22747	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	—	89	1	no	—
145	#446	—	89	4	no	—
146	#446	—	89	4	no	—
147	#446	—	178	2	no	—
148	#446	—	178	4	no	—
149	C₈×Q₈	—	178	2	no	—
150	C₂×#108	—	89	4	no	—
151	C₂×#108	—	89	2	no	—
152	#423	—	89	4	no	—
153	#423	C₂×D₈₉	1	2	yes	—
154	C₂×C₄⋊C₈	—	89	2	no	—
155	C₂×C₄⋊C₈	—	89	1	no	—
156	C₄×M₁₆	—	89	4	no	—
157	C₄×M₁₆	—	89	2	no	—
158	C₂×#97	—	89	2	no	—
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	#341	—	89	4	no	—
161	#341	—	89	1	no	—
162	#341	—	89	1	no	—
163	#341	C₂×D₈₉	1	2	yes	—
164	#341	C₂×D₈₉	1	2	yes	—
165	C₂²×Q₈	—	89	1	no	—
166	C₂×M₁₆	—	89	2	no	—
167	C₂×M₁₆	—	89	2	no	—
168	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
169	C₂²×C₈	—	89	2	no	—
170	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
172	C₄×Q₈	—	89	2	no	—
173	C₄×Q₈	—	89	2	no	—
174	C₄×Q₈	—	89	2	no	—
175	C₄×Q₈	—	89	2	no	—
176	C₂×C₄⋊C₄	—	89	2	no	—
177	C₂×C₄⋊C₄	—	89	1	no	—
178	C₂×C₄²	—	89	2	no	—
179	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
180	#108	—	178	4	no	—
181	#108	—	178	8	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	89	8	no	—
185	C₄⋊C₈	—	89	2	no	—
186	C₄⋊C₈	—	89	2	no	—
187	C₄⋊C₈	—	178	4	no	—
188	C₄⋊C₈	—	178	4	no	—
189	#97	—	89	8	no	—
190	#97	—	178	4	no	—
191	#97	—	178	2	no	—
192	C₄×C₈	—	178	4	no	—
193	C₄×C₈	—	178	2	no	—
194	C₄×C₈	C₄×D₈₉	1	2	yes	—
195	C₄×C₈	C₄×D₈₉	1	2	yes	—
196	C₂×Q₈	—	89	2	no	—
197	C₂×Q₈	—	89	2	no	—
198	C₂×Q₈	—	356	2	no	—
199	C₂²×C₄	—	89	2	no	—
200	C₂²×C₄	—	89	2	no	—
201	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	89	4	no	—
207	M₁₆	—	178	2	no	—
208	M₁₆	—	178	2	no	—
209	M₁₆	—	178	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	89	4	no	—
214	C₂×C₈	—	178	2	no	—
215	C₂×C₈	—	178	2	no	—
216	C₂×C₈	—	178	4	no	—
217	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
219	C₄⋊C₄	—	89	4	no	—
220	C₄⋊C₄	—	89	4	no	—
221	C₄⋊C₄	—	89	2	no	—
222	C₄⋊C₄	—	89	2	no	—
223	C₄²	—	89	4	no	—
224	C₄²	—	89	4	no	—
225	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₄²	D₄×D₈₉	1	1	yes	characteristic
228	C₄²	D₄×D₈₉	1	1	yes	characteristic
229	C₂³	C₂×Q₈×D₈₉	1	1	yes	characteristic
230	Q₈	—	356	2	no	—
231	Q₈	—	178	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	89	4	no	—
235	C₂×C₄	—	89	4	no	—
236	C₂×C₄	—	356	2	no	—
237	C₂×C₄	—	356	2	no	—
238	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
244	C₂×C₄	#41×D₈₉	1	2	yes	—
245	C₂×C₄	#15668059	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	4	4	no	—
249	C₈	—	356	2	no	—
250	C₈	—	178	4	no	—
251	C₈	—	178	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
255	C₂²	#19152325	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	178	4	no	—
260	C₄	—	178	4	no	—
261	C₄	#19151645	1	2	yes	—
262	C₄	#19152066	1	1	yes	characteristic
263	C₄	#19152075	1	1	yes	characteristic
264	C₄	#19152256	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#26023169	1	1	yes	characteristic
267	C₂	C₂×#19151645	1	1	yes	characteristic
268	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1837	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₉	C₂²×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1837	D₈₉
hypocenter	C₈₉	#22747
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
143. C₈₉	#22747	semidirect

Maximal subgroups

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

subgroup	index
#26029203	2
C₂×#19151728	2
#26021026	2
#26021025	2
#26020391	2
#26020157	2
C₄×#15667948	2
C₈₉×#1837	2
#22747	89