Subgroups — #36573331

Order 22784.

876 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26020970	C₂	1	3	yes	maximal
4	#26020968	C₂	1	3	yes	maximal
5	#26020368	C₂	1	3	yes	maximal
6	C₄×#15667941	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	C₄	1	1	yes	characteristic
9	C₂×Q₈×Dic₈₉	C₄	1	1	yes	characteristic
10	Q₈×#10623494	—	2	4	no	—
11	C₂×C₄⋊#10623494	C₂²	1	3	yes	—
12	C₂×#15667941	C₂²	1	3	yes	—
13	C₂×#15667941	C₂²	1	3	yes	—
14	C₂×C₄×#10623494	C₂²	1	3	yes	—
15	C₄⋊C₄×Dic₈₉	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₈₉	C₄	1	3	yes	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	3	yes	—
18	C₄²×Dic₈₉	C₂²	1	3	yes	—
19	#19151352	C₂²	1	1	yes	characteristic
20	#19151352	C₂²	1	3	yes	—
21	#19151352	C₂²	1	6	yes	—
22	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₈×Q₈×C₈₉	—	2	4	no	—
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
26	C₂²×#10623494	C₂³	1	3	yes	—
27	C₂²×#10623494	C₂³	1	1	yes	characteristic
28	Q₈×Dic₈₉	—	2	4	no	—
29	Q₈×Dic₈₉	C₈	1	8	yes	complemented
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₈₉	C₂×C₄	1	3	yes	—
34	C₂×C₄×Dic₈₉	C₂×C₄	1	3	yes	—
35	C₂×C₄×Dic₈₉	C₂×C₄	1	3	yes	—
36	C₂×C₄×Dic₈₉	C₂³	1	3	yes	—
37	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
38	C₄⋊#10623494	—	2	6	no	—
39	#15667941	—	2	6	no	—
40	#15667941	—	2	6	no	—
41	C₄×#10623494	—	2	6	no	—
42	#15667919	C₂×C₄	1	6	yes	—
43	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₈₉	D₄	1	4	yes	—
47	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
49	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
50	C₄⋊C₈×C₈₉	—	2	6	no	—
51	C₄×C₈×C₈₉	—	2	6	no	—
52	C₂²×Dic₈₉	C₂²×C₄	1	3	yes	—
53	C₂²×Dic₈₉	C₂²×C₄	1	3	yes	—
54	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×#10623494	—	2	1	no	—
57	C₂×#10623494	—	2	1	no	—
58	C₂×#10623494	—	2	3	no	—
59	C₂×#10623494	—	2	3	no	—
60	C₂×#10623494	—	2	3	no	—
61	C₂×#10623494	#41	1	2	yes	—
62	C₄⋊Dic₈₉	—	2	6	no	—
63	C₄⋊Dic₈₉	C₂×C₈	1	12	yes	—
64	C₄×Dic₈₉	—	2	6	no	—
65	C₄×Dic₈₉	—	2	3	no	—
66	C₄×Dic₈₉	—	2	3	no	—
67	C₄×Dic₈₉	—	2	1	no	—
68	C₄×Dic₈₉	—	2	1	no	—
69	C₄×Dic₈₉	—	2	6	no	—
70	C₄×Dic₈₉	—	4	3	no	—
71	C₄×Dic₈₉	C₂×C₈	1	12	yes	—
72	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
73	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
74	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	1	no	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	3	no	—
81	C₂×C₈×C₈₉	—	2	3	no	—
82	C₂×C₈×C₈₉	—	2	3	no	—
83	C₂×C₈×C₈₉	#41	1	2	yes	—
84	C₄⋊C₄×C₈₉	C₄⋊C₄	1	12	yes	—
85	C₄²×C₈₉	C₂×D₄	1	6	yes	—
86	C₄²×C₈₉	C₂×Q₈	1	6	yes	—
87	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₈₉	C₄⋊C₈	1	16	yes	complemented
89	C₂×C₄×C₈₉	—	2	3	no	—
90	C₂×C₄×C₈₉	—	2	3	no	—
91	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
92	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
93	C₂×C₄×C₈₉	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₈₉	C₂×M₁₆	1	6	yes	—
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
