Subgroups — #36523902

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029333	C₂	1	2	yes	maximal
2	#26019071	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#10623494	C₂	1	4	yes	maximal, complemented
4	#26019000	C₂	1	1	yes	maximal, characteristic
5	#26018647	C₂	1	2	yes	maximal
6	#26018535	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×#10623494	—	2	4	no	—
9	Q₈×#10623494	C₂²	1	8	yes	—
10	#19151651	—	2	4	no	—
11	#19151651	C₂²	1	2	yes	—
12	#19151651	C₂²	1	1	yes	characteristic
13	#19151651	C₂²	1	1	yes	characteristic
14	#19151351	C₂²	1	2	yes	—
15	#19151349	C₂²	1	2	yes	—
16	Q₈⋊#10623494	C₂²	1	8	yes	—
17	#19151319	C₂²	1	2	yes	—
18	C₈×#10623494	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
22	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
23	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
25	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
26	C₂×#103×C₈₉	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊#10623494	—	2	8	no	—
29	C₄⋊#10623494	—	2	1	no	—
30	C₄⋊#10623494	D₄	1	2	yes	—
31	C₄⋊#10623494	C₂³	1	4	yes	—
32	C₂²⋊#10623494	—	2	2	no	—
33	C₂²⋊#10623494	—	2	1	no	—
34	C₂²⋊#10623494	D₄	1	2	yes	—
35	C₄×#10623494	—	2	8	no	—
36	C₄×#10623494	—	2	2	no	—
37	C₄×#10623494	C₂³	1	2	yes	—
38	C₄×#10623494	C₂³	1	2	yes	—
39	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₂×Q₁₆×C₈₉	C₈	1	8	yes	complemented
41	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₈₉	—	2	2	no	—
46	C₄×Q₈×C₈₉	—	2	2	no	—
47	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₈₉	—	2	2	no	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	#107×C₈₉	C₂×C₄	1	2	yes	—
54	#107×C₈₉	C₂×C₄	1	2	yes	—
55	#103×C₈₉	C₂×C₄	1	4	yes	—
56	#103×C₈₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	C₂×D₄	1	4	yes	—
64	C₂×Q₈×C₈₉	—	2	2	no	—
65	C₂×Q₈×C₈₉	—	2	4	no	—
66	C₂×Q₈×C₈₉	C₂×C₈	1	8	yes	—
67	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₈₉	—	2	2	no	—
69	C₂²×C₄×C₈₉	—	2	2	no	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
73	Q₁₆×C₈₉	C₂×C₈	1	8	yes	—
74	Q₁₆×C₈₉	C₂×C₈	1	8	yes	—
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
77	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₈₉	—	2	4	no	—
82	C₄⋊C₄×C₈₉	—	2	2	no	—
83	C₄⋊C₄×C₈₉	—	2	4	no	—
84	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
85	C₄²×C₈₉	—	2	4	no	—
86	C₄²×C₈₉	—	2	4	no	—
87	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	#41	1	1	yes	characteristic
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₈₉	—	2	4	no	—
93	Q₈×C₈₉	—	2	8	no	—
94	Q₈×C₈₉	C₂²×C₈	1	8	yes	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	2	8	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	—	8	2	no	—
101	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₈₉	#131	1	2	yes	—
106	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	C₂²×C₈	1	2	yes	—
112	C₈×C₈₉	C₂²×C₈	1	2	yes	—
113	#10623494	—	2	4	no	—
114	#10623494	—	4	2	no	—
115	#10623494	—	4	4	no	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	C₈×D₄	1	2	yes	—
119	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₈₉	—	2	8	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	4	4	no	—
125	C₄×C₈₉	C₈×D₄	1	2	yes	—
126	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
128	C₄×C₈₉	#578	1	2	yes	—
129	#17856	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	2	no	—
131	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
132	C₂×C₈₉	#2950	1	1	yes	characteristic
133	C₂×C₈₉	#2957	1	1	yes	characteristic
134	#2982	—	89	2	no	—
