Subgroups — #36470946

Order 22784.

404 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	C₂²×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151428	C₂	1	2	yes	maximal, complemented
3	#26020290	C₂	1	1	yes	maximal, characteristic
4	#26018346	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	—	2	1	no	—
7	C₂×#15667948	—	2	4	no	—
8	C₂×#15667948	—	2	1	no	—
9	C₂×#15667948	C₂²	1	1	yes	characteristic
10	C₂×#15667948	C₂²	1	1	yes	characteristic
11	C₂×#15667948	C₂²	1	2	yes	—
12	#19151562	—	2	1	no	—
13	#19151428	—	2	4	no	—
14	#19151363	—	2	1	no	—
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂²⋊C₁₆×C₈₉	C₄	1	2	yes	—
18	C₂²⋊C₁₆×C₈₉	C₄	1	2	yes	—
19	C₂×#13083558	—	2	4	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	—	2	1	no	—
23	C₂×#13083558	—	4	2	no	—
24	C₂×#13083558	—	4	1	no	—
25	C₂²×#10623494	—	4	1	no	—
26	#15667948	—	2	4	no	—
27	#15667948	—	2	2	no	—
28	#15667948	—	2	2	no	—
29	#15667948	—	2	2	no	—
30	#15667948	D₄	1	4	yes	—
31	C₂²⋊#10623494	—	4	1	no	—
32	C₂²⋊#10623494	—	4	1	no	—
33	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	—	2	2	no	—
35	C₂²×C₈×C₈₉	—	2	2	no	—
36	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₈₉	—	2	2	no	—
43	C₂×C₁₆×C₈₉	—	2	2	no	—
44	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
45	#13083558	—	2	4	no	—
46	#13083558	—	4	4	no	—
47	#13083558	—	4	2	no	—
48	#13083558	—	4	1	no	—
49	#13083558	—	4	1	no	—
50	#13083558	—	4	1	no	—
51	#13083558	—	4	1	no	—
52	C₂×#10623494	—	4	4	no	—
53	C₂×#10623494	—	4	2	no	—
54	C₂×#10623494	—	4	1	no	—
55	C₂×#10623494	—	4	1	no	—
56	C₂⁴×C₈₉	Q₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	—	2	1	no	—
58	C₂²×C₄×C₈₉	—	2	1	no	—
59	C₂²×C₄×C₈₉	—	2	1	no	—
60	C₂²×C₄×C₈₉	—	2	1	no	—
61	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	1	no	—
68	C₂×C₈×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	2	2	no	—
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₈₉	#41	1	2	yes	—
79	C₁₆×C₈₉	—	2	4	no	—
80	C₂³×C₈₉	—	2	2	no	—
81	C₂³×C₈₉	—	2	1	no	—
82	C₂³×C₈₉	—	2	1	no	—
83	C₂³×C₈₉	—	4	1	no	—
84	C₂³×C₈₉	#107	1	1	yes	characteristic
85	C₂³×C₈₉	#107	1	1	yes	characteristic
86	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
105	C₈×C₈₉	—	2	1	no	—
106	C₈×C₈₉	—	2	1	no	—
107	C₈×C₈₉	C₄×D₄	1	2	yes	—
108	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
109	#10623494	—	4	4	no	—
110	#10623494	—	8	1	no	—
111	#10623494	—	8	1	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	—	4	2	no	—
120	C₂²×C₈₉	—	4	2	no	—
121	C₂²×C₈₉	#368	1	2	yes	complemented
122	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
123	C₂²×C₈₉	#451	1	1	yes	characteristic
124	C₂²×C₈₉	#484	1	1	yes	characteristic
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	#385	1	1	yes	characteristic
128	C₄×C₈₉	#428	1	1	yes	characteristic
129	C₄×C₈₉	#447	1	1	yes	characteristic
130	C₄×C₈₉	#480	1	1	yes	characteristic
131	#12922	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	—	2	2	no	—
133	C₂×C₈₉	—	2	1	no	—
134	C₂×C₈₉	—	2	1	no	—
135	C₂×C₈₉	—	4	1	no	—
136	C₂×C₈₉	#1852	1	1	yes	characteristic
137	C₂×C₈₉	C₂×#368	1	1	yes	characteristic
138	C₂×C₈₉	#2161	1	1	yes	characteristic
139	C₂²×#108	—	89	1	no	—
140	C₂×#368	—	89	2	no	—
141	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
142	#1810	—	89	1	no	—
143	#1381	—	89	2	no	—
144	C₈₉	#12922	1	1	yes	characteristic, complemented, Sylow
145	C₂²×M₁₆	—	178	1	no	—
146	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
