Subgroups — #36553080

Order 22784.

938 conjugacy classes of subgroups.

The subgroup lattice has 276 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	#26020127	C₂	1	4	yes	maximal
3	C₄⋊C₄×#10623494	C₂	1	2	yes	maximal
4	#26020105	C₂	1	1	yes	maximal, characteristic
5	#26020105	C₂	1	2	yes	maximal
6	C₄×#15667923	C₂	1	2	yes	maximal
7	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19151729	C₄	1	16	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
12	C₂×#15667923	C₂²	1	2	yes	—
13	C₂×#15667923	C₂²	1	1	yes	characteristic
14	C₂×#15667923	C₂²	1	2	yes	—
15	C₂×#15667923	C₂²	1	4	yes	—
16	C₂×#15667923	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	4	yes	—
18	C₂×C₄×#10623494	C₂²	1	2	yes	—
19	#19151353	C₂²	1	4	yes	—
20	#19151353	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₈₉	C₄	1	2	yes	—
22	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₈₉	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
27	C₄³×C₈₉	C₂²	1	2	yes	—
28	C₄³×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×#10623494	C₂³	1	4	yes	—
30	C₂²×#10623494	C₂³	1	2	yes	—
31	C₂²×#10623494	C₂³	1	2	yes	—
32	C₄⋊#10623494	C₂×C₄	1	8	yes	—
33	C₄⋊#10623494	C₂×C₄	1	16	yes	—
34	#15667923	—	2	4	no	—
35	#15667923	—	2	4	no	—
36	#15667923	—	2	2	no	—
37	#15667923	C₂×C₄	1	16	yes	—
38	#15667923	Q₈	1	4	yes	complemented
39	C₄×#10623494	—	2	8	no	—
40	C₄×#10623494	C₂×C₄	1	8	yes	—
41	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	—	2	8	no	—
43	C₄×Q₈×C₈₉	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	#95×C₈₉	C₂×C₄	1	4	yes	—
58	#95×C₈₉	C₂×C₄	1	2	yes	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	C₂²×C₄	1	4	yes	—
64	C₂×#10623494	C₂²×C₄	1	8	yes	—
65	C₂×#10623494	C₂²×C₄	1	4	yes	—
66	C₂×#10623494	C₂×Q₈	1	4	yes	—
67	C₂×#10623494	#41	1	4	yes	—
68	C₂×Q₈×C₈₉	—	2	4	no	—
69	C₂×Q₈×C₈₉	C₄²	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₈₉	—	2	8	no	—
79	C₄⋊C₄×C₈₉	C₄²	1	8	yes	—
80	C₄⋊C₄×C₈₉	C₄²	1	4	yes	—
81	C₄⋊C₄×C₈₉	M₁₆	1	8	yes	—
82	C₄²×C₈₉	—	2	8	no	—
83	C₄²×C₈₉	—	2	4	no	—
84	C₄²×C₈₉	—	2	4	no	—
85	C₄²×C₈₉	—	2	2	no	—
86	C₄²×C₈₉	—	4	4	no	—
87	C₄²×C₈₉	—	4	2	no	—
88	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
89	C₄²×C₈₉	C₂²×C₄	1	8	yes	—
90	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₈₉	#41	1	2	yes	—
92	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₈₉	—	2	8	no	—
94	C₂×C₄×C₈₉	—	2	4	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₈₉	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
115	#10623494	—	2	8	no	—
116	#10623494	—	2	4	no	—
117	#10623494	C₄×Q₈	1	8	yes	complemented
118	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
121	C₂²×C₈₉	C₂²×M₁₆	1	2	yes	—
122	C₂²×C₈₉	#568	1	2	yes	—
123	C₄×C₈₉	—	2	4	no	—
124	C₄×C₈₉	—	2	2	no	—
125	C₄×C₈₉	—	2	4	no	—
126	C₄×C₈₉	—	2	4	no	—
127	C₄×C₈₉	—	2	8	no	—
128	C₄×C₈₉	—	4	2	no	—
129	C₄×C₈₉	C₄×M₁₆	1	4	yes	—
130	C₄×C₈₉	C₂×C₄×Q₈	1	2	yes	—
131	C₄×C₈₉	C₄×#41	1	2	yes	—
132	#20520	—	89	1	no	maximal, Sylow
133	C₂×C₈₉	C₄²×Q₈	1	1	yes	characteristic
134	C₂×C₈₉	C₂×C₄×M₁₆	1	1	yes	characteristic
135	C₂×C₈₉	#2875	1	1	yes	characteristic
136	C₂×C₈₉	Q₈×M₁₆	1	2	yes	—
137	C₂×C₈₉	#2970	1	2	yes	—
138	C₂×#446	—	89	2	no	—
139	C₄²×Q₈	D₈₉	1	1	yes	characteristic
140	#1771	—	89	4	no	—
141	C₈×C₄⋊C₄	—	89	2	no	—
142	#1767	—	89	2	no	—
143	#1767	—	89	1	no	—
144	C₄×#97	—	89	1	no	—
145	C₄×#97	—	89	2	no	—
146	C₈₉	#20520	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄×Q₈	Dic₈₉	1	2	yes	—
148	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
149	#446	—	89	16	no	—
150	C₂×C₄⋊C₈	—	89	2	no	—
151	C₂×C₄⋊C₈	—	89	4	no	—
152	C₂×#97	—	89	4	no	—
153	C₂×#97	—	89	1	no	—
154	C₂×#97	—	89	1	no	—
155	C₂×#97	—	89	2	no	—
156	C₂×#97	—	89	2	no	—
157	C₂×C₄×C₈	—	89	2	no	—
158	C₂×C₄×C₈	—	89	4	no	—
159	C₄×C₄⋊C₄	Dic₈₉	1	4	yes	—
160	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
161	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
162	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
163	C₄³	C₂×D₈₉	1	2	yes	—
164	C₄³	C₂×D₈₉	1	1	yes	characteristic
165	#336	—	89	4	no	—
166	#336	—	89	2	no	—
167	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
168	C₂²×C₈	—	89	2	no	—
169	C₂²×C₈	—	89	2	no	—
170	C₂²×C₈	—	89	4	no	—
171	C₄×Q₈	—	2	8	no	—
172	C₄×Q₈	C₄×D₈₉	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
177	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
178	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
179	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
180	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
181	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
182	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
185	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	C₄⋊C₈	—	89	8	no	—
187	C₄⋊C₈	—	89	16	no	—
188	#97	—	89	16	no	—
189	#97	—	89	4	no	—
190	#97	—	178	4	no	—
191	#97	—	178	2	no	—
192	#97	—	178	4	no	—
193	C₄×C₈	—	89	8	no	—
194	C₄×C₈	—	178	8	no	—
195	#95	C₂×Dic₈₉	1	4	yes	—
196	#95	C₂×Dic₈₉	1	2	yes	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₄×Dic₈₉	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
202	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
207	C₂×C₈	—	89	4	no	—
208	C₂×C₈	—	89	4	no	—
209	C₂×C₈	—	89	4	no	—
210	C₂×C₈	—	89	4	no	—
211	C₂×C₈	—	89	8	no	—
212	C₂×C₈	—	178	4	no	—
213	C₂×C₈	—	178	2	no	—
214	C₂×C₈	—	178	2	no	—
215	C₂×C₈	—	178	4	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	C₄×Dic₈₉	1	8	yes	—
218	C₄⋊C₄	C₄×Dic₈₉	1	4	yes	—
219	C₄⋊C₄	#13083558	1	8	yes	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	4	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₂×C₄×D₈₉	1	4	yes	—
227	C₄²	C₂×C₄×D₈₉	1	8	yes	—
228	C₄²	Q₈×D₈₉	1	2	yes	—
229	C₄²	#13083575	1	2	yes	—
230	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
244	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
245	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
246	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
247	C₂×C₄	C₂×#13083558	1	8	yes	—
248	C₂×C₄	C₂×#13083558	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
250	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
251	C₂×C₄	#41×D₈₉	1	2	yes	—
252	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
253	C₈	—	89	8	no	—
254	C₈	—	178	4	no	—
255	C₈	—	178	8	no	—
256	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
257	C₂²	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
258	C₂²	#41×Dic₈₉	1	1	yes	characteristic
259	C₂²	C₂²×#13083558	1	2	yes	—
260	C₂²	#19152365	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	8	no	—
266	C₄	—	4	2	no	—
267	C₄	C₄×#13083558	1	4	yes	—
268	C₄	C₄×Q₈×D₈₉	1	2	yes	—
269	C₄	C₄×#13083575	1	2	yes	—
270	C₂	C₄×Q₈×Dic₈₉	1	1	yes	characteristic
271	C₂	C₂×C₄×#13083558	1	1	yes	characteristic
272	C₂	#26028680	1	1	yes	characteristic
273	C₂	Q₈×#13083558	1	2	yes	—
274	C₂	#26029266	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Fitting	C₄²×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×Q₈	D₈₉
hypocenter	C₈₉	#20520
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
9. #19151729	C₄	semidirect
38. #15667923	Q₈	semidirect
117. #10623494	C₄×Q₈	semidirect
146. C₈₉	#20520	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
#26020127	2
C₄⋊C₄×#10623494	2
#26020105	2
C₄×#15667923	2
C₄²×Q₈×C₈₉	2
#20520	89