Subgroups — #36563096

Order 22784.

822 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	C₂×#19151729	C₂	1	2	yes	maximal
2	#26021189	C₂	1	2	yes	maximal
3	#26020971	C₂	1	4	yes	maximal
4	#26020404	C₂	1	1	yes	maximal, characteristic
5	#26020121	C₂	1	4	yes	maximal
6	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19151729	—	2	8	no	—
9	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
12	C₂×#15667923	C₂²	1	4	yes	—
13	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10623494	C₂²	1	2	yes	—
17	#19151353	C₂²	1	8	yes	—
18	C₂×C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
20	#391×C₈₉	C₄	1	2	yes	—
21	#389×C₈₉	C₄	1	4	yes	—
22	#384×C₈₉	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
24	C₄³×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×#10623494	C₂³	1	4	yes	—
26	C₂²×#10623494	C₂³	1	1	yes	characteristic
27	C₂²×#10623494	C₂³	1	2	yes	—
28	C₂²×#10623494	C₂³	1	1	yes	characteristic
29	C₄⋊#10623494	—	2	8	no	—
30	C₄⋊#10623494	—	2	4	no	—
31	C₄⋊#10623494	—	2	8	no	—
32	#15667923	—	2	8	no	—
33	C₄×#10623494	—	2	2	no	—
34	C₄×#10623494	—	2	4	no	—
35	C₄×#10623494	D₄	1	4	yes	—
36	C₄×#10623494	Q₈	1	8	yes	complemented
37	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄⋊Q₈×C₈₉	—	2	4	no	—
39	C₄×Q₈×C₈₉	—	2	4	no	—
40	C₄×Q₈×C₈₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	#95×C₈₉	C₂×C₄	1	8	yes	—
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	—	2	2	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	8	no	—
57	C₂×#10623494	C₂×D₄	1	2	yes	—
58	C₂×#10623494	C₂×D₄	1	2	yes	—
59	C₂×#10623494	C₂×Q₈	1	4	yes	—
60	C₂×#10623494	C₂×Q₈	1	4	yes	—
61	C₂×Q₈×C₈₉	—	2	4	no	—
62	C₂×Q₈×C₈₉	—	2	4	no	—
63	C₂×Q₈×C₈₉	—	2	4	no	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₈₉	—	2	4	no	—
72	C₄⋊C₄×C₈₉	—	2	8	no	—
73	C₄⋊C₄×C₈₉	—	2	8	no	—
74	C₄⋊C₄×C₈₉	—	2	8	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	4	no	—
77	C₄²×C₈₉	—	2	4	no	—
78	C₄²×C₈₉	—	2	4	no	—
79	C₄²×C₈₉	M₁₆	1	4	yes	—
80	C₄²×C₈₉	C₂×D₄	1	2	yes	—
81	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
82	C₄²×C₈₉	#41	1	2	yes	—
83	C₄²×C₈₉	#41	1	4	yes	—
84	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₈₉	—	4	8	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	4	no	—
92	C₂×C₄×C₈₉	—	2	8	no	—
93	C₂×C₄×C₈₉	—	2	4	no	—
94	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
97	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
98	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₈₉	#131	1	2	yes	—
101	C₂×C₄×C₈₉	#131	1	4	yes	—
102	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
106	#10623494	—	2	2	no	—
107	#10623494	—	2	4	no	—
108	#10623494	—	4	4	no	—
109	#10623494	C₄⋊Q₈	1	4	yes	complemented
110	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₈₉	C₂×C₄×Q₈	1	2	yes	—
112	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
113	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#131	1	2	yes	—
115	C₄×C₈₉	—	2	2	no	—
116	C₄×C₈₉	—	2	4	no	—
117	C₄×C₈₉	—	4	4	no	—
118	C₄×C₈₉	—	4	4	no	—
119	C₄×C₈₉	#436	1	4	yes	—
120	C₄×C₈₉	#446	1	8	yes	complemented
121	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₈₉	#532	1	1	yes	characteristic
123	C₄×C₈₉	#533	1	2	yes	—
124	#21712	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₈₉	C₂×#436	1	1	yes	characteristic
127	C₂×C₈₉	#2931	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#446	1	2	yes	—
129	C₂×C₈₉	#2960	1	2	yes	—
130	C₂×#446	—	89	2	no	—
131	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
132	#1984	—	89	2	no	—
133	#1939	—	89	4	no	—
134	#1842	—	89	1	no	—
