Subgroups — #36563090

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×#19151725	C₂	1	2	yes	maximal
2	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
3	#26021187	C₂	1	2	yes	maximal
4	#26020968	C₂	1	4	yes	maximal
5	#26020122	C₂	1	4	yes	maximal
6	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668012	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
10	#19151725	—	2	8	no	—
11	C₂×#15667941	C₂²	1	4	yes	—
12	C₂×#15667922	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	#19151504	C₄	1	2	yes	—
18	#19151496	C₄	1	4	yes	—
19	#19151468	C₄	1	1	yes	characteristic
20	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
21	#19151352	C₂²	1	8	yes	—
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
24	C₄³×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₇₈	C₂×C₄	1	2	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	#15668012	—	2	4	no	—
30	C₄×Dic₁₇₈	—	2	4	no	—
31	C₄×Dic₁₇₈	—	2	4	no	—
32	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	C₂×#13083553	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₈₉	C₂³	1	4	yes	—
37	#15667941	—	2	8	no	—
38	#15667922	—	2	8	no	—
39	C₄×#10623494	—	2	4	no	—
40	C₄×#10623494	—	2	2	no	—
41	C₄×#10623494	D₄	1	4	yes	—
42	C₄×#10623494	Q₈	1	8	yes	complemented
43	#15667919	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₈₉	C₂³	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	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
50	C₄⋊C₈×C₈₉	—	2	4	no	—
51	C₄⋊C₈×C₈₉	—	2	8	no	—
52	C₂×Dic₁₇₈	—	2	4	no	—
53	C₂×Dic₁₇₈	—	2	4	no	—
54	C₂×Dic₁₇₈	—	2	4	no	—
55	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
57	C₂×#10623494	—	2	2	no	—
58	C₂×#10623494	—	2	2	no	—
59	C₂×#10623494	—	2	2	no	—
60	C₂×#10623494	C₂×D₄	1	2	yes	—
61	C₂×#10623494	C₂×D₄	1	2	yes	—
62	C₂×#10623494	C₂×Q₈	1	4	yes	—
63	C₂×#10623494	C₂×Q₈	1	4	yes	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	2	8	no	—
66	C₄⋊Dic₈₉	—	2	8	no	—
67	#13083553	—	2	8	no	—
68	C₄×Dic₈₉	—	2	4	no	—
69	C₄×Dic₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	2	4	no	—
76	C₂×C₈×C₈₉	—	2	8	no	—
77	C₄²×C₈₉	—	2	2	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	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
91	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
94	C₂×C₄×C₈₉	#131	1	2	yes	—
95	C₂×C₄×C₈₉	#131	1	4	yes	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
100	C₂×Dic₈₉	—	2	4	no	—
101	C₂×Dic₈₉	—	2	4	no	—
102	C₂×Dic₈₉	—	2	4	no	—
103	C₂×Dic₈₉	—	2	8	no	—
104	C₂×Dic₈₉	—	2	4	no	—
105	Dic₁₇₈	—	4	8	no	—
106	C₈×C₈₉	—	4	4	no	—
107	#10623494	—	2	2	no	—
108	#10623494	—	2	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	#436	1	4	yes	—
118	C₄×C₈₉	#446	1	8	yes	complemented
119	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₄×C₈₉	#532	1	1	yes	characteristic
121	C₄×C₈₉	#533	1	2	yes	—
122	Dic₈₉	—	4	4	no	—
123	Dic₈₉	—	4	4	no	—
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	—	89	1	no	—
132	#1984	—	89	2	no	—
133	#1939	—	89	4	no	—
134	#1842	D₈₉	1	1	yes	characteristic
135	#1770	—	89	4	no	—
136	C₄²×C₈	—	89	1	no	—
137	C₈₉	#21712	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄⋊Q₈	—	89	1	no	—
139	C₂×C₄×Q₈	—	89	2	no	—
140	#446	—	178	8	no	—
141	C₂×C₄⋊C₈	—	89	4	no	—
142	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
