Subgroups — #36453582

Order 22784.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic
2	#26021195	C₂	1	2	yes	maximal
3	#26018750	C₂	1	2	yes	maximal
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
5	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	2	yes	—
7	#19152003	C₄	1	1	yes	characteristic, complemented
8	#19151978	C₄	1	1	yes	characteristic, complemented
9	#19151949	C₄	1	2	yes	complemented
10	#19151839	—	2	1	no	—
11	#19151839	—	2	1	no	—
12	C₄×Dic₃₅₆	—	2	1	no	—
13	C₄×Dic₃₅₆	—	2	1	no	—
14	#19151651	C₂²	1	1	yes	characteristic
15	C₄×#13083558	C₂²	1	1	yes	characteristic
16	#19151358	—	2	4	no	—
17	#19151334	—	2	4	no	—
18	#19151320	—	2	1	no	—
19	C₈×#10623494	—	2	1	no	—
20	C₂×#107×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	#340×C₈₉	—	2	1	no	—
24	C₂×Dic₃₅₆	—	2	1	no	—
25	C₂×Dic₃₅₆	—	2	1	no	—
26	C₂×#13083558	D₄	1	2	yes	—
27	#15668012	C₂×C₄	1	2	yes	—
28	#15668012	C₂×C₄	1	2	yes	—
29	#15668004	—	2	2	no	—
30	#15667991	—	2	2	no	—
31	#15667991	—	2	2	no	—
32	C₄×Dic₁₇₈	—	2	2	no	—
33	C₄×Dic₁₇₈	—	2	2	no	—
34	#15667971	—	2	2	no	—
35	#15667946	—	2	1	no	—
36	#15667946	—	2	1	no	—
37	C₄⋊#10623494	—	2	1	no	—
38	#15667937	—	2	2	no	—
39	#15667937	—	2	2	no	—
40	C₂²⋊#10623494	D₄	1	2	yes	—
41	#15667923	—	2	1	no	—
42	C₄×#10623494	—	2	1	no	—
43	C₄×#10623494	—	2	1	no	—
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	—	2	2	no	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	#107×C₈₉	—	2	4	no	—
50	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	#95×C₈₉	—	2	2	no	—
55	C₂×Dic₁₇₈	—	2	2	no	—
56	C₂×Dic₁₇₈	—	2	2	no	—
57	Dic₃₅₆	—	4	2	no	—
58	Dic₃₅₆	—	4	2	no	—
59	#13083558	—	2	2	no	—
60	#13083558	Q₁₆	1	4	yes	complemented
61	C₂×#10623494	—	2	2	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₄⋊Dic₈₉