Subgroups — #36573329

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029203	C₂	1	2	yes	maximal
2	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
3	#26021025	C₂	1	2	yes	maximal
4	#26021024	C₂	1	4	yes	maximal
5	#26020425	C₂	1	2	yes	maximal
6	#26020160	C₂	1	2	yes	maximal
7	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#19151728	C₄	1	4	yes	—
11	#19151728	C₄	1	4	yes	—
12	#19151727	—	2	4	no	—
13	#19151725	—	2	2	no	—
14	#19151722	—	2	2	no	—
15	C₂×#15667948	C₂²	1	1	yes	characteristic
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	C₂×#15667947	C₂²	1	4	yes	—
18	#19151621	C₂²	1	4	yes	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667941	C₂²	1	2	yes	—
21	C₄×#13083558	C₂²	1	2	yes	—
22	M₁₆×Dic₈₉	C₂²	1	4	yes	—
23	C₂×#15667922	C₂²	1	2	yes	—
24	#19151363	C₂²	1	2	yes	—
25	#19151362	C₂²	1	4	yes	—
26	C₂×#108×C₈₉	C₂²	1	2	yes	—
27	C₂×#108×C₈₉	C₂²	1	2	yes	—
28	#423×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₇₈	Q₈	1	1	yes	characteristic
31	C₂×#13083558	C₂³	1	2	yes	—
32	C₂×#13083558	C₂³	1	2	yes	—
33	C₂²×#10623494	C₂³	1	2	yes	—
34	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₇₈	D₄	1	2	yes	—
37	C₄×Dic₁₇₈	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₈₉	Q₈	1	1	yes	characteristic
39	C₂×#13083553	Q₈	1	2	yes	—
40	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
41	#15667948	—	2	2	no	—
42	#15667948	Q₈	1	4	yes	complemented
43	#15667947	—	4	4	no	—
44	C₄⋊#10623494	C₂×C₄	1	2	yes	—
45	C₄⋊#10623494	C₂×C₄	1	2	yes	—
46	#15667941	—	2	4	no	—
47	#15667941	—	2	4	no	—
48	#15667941	C₂×C₄	1	8	yes	—
49	#15667923	—	2	2	no	—
50	#15667922	—	2	4	no	—
51	#15667922	C₂×C₄	1	8	yes	—
52	C₄×#10623494	—	2	2	no	—
53	C₈×Dic₈₉	—	2	4	no	—
54	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	#108×C₈₉	—	2	4	no	—
59	#108×C₈₉	—	4	2	no	—
60	C₄⋊C₈×C₈₉	—	2	2	no	—
61	C₄⋊C₈×C₈₉	—	2	2	no	—
62	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
64	C₂×Dic₁₇₈	—	4	2	no	—
65	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₁₇₈	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₈₉	C₂×Q₈	1	2	yes	—
68	C₂²×Dic₈₉	C₂×Q₈	1	2	yes	—
69	#13083558	—	2	2	no	—
70	#13083558	—	2	2	no	—
71	#13083558	—	2	2	no	—
72	#13083558	C₂×Q₈	1	4	yes	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	C₂²×C₄	1	4	yes	—
78	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
79	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
80	#13083553	C₄⋊C₄	1	4	yes	—
81	#13083553	C₄⋊C₄	1	4	yes	—
82	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
83	C₄×Dic₈₉	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₈₉	—	2	4	no	—
88	M₁₆×C₈₉	—	2	4	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	4	no	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
117	Dic₁₇₈	—	2	4	no	—
118	Dic₁₇₈	—	4	2	no	—
119	C₈×C₈₉	—	2	2	no	—
120	C₈×C₈₉	—	4	4	no	—
121	C₈×C₈₉	C₄×Q₈	1	4	yes	—
122	#10623494	—	2	4	no	—
123	#10623494	—	2	4	no	—
124	#10623494	—	4	2	no	—
125	C₂²×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₈₉	#557	1	1	yes	characteristic
129	C₂²×C₈₉	Q₈²	1	1	yes	characteristic
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	#430	1	2	yes	—
134	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₈₉	#520	1	1	yes	characteristic
136	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
137	C₄×C₈₉	#550	1	1	yes	characteristic
138	Dic₈₉	—	2	4	no	—
139	Dic₈₉	—	2	4	no	—
140	#22733	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
145	C₂×C₈₉	#2916	1	1	yes	characteristic
146	Q₈×M₁₆	—	89	2	no	—
147	C₂×#446	—	89	1	no	—
148	#1950	—	89	4	no	—
149	#1950	—	89	2	no	—
150	#1849	—	89	2	no	—
151	#1848	D₈₉	1	1	yes	characteristic