135	C₂×C₄×Q₁₆	D₈₉	1	1	yes	characteristic
136	#1579	—	89	4	no	—
137	C₈×Q₁₆	—	89	4	no	—
138	#1564	—	89	1	no	—
139	#1479	—	89	2	no	—
140	#1450	—	89	1	no	—
141	C₈₉	#17856	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₁₆	Dic₈₉	1	1	yes	characteristic
143	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
144	C₈×Q₈	—	89	8	no	—
145	C₈×Q₈	—	178	4	no	—
146	C₄×Q₁₆	Dic₈₉	1	2	yes	—
147	C₄×Q₁₆	Dic₈₉	1	2	yes	—
148	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
149	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
150	#431	—	89	1	no	—
151	#431	—	89	1	no	—
152	#431	—	89	2	no	—
153	#431	—	178	4	no	—
154	C₂×#107	Dic₈₉	1	1	yes	characteristic
155	C₂×#103	Dic₈₉	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
157	#335	—	89	2	no	—
158	#334	—	89	2	no	—
159	Q₈⋊C₈	—	89	8	no	—
160	#322	—	89	2	no	—
161	C₈²	—	89	2	no	—
162	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
163	C₂×Q₁₆	#10623494	1	8	yes	complemented
164	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
165	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
166	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
171	C₄×Q₈	C₂²×D₈₉	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
176	#107	C₂×Dic₈₉	1	2	yes	—
177	#107	C₂×Dic₈₉	1	2	yes	—
178	C₄⋊C₈	—	89	2	no	—
179	C₄⋊C₈	—	89	4	no	—
180	C₄⋊C₈	—	178	8	no	—
181	C₄⋊C₈	—	178	1	no	—
182	#103	C₂×Dic₈₉	1	4	yes	—
183	#103	C₂×Dic₈₉	1	4	yes	—
184	C₂²⋊C₈	—	89	2	no	—
185	C₂²⋊C₈	—	178	2	no	—
186	C₂²⋊C₈	—	178	1	no	—
187	C₄×C₈	—	89	2	no	—
188	C₄×C₈	—	89	2	no	—
189	C₄×C₈	—	178	8	no	—
190	C₄×C₈	—	178	2	no	—
191	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂×#10623494	1	8	yes	—
198	C₂×Q₈	C₂²×Dic₈₉	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
203	C₂²×C₄	#13083573	1	1	yes	characteristic
204	Q₁₆	C₂×#10623494	1	8	yes	—
205	Q₁₆	C₂×#10623494	1	8	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	89	4	no	—
208	C₂×C₈	—	178	4	no	—
209	C₂×C₈	—	178	2	no	—
210	C₂×C₈	C₂×#10623494	1	4	yes	—
211	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
222	C₄²	D₄×D₈₉	1	1	yes	characteristic
223	C₄²	#13083573	1	1	yes	characteristic
224	C₄²	C₂³×D₈₉	1	1	yes	characteristic
225	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	Q₈	—	2	4	no	—
228	Q₈	C₂²×#10623494	1	8	yes	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
238	C₂×C₄	C₂²×#10623494	1	2	yes	—
239	C₂×C₄	#15668019	1	2	yes	—
240	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	356	2	no	—
246	C₈	—	356	4	no	—
247	C₈	—	178	4	no	—
248	C₈	C₂²×#10623494	1	2	yes	—
249	C₈	C₂²×#10623494	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	D₄×#10623494	1	2	yes	—
253	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	8	no	—
258	C₄	—	4	4	no	—
259	C₄	D₄×#10623494	1	2	yes	—
260	C₄	C₂³×#10623494	1	1	yes	characteristic
261	C₄	C₂×#15668019	1	1	yes	characteristic
262	C₄	#19152411	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×D₄×#10623494	1	1	yes	characteristic
265	C₂	#26029120	1	1	yes	characteristic
266	C₂	#26029169	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₄×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₈₉
hypocenter	C₈₉	#17856
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #26019071	C₂	semidirect
3. Q₁₆×#10623494	C₂	semidirect
40. C₂×Q₁₆×C₈₉	C₈	semidirect
141. C₈₉	#17856	semidirect
163. C₂×Q₁₆	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26029333	2
#26019071	2
Q₁₆×#10623494	2
#26019000	2
#26018647	2
#26018535	2
C₂×C₄×Q₁₆×C₈₉	2
#17856	89