Subgroups — #36454546

Order 22784.

538 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020404	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018135	C₂	1	4	yes	maximal
4	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	—	2	1	no	—
6	C₂×C₄⋊#10623494	—	2	1	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₄×#10623494	—	2	1	no	—
10	#19151353	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₄	1	2	yes	—
14	#384×C₈₉	—	2	1	no	—
15	#384×C₈₉	—	2	1	no	—
16	#384×C₈₉	C₄	1	2	yes	—
17	#384×C₈₉	C₂²	1	2	yes	—
18	C₄³×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	—	2	2	no	—
20	C₂²×#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	4	2	no	—
24	C₄⋊#10623494	—	4	2	no	—
25	C₄⋊#10623494	Q₈	1	4	yes	complemented
26	C₄×#10623494	—	4	2	no	—
27	C₂²×Q₈×C₈₉	—	2	2	no	—
28	C₄⋊Q₈×C₈₉	—	4	4	no	—
29	C₄⋊Q₈×C₈₉	—	4	4	no	—
30	C₄⋊Q₈×C₈₉	—	4	2	no	—
31	C₄⋊Q₈×C₈₉	—	4	2	no	—
32	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×Q₈×C₈₉	—	4	4	no	—
52	C₂×Q₈×C₈₉	—	4	4	no	—
53	C₂×Q₈×C₈₉	—	4	4	no	—
54	C₂²×C₄×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₈₉	—	4	4	no	—
63	C₄⋊C₄×C₈₉	—	4	4	no	—
64	C₄⋊C₄×C₈₉	—	4	4	no	—
65	C₄⋊C₄×C₈₉	—	4	4	no	—
66	C₄⋊C₄×C₈₉	—	4	2	no	—
67	C₄⋊C₄×C₈₉	—	4	2	no	—
68	C₄⋊C₄×C₈₉	—	4	4	no	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₄²×C₈₉	—	2	2	no	—
71	C₄²×C₈₉	—	2	2	no	—
72	C₄²×C₈₉	D₈	1	2	yes	—
73	C₄²×C₈₉	SD₁₆	1	2	yes	—
74	C₄²×C₈₉	SD₁₆	1	2	yes	—
75	C₄²×C₈₉	SD₁₆	1	2	yes	—
76	C₄²×C₈₉	Q₁₆	1	2	yes	—
77	C₄²×C₈₉	Q₁₆	1	2	yes	—
78	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
79	C₄²×C₈₉	#41	1	2	yes	—
80	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₈₉	—	8	8	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	4	4	no	—
89	C₂×C₄×C₈₉	—	4	4	no	—
90	C₂×C₄×C₈₉	—	4	4	no	—
91	C₂×C₄×C₈₉	#100	1	2	yes	—
92	C₂×C₄×C₈₉	#101	1	2	yes	—
93	C₂×C₄×C₈₉	#102	1	2	yes	—
94	C₂×C₄×C₈₉	#102	1	2	yes	—
95	C₂×C₄×C₈₉	#103	1	2	yes	—
96	C₂×C₄×C₈₉	#103	1	2	yes	—
97	C₂×C₄×C₈₉	#103	1	2	yes	—
98	C₂×C₄×C₈₉	#103	1	2	yes	—
99	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
110	#10623494	—	4	4	no	—
111	#10623494	—	8	2	no	—
112	C₂²×C₈₉	#391	1	1	yes	characteristic
113	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
116	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
117	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
118	C₂²×C₈₉	#418	1	1	yes	characteristic
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	8	4	no	—
123	C₄×C₈₉	#332	1	4	yes	—
124	C₄×C₈₉	#333	1	4	yes	—
125	C₄×C₈₉	#476	1	2	yes	—
126	C₄×C₈₉	#477	1	2	yes	—
127	C₄×C₈₉	#486	1	1	yes	characteristic
128	C₄×C₈₉	#487	1	1	yes	characteristic
129	C₄×C₈₉	#498	1	1	yes	characteristic
130	C₄×C₈₉	#500	1	1	yes	characteristic
131	#9896	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	C₂×#332	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#333	1	1	yes	characteristic
134	C₂×C₈₉	#1545	1	1	yes	characteristic
135	C₂×C₈₉	#1549	1	1	yes	characteristic
136	C₂×C₈₉	#1987	1	1	yes	characteristic
137	C₂×C₈₉	#1990	1	1	yes	characteristic
138	C₂×C₈₉	#1997	1	1	yes	characteristic
139	#2868	D₈₉	1	1	yes	characteristic
140	#1842	—	89	1	no	—
141	C₄×C₄⋊C₈	—	89	1	no	—
142	#1296	—	89	4	no	—
143	C₈₉	#9896	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	2	1	no	—
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×C₄⋊Q₈	Dic₈₉	1	2	yes	—