152	#1778	—	89	2	no	—
153	C₄×#108	—	89	1	no	—
154	C₈₉	#22733	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄×Q₈	—	89	1	no	—
156	#446	—	89	4	no	—
157	#446	—	89	4	no	—
158	#446	—	178	2	no	—
159	#446	—	178	4	no	—
160	C₈×Q₈	—	178	2	no	—
161	C₂×#108	—	89	1	no	—
162	C₂×#108	—	89	4	no	—
163	C₂×#108	—	89	1	no	—
164	C₂×#108	C₂×D₈₉	1	2	yes	—
165	C₂×#108	C₂×D₈₉	1	2	yes	—
166	#423	—	89	4	no	—
167	#423	C₂×D₈₉	1	2	yes	—
168	C₂×C₄⋊C₈	—	89	2	no	—
169	C₂×C₄⋊C₈	—	89	1	no	—
170	C₄×M₁₆	—	89	4	no	—
171	C₄×M₁₆	—	89	2	no	—
172	C₂×#97	—	89	2	no	—
173	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
174	#341	—	89	4	no	—
175	#341	—	89	2	no	—
176	C₂²×Q₈	—	89	1	no	—
177	C₂×M₁₆	—	89	2	no	—
178	C₂×M₁₆	—	89	2	no	—
179	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
180	C₂²×C₈	—	89	2	no	—
181	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
183	C₄×Q₈	—	89	2	no	—
184	C₄×Q₈	—	89	2	no	—
185	C₄×Q₈	—	89	2	no	—
186	C₄×Q₈	—	89	2	no	—
187	C₂×C₄⋊C₄	—	89	2	no	—
188	C₂×C₄⋊C₄	—	89	1	no	—
189	C₂×C₄²	—	89	2	no	—
190	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
191	#108	—	2	4	no	—
192	#108	—	4	2	no	—
193	#108	—	89	4	no	—
194	#108	—	356	4	no	—
195	#108	—	178	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	89	8	no	—
199	C₄⋊C₈	—	89	2	no	—
200	C₄⋊C₈	—	89	2	no	—
201	C₄⋊C₈	—	178	4	no	—
202	C₄⋊C₈	—	178	4	no	—
203	#97	—	89	8	no	—
204	#97	—	178	4	no	—
205	#97	—	178	2	no	—
206	C₄×C₈	—	178	4	no	—
207	C₄×C₈	—	178	2	no	—
208	C₄×C₈	C₄×D₈₉	1	2	yes	—
209	C₄×C₈	C₄×D₈₉	1	2	yes	—
210	C₂×Q₈	—	89	2	no	—
211	C₂×Q₈	—	89	2	no	—
212	C₂×Q₈	—	356	2	no	—
213	C₂²×C₄	—	89	2	no	—
214	C₂²×C₄	—	89	2	no	—
215	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	89	4	no	—
221	M₁₆	—	178	2	no	—
222	M₁₆	—	178	2	no	—
223	M₁₆	—	178	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	89	4	no	—
228	C₂×C₈	—	178	2	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
234	C₂×C₈	Q₈×D₈₉	1	2	yes	—
235	C₂×C₈	#13083573	1	2	yes	—
236	C₄⋊C₄	—	89	4	no	—
237	C₄⋊C₄	—	89	4	no	—
238	C₄⋊C₄	—	89	2	no	—
239	C₄⋊C₄	—	89	2	no	—
240	C₄²	—	89	4	no	—
241	C₄²	—	89	4	no	—
242	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₄²	D₄×D₈₉	1	1	yes	characteristic
245	C₄²	D₄×D₈₉	1	1	yes	characteristic
246	C₂³	C₂×Q₈×D₈₉	1	1	yes	characteristic
247	Q₈	—	356	2	no	—
248	Q₈	—	178	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	89	4	no	—
252	C₂×C₄	—	89	4	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
263	C₂×C₄	#15668057	1	1	yes	characteristic
264	C₂×C₄	#15668060	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	4	4	no	—
267	C₈	—	356	2	no	—
268	C₈	—	178	4	no	—
269	C₈	—	178	4	no	—
270	C₈	#15667977	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
274	C₂²	#19152323	1	1	yes	characteristic
275	C₂²	#19152328	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	178	4	no	—
280	C₄	—	178	4	no	—
281	C₄	#19151645	1	2	yes	—
282	C₄	C₂×#15667977	1	1	yes	characteristic
283	C₄	#19152079	1	1	yes	characteristic
284	C₄	#19152263	1	1	yes	characteristic
285	C₄	#19152268	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	#26023164	1	1	yes	characteristic
289	C₂	C₂×#19151645	1	1	yes	characteristic
290	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₉	C₂²×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#22733
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
42. #15667948	Q₈	semidirect
154. C₈₉	#22733	semidirect

Maximal subgroups

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

subgroup	index
#26029203	2
C₂×#19151728	2
#26021025	2
#26021024	2
#26020425	2
#26020160	2
C₄×#15667948	2
C₈₉×#1848	2
#22733	89