143	C₂×C₄⋊C₈	C₂×D₈₉	1	4	yes	—
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	—	89	2	no	—
150	#389	—	89	4	no	—
151	#384	—	89	1	no	—
152	C₄×C₄⋊C₄	—	89	4	no	—
153	C₄³	C₂×D₈₉	1	1	yes	characteristic
154	#336	—	89	8	no	—
155	C₂²×Q₈	—	89	2	no	—
156	C₂²×C₈	—	89	1	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₂²×C₈	—	89	1	no	—
159	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
160	C₄⋊Q₈	—	178	4	no	—
161	C₄×Q₈	—	178	4	no	—
162	C₄×Q₈	—	178	4	no	—
163	C₂×C₄⋊C₄	—	89	2	no	—
164	C₂×C₄⋊C₄	—	89	4	no	—
165	C₂×C₄⋊C₄	—	89	4	no	—
166	C₂×C₄⋊C₄	—	89	4	no	—
167	C₂×C₄²	—	89	4	no	—
168	C₂×C₄²	C₄×D₈₉	1	2	yes	—
169	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	8	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	4	no	—
180	C₄×C₈	—	178	2	no	—
181	#95	—	89	8	no	—
182	C₂×Q₈	—	178	4	no	—
183	C₂×Q₈	—	178	4	no	—
184	C₂×Q₈	—	178	4	no	—
185	C₂²×C₄	—	89	4	no	—
186	C₂²×C₄	—	89	4	no	—
187	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
191	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	89	2	no	—
195	C₂×C₈	—	89	4	no	—
196	C₂×C₈	—	89	2	no	—
197	C₂×C₈	—	89	4	no	—
198	C₂×C₈	—	178	2	no	—
199	C₂×C₈	—	178	2	no	—
200	C₂×C₈	—	178	2	no	—
201	C₄⋊C₄	—	178	8	no	—
202	C₄⋊C₄	—	178	4	no	—
203	C₄⋊C₄	—	178	8	no	—
204	C₄⋊C₄	—	178	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	178	4	no	—
208	C₄²	—	178	4	no	—
209	C₄²	#13083557	1	4	yes	—
210	C₄²	D₄×D₈₉	1	2	yes	—
211	C₄²	Q₈×D₈₉	1	4	yes	—
212	C₄²	#13083573	1	4	yes	—
213	C₄²	#13083575	1	2	yes	—
214	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
215	Q₈	—	356	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	178	4	no	—
220	C₂×C₄	—	178	4	no	—
221	C₂×C₄	—	178	4	no	—
222	C₂×C₄	—	178	8	no	—
223	C₂×C₄	—	178	4	no	—
224	C₂×C₄	#15667974	1	2	yes	—
225	C₂×C₄	#15667974	1	2	yes	—
226	C₂×C₄	#15667977	1	4	yes	—
227	C₂×C₄	#15667977	1	4	yes	—
228	C₂×C₄	C₂×#13083557	1	2	yes	—
229	C₂×C₄	C₂×#13083557	1	4	yes	—
230	C₂×C₄	#15668020	1	2	yes	—
231	C₂×C₄	#15668020	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	2	yes	—
235	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
236	C₈	—	4	4	no	—
237	C₈	—	89	4	no	—
238	C₈	—	178	2	no	—
239	C₈	—	178	4	no	—
240	C₂²	C₂×#15667974	1	1	yes	characteristic
241	C₂²	C₂×#15667977	1	2	yes	—
242	C₂²	C₂²×#13083557	1	1	yes	characteristic
243	C₂²	C₂×#15668020	1	2	yes	—
244	C₂²	C₂×#15668020	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	356	4	no	—
248	C₄	—	356	4	no	—
249	C₄	#19151675	1	4	yes	—
250	C₄	#19151725	1	8	yes	complemented
251	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
252	C₄	#19152146	1	1	yes	characteristic
253	C₄	#19152153	1	2	yes	—
254	C₂	#26022819	1	1	yes	characteristic
255	C₂	C₂×#19151675	1	1	yes	characteristic
256	C₂	#26028963	1	1	yes	characteristic
257	C₂	C₂×#19151725	1	2	yes	—
258	C₂	#26029182	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#1842	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#21712
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
42. C₄×#10623494	Q₈	semidirect
109. #10623494	C₄⋊Q₈	semidirect
118. C₄×C₈₉	#446	semidirect
137. C₈₉	#21712	semidirect
250. C₄	#19151725	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151725	2
C₄×#15668012	2
#26021187	2
#26020968	2
#26020122	2
C₄²×#10623494	2
C₈₉×#1842	2
#21712	89