101	C₂×Dic₈₉	—	2	3	no	—
102	C₂×Dic₈₉	—	2	1	no	—
103	C₂×Dic₈₉	—	2	1	no	—
104	C₂×Dic₈₉	—	2	1	no	—
105	C₂×Dic₈₉	—	2	3	no	—
106	C₂×Dic₈₉	—	2	1	no	—
107	C₂×Dic₈₉	—	2	3	no	—
108	C₂×Dic₈₉	—	2	3	no	—
109	C₂×Dic₈₉	—	2	3	no	—
110	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
111	C₂×Dic₈₉	C₂²×C₈	1	6	yes	—
112	C₂×Dic₈₉	#131	1	2	yes	—
113	C₈×C₈₉	—	2	2	no	—
114	C₈×C₈₉	—	4	3	no	—
115	#10623494	—	2	2	no	—
116	#10623494	—	4	3	no	—
117	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
119	C₂²×C₈₉	#519	1	1	yes	characteristic
120	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
121	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
123	C₂²×C₈₉	#568	1	1	yes	characteristic
124	C₄×C₈₉	—	2	6	no	—
125	C₄×C₈₉	C₂×C₄⋊C₈	1	12	yes	—
126	C₄×C₈₉	#548	1	1	yes	characteristic
127	C₄×C₈₉	#548	1	1	yes	characteristic
128	C₄×C₈₉	#557	1	1	yes	characteristic
129	C₄×C₈₉	#557	1	1	yes	characteristic
130	Dic₈₉	—	2	6	no	—
131	Dic₈₉	—	2	2	no	—
132	Dic₈₉	—	2	2	no	—
133	Dic₈₉	—	4	3	no	—
134	#22734	—	89	1	no	maximal, Sylow
135	C₂×C₈₉	#2351	1	1	yes	characteristic
136	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₈₉	#2906	1	1	yes	characteristic
138	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
139	C₂×C₈₉	#2971	1	1	yes	characteristic
140	C₂×C₈₉	#2982	1	1	yes	characteristic
141	C₂×C₈₉	#2984	1	1	yes	characteristic
142	C₂×C₈×Q₈	—	89	1	no	—
143	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
144	C₄²×Q₈	—	89	1	no	—
145	#1939	—	89	3	no	—
146	#1939	—	89	3	no	—
147	#1831	—	89	3	no	—
148	C₄×C₄⋊C₈	—	89	3	no	—
149	C₈₉	#22734	1	1	yes	characteristic, complemented, Sylow
150	C₂×C₄×Q₈	—	89	1	no	—
151	C₂×C₄×Q₈	—	89	1	no	—
152	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
153	C₈×Q₈	—	2	4	no	—
154	C₈×Q₈	—	178	4	no	—
155	C₂×C₄⋊C₈	—	89	3	no	—
156	C₂×C₄⋊C₈	—	89	3	no	—
157	C₂×C₄⋊C₈	—	89	3	no	—
158	C₂×C₄⋊C₈	C₂×D₈₉	1	3	yes	—
159	C₂×C₄×C₈	—	89	3	no	—
160	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
161	C₄×C₄⋊C₄	—	89	3	no	—
162	C₄×C₄⋊C₄	—	89	3	no	—
163	C₄×C₄⋊C₄	—	89	3	no	—
164	C₄³	—	89	3	no	—
165	#336	—	89	3	no	—
166	#336	—	89	6	no	—
167	#336	—	89	1	no	—
168	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
169	C₂²×C₈	—	89	1	no	—
170	C₂²×C₈	—	89	3	no	—
171	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
172	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
173	C₄×Q₈	—	89	8	no	—
174	C₄×Q₈	—	178	4	no	—
175	C₄×Q₈	Dic₁₇₈	1	4	yes	—
176	C₄×Q₈	C₂²⋊D₈₉	1	4	yes	—
177	C₂×C₄⋊C₄	—	89	3	no	—
178	C₂×C₄⋊C₄	—	89	3	no	—
179	C₂×C₄⋊C₄	C₄×D₈₉	1	3	yes	—
180	C₂×C₄²	—	89	3	no	—
181	C₂×C₄²	—	89	3	no	—
182	C₂×C₄²	—	89	3	no	—
183	C₂×C₄²	—	89	3	no	—
184	C₂×C₄²	—	89	1	no	—
185	C₂×C₄²	—	89	3	no	—
186	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
187	C₄⋊C₈	—	2	6	no	—
188	C₄⋊C₈	—	178	6	no	—
189	C₄⋊C₈	—	178	6	no	—
190	C₄⋊C₈	—	178	6	no	—
191	C₄×C₈	—	2	6	no	—
192	C₄×C₈	—	178	6	no	—
193	#95	—	89	6	no	—
194	C₂×Q₈	#13083553	1	4	yes	—
195	C₂×Q₈	C₈×D₈₉	1	4	yes	—