147	C₂×C₄⋊C₈	—	89	2	no	—
148	C₂×C₄⋊C₈	—	89	2	no	—
149	C₂×C₄⋊C₈	—	178	1	no	—
150	C₂×C₄⋊C₈	—	178	1	no	—
151	C₂×C₄×C₈	—	178	1	no	—
152	#384	—	2	1	no	—
153	#384	—	2	1	no	—
154	#384	Dic₈₉	1	2	yes	—
155	#384	C₂×D₈₉	1	2	yes	—
156	C₄³	C₂×D₈₉	1	1	yes	characteristic
157	#336	—	178	1	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×C₈	—	178	2	no	—
160	C₂²×C₈	—	178	2	no	—
161	C₄⋊Q₈	—	4	4	no	—
162	C₄⋊Q₈	—	4	4	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
171	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
178	C₄⋊C₈	—	89	4	no	—
179	C₄⋊C₈	—	356	2	no	—
180	C₄⋊C₈	—	356	2	no	—
181	C₄⋊C₈	—	178	4	no	—
182	C₄⋊C₈	—	178	2	no	—
183	C₄×C₈	—	356	2	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
195	C₂×C₈	—	356	2	no	—
196	C₂×C₈	—	356	2	no	—
197	C₂×C₈	—	356	2	no	—
198	C₂×C₈	—	356	2	no	—
199	C₂×C₈	—	356	2	no	—
200	C₂×C₈	—	178	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	#13083559	1	2	yes	—
212	C₄²	#13083561	1	2	yes	—
213	C₄²	#13083561	1	2	yes	—
214	C₄²	#13083562	1	2	yes	—
215	C₄²	#13083564	1	2	yes	—
216	C₄²	#13083564	1	2	yes	—
217	C₄²	Q₈×D₈₉	1	2	yes	—
218	C₄²	#13083575	1	2	yes	—
219	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
220	Q₈	—	8	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	#15667929	1	2	yes	—
231	C₂×C₄	#15667931	1	2	yes	—
232	C₂×C₄	#15667933	1	2	yes	—
233	C₂×C₄	#15667933	1	2	yes	—
234	C₂×C₄	#15667936	1	2	yes	—
235	C₂×C₄	#15667936	1	2	yes	—
236	C₂×C₄	#15667936	1	2	yes	—
237	C₂×C₄	#15667936	1	2	yes	—
238	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
239	C₂×C₄	#15667988	1	2	yes	—
240	C₂×C₄	#15668001	1	1	yes	characteristic
241	C₂×C₄	#15668010	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
243	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
244	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
245	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
246	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
247	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
248	C₂×C₄	#15668038	1	2	yes	—
249	C₈	—	356	4	no	—
250	C₈	—	712	2	no	—
251	C₂²	#19151505	1	1	yes	characteristic
252	C₂²	C₂×#15667929	1	1	yes	characteristic
253	C₂²	C₂×#15667931	1	1	yes	characteristic
254	C₂²	C₂×#15667933	1	1	yes	characteristic
255	C₂²	C₂×#15667936	1	1	yes	characteristic
256	C₂²	C₂×#15667936	1	1	yes	characteristic
257	C₂²	#19151603	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	8	4	no	—
262	C₄	#19151344	1	4	yes	—
263	C₄	#19151347	1	4	yes	—
264	C₄	#19151885	1	2	yes	—
265	C₄	#19151890	1	2	yes	—
266	C₄	#19151946	1	1	yes	characteristic
267	C₄	#19151950	1	1	yes	characteristic
268	C₄	#19151994	1	1	yes	characteristic
269	C₄	#19152001	1	1	yes	characteristic
270	C₂	C₂×#19151344	1	1	yes	characteristic
271	C₂	C₂×#19151347	1	1	yes	characteristic
272	C₂	#26018920	1	1	yes	characteristic
273	C₂	#26018937	1	1	yes	characteristic
274	C₂	#26021201	1	1	yes	characteristic
275	C₂	#26021214	1	1	yes	characteristic
276	C₂	#26021242	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2868	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2868	D₈₉
hypocenter	C₈₉	#9896
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
25. C₄⋊#10623494	Q₈	semidirect
143. C₈₉	#9896	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
C₄×C₄⋊#10623494	2
#26018135	2
C₈₉×#2868	2
#9896	89