147	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
148	C₂×#108	—	89	2	no	—
149	C₂×#108	—	89	1	no	—
150	C₂×#108	—	89	1	no	—
151	C₂×#108	—	178	4	no	—
152	C₂×#108	—	178	1	no	—
153	#407	—	178	1	no	—
154	#368	—	178	4	no	—
155	C₂²⋊C₁₆	Dic₈₉	1	2	yes	—
156	C₂²⋊C₁₆	Dic₈₉	1	2	yes	—
157	#341	—	178	1	no	—
158	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
159	C₂×M₁₆	—	356	2	no	—
160	C₂×M₁₆	—	356	1	no	—
161	C₂×M₁₆	—	178	1	no	—
162	C₂×M₁₆	—	178	1	no	—
163	C₂×M₁₆	—	178	4	no	—
164	C₂×M₁₆	—	178	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	356	1	no	—
168	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
169	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
170	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
177	#108	—	89	4	no	—
178	#108	—	178	4	no	—
179	#108	—	178	2	no	—
180	#108	—	178	2	no	—
181	#108	—	178	2	no	—
182	C₂²⋊C₈	—	356	1	no	—
183	C₂²⋊C₈	—	356	1	no	—
184	C₂⁴	Dic₃₅₆	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
191	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
192	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
193	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
195	M₁₆	—	356	4	no	—
196	M₁₆	—	356	2	no	—
197	M₁₆	—	356	1	no	—
198	M₁₆	—	356	1	no	—
199	M₁₆	—	356	1	no	—
200	M₁₆	—	356	1	no	—
201	M₁₆	—	178	4	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	356	4	no	—
209	C₂×C₈	—	356	2	no	—
210	C₂×C₈	—	356	1	no	—
211	C₂×C₈	—	356	1	no	—
212	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
213	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
215	C₂×C₈	D₄×D₈₉	1	2	yes	—
216	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
217	C₂×C₈	#13083573	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	#15667946	1	1	yes	characteristic
224	C₂³	#15667946	1	1	yes	characteristic
225	C₂³	C₂×Dic₃₅₆	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	#15667946	1	1	yes	characteristic
237	C₂×C₄	#15667946	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
239	C₂×C₄	#15667982	1	1	yes	characteristic
240	C₂×C₄	#15667996	1	1	yes	characteristic
241	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
242	C₂×C₄	#15668040	1	1	yes	characteristic
243	C₂×C₄	#15668045	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	356	4	no	—
247	C₈	—	712	1	no	—
248	C₈	—	712	1	no	—
249	C₈	D₄×Dic₈₉	1	2	yes	—
250	C₈	#15667989	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#19151428	1	2	yes	complemented
261	C₂²	C₂×#15667946	1	1	yes	characteristic
262	C₂²	#19151756	1	1	yes	characteristic
263	C₂²	#19151934	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	#19151473	1	1	yes	characteristic
267	C₄	#19151640	1	1	yes	characteristic
268	C₄	#19151734	1	1	yes	characteristic
269	C₄	#19151912	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	2	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	1	no	—
274	C₂	#26020442	1	1	yes	characteristic
275	C₂	C₂×#19151428	1	1	yes	characteristic
276	C₂	#26022047	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂⁴×C₈₉	Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	#12922
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151428	C₂	semidirect
4. #26018346	C₂	semidirect
121. C₂²×C₈₉	#368	semidirect
144. C₈₉	#12922	semidirect
260. C₂²	#19151428	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667948	2
C₂×#19151428	2
#26020290	2
#26018346	2
C₂×C₂²⋊C₁₆×C₈₉	2
#12922	89