135	#1770	—	89	4	no	—
136	C₄²×C₈	—	89	1	no	—
137	C₈₉	#21712	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
140	#446	—	178	8	no	—
141	C₂×C₄⋊C₈	—	89	2	no	—
142	C₂×C₄⋊C₈	—	89	4	no	—
143	C₂×C₄⋊C₈	—	89	4	no	—
144	C₂×#97	—	89	4	no	—
145	C₂×C₄×C₈	—	89	1	no	—
146	C₂×C₄×C₈	—	89	2	no	—
147	C₂×C₄×C₈	—	89	2	no	—
148	C₂×C₄×C₈	—	89	1	no	—
149	#391	Dic₈₉	1	2	yes	—
150	#389	Dic₈₉	1	4	yes	—
151	#384	Dic₈₉	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
153	C₄³	C₂×D₈₉	1	1	yes	characteristic
154	#336	—	89	8	no	—
155	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
156	C₂²×C₈	—	89	1	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₂²×C₈	—	89	1	no	—
159	C₂²×C₈	—	89	4	no	—
160	C₄⋊Q₈	—	2	4	no	—
161	C₄×Q₈	—	2	4	no	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
167	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
168	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
171	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
173	C₄⋊C₈	—	178	8	no	—
174	C₄⋊C₈	—	178	4	no	—
175	C₄⋊C₈	—	178	8	no	—
176	#97	—	178	8	no	—
177	C₄×C₈	—	89	4	no	—
178	C₄×C₈	—	89	8	no	—
179	C₄×C₈	—	178	2	no	—
180	C₄×C₈	—	178	4	no	—
181	#95	C₂×Dic₈₉	1	8	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
186	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
187	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
191	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
192	C₂×C₈	—	89	2	no	—
193	C₂×C₈	—	89	4	no	—
194	C₂×C₈	—	89	2	no	—
195	C₂×C₈	—	89	4	no	—
196	C₂×C₈	—	178	4	no	—
197	C₂×C₈	—	178	8	no	—
198	C₂×C₈	—	178	2	no	—
199	C₂×C₈	—	178	2	no	—
200	C₂×C₈	—	178	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	#13083558	1	4	yes	—
210	C₄²	D₄×D₈₉	1	2	yes	—
211	C₄²	Q₈×D₈₉	1	4	yes	—
212	C₄²	#13083573	1	2	yes	—
213	C₄²	#13083575	1	4	yes	—
214	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
215	Q₈	—	4	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
225	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
226	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
227	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
228	C₂×C₄	C₂×#13083558	1	2	yes	—
229	C₂×C₄	C₂×#13083558	1	4	yes	—
230	C₂×C₄	#15668019	1	2	yes	—
231	C₂×C₄	#15668019	1	4	yes	—
232	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₈×D₈₉	1	2	yes	—
234	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
235	C₂×C₄	C₂×#13083575	1	2	yes	—
236	C₈	—	89	4	no	—
237	C₈	—	356	4	no	—
238	C₈	—	178	2	no	—
239	C₈	—	178	4	no	—
240	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
241	C₂²	C₂×Q₈×Dic₈₉	1	2	yes	—
242	C₂²	C₂²×#13083558	1	1	yes	characteristic
243	C₂²	C₂×#15668019	1	2	yes	—
244	C₂²	C₂×#15668019	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	4	4	no	—
249	C₄	#19151676	1	4	yes	—
250	C₄	#19151729	1	8	yes	complemented
251	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
252	C₄	#19152149	1	1	yes	characteristic
253	C₄	#19152153	1	2	yes	—
254	C₂	C₄⋊Q₈×Dic₈₉	1	1	yes	characteristic
255	C₂	C₂×#19151676	1	1	yes	characteristic
256	C₂	#26028964	1	1	yes	characteristic
257	C₂	C₂×#19151729	1	2	yes	—
258	C₂	#26029187	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#21712
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
36. C₄×#10623494	Q₈	semidirect
109. #10623494	C₄⋊Q₈	semidirect
120. C₄×C₈₉	#446	semidirect
137. C₈₉	#21712	semidirect
250. C₄	#19151729	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151729	2
#26021189	2
#26020971	2
#26020404	2
#26020121	2
C₄²×#10623494	2
C₄×C₄⋊Q₈×C₈₉	2
#21712	89