196	C₂×Q₈	#13083557	1	4	yes	—
197	C₂²×C₄	—	89	3	no	—
198	C₂²×C₄	—	89	1	no	—
199	C₂²×C₄	—	89	1	no	—
200	C₂²×C₄	—	89	3	no	—
201	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
202	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
203	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	3	no	—
207	C₂×C₈	—	2	3	no	—
208	C₂×C₈	—	2	3	no	—
209	C₂×C₈	—	89	2	no	—
210	C₂×C₈	—	178	3	no	—
211	C₂×C₈	—	178	3	no	—
212	C₂×C₈	—	178	3	no	—
213	C₂×C₈	—	178	1	no	—
214	C₂×C₈	—	178	1	no	—
215	C₂×C₈	#13083575	1	2	yes	—
216	C₄⋊C₄	—	89	12	no	—
217	C₄⋊C₄	—	178	6	no	—
218	C₄⋊C₄	#13083553	1	12	yes	—
219	C₄²	—	89	12	no	—
220	C₄²	—	356	3	no	—
221	C₄²	—	178	6	no	—
222	C₄²	—	178	6	no	—
223	C₄²	—	178	3	no	—
224	C₄²	—	178	3	no	—
225	C₄²	—	178	1	no	—
226	C₄²	—	178	1	no	—
227	C₄²	C₂×C₂²⋊D₈₉	1	6	yes	—
228	C₄²	C₂×Dic₁₇₈	1	6	yes	—
229	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
230	Q₈	#15667941	1	16	yes	complemented
231	C₂×C₄	—	2	3	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₄	—	89	2	no	—
234	C₂×C₄	—	89	6	no	—
235	C₂×C₄	—	89	2	no	—
236	C₂×C₄	—	178	3	no	—
237	C₂×C₄	—	178	3	no	—
238	C₂×C₄	—	178	3	no	—
239	C₂×C₄	—	178	3	no	—
240	C₂×C₄	—	178	1	no	—
241	C₂×C₄	—	178	1	no	—
242	C₂×C₄	—	178	1	no	—
243	C₂×C₄	—	178	3	no	—
244	C₂×C₄	—	178	1	no	—
245	C₂×C₄	C₂×#13083553	1	6	yes	—
246	C₂×C₄	C₂×#13083553	1	6	yes	—
247	C₂×C₄	C₂×C₈×D₈₉	1	6	yes	—
248	C₂×C₄	C₂×#13083557	1	6	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
251	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
253	C₂×C₄	#15668059	1	1	yes	characteristic
254	C₂×C₄	#15668060	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	3	no	—
257	C₈	—	356	3	no	—
258	C₈	—	178	2	no	—
259	C₂²	C₂²×#13083553	1	1	yes	characteristic
260	C₂²	C₄×#13083575	1	1	yes	characteristic
261	C₂²	#19152074	1	1	yes	characteristic
262	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
263	C₂²	C₂²×#13083557	1	1	yes	characteristic
264	C₂²	#131×D₈₉	1	1	yes	characteristic
265	C₂²	#19152363	1	1	yes	characteristic
266	C₄	—	2	6	no	—
267	C₄	—	356	3	no	—
268	C₄	—	178	2	no	—
269	C₄	—	178	2	no	—
270	C₄	—	178	6	no	—
271	C₄	C₂×#15667941	1	12	yes	—
272	C₄	#19152255	1	1	yes	characteristic
273	C₄	#19152259	1	1	yes	characteristic
274	C₄	#19152324	1	1	yes	characteristic
275	C₄	#19152327	1	1	yes	characteristic
276	C₂	#26023170	1	1	yes	characteristic
277	C₂	C₂²×#15667941	1	1	yes	characteristic
278	C₂	#26028818	1	1	yes	characteristic
279	C₂	C₈×#13083575	1	1	yes	characteristic
280	C₂	#26029271	1	1	yes	characteristic
281	C₂	#26029331	1	1	yes	characteristic
282	C₂	#26029340	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#22734
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
29. Q₈×Dic₈₉	C₈	semidirect
88. Q₈×C₈₉	C₄⋊C₈	semidirect
149. C₈₉	#22734	semidirect
230. Q₈	#15667941	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10623494	2
C₄×Q₈×Dic₈₉	2
#26020970	2
#26020968	2
#26020368	2
C₄×#15667941	2
C₂×C₈×Q₈×C₈₉	2
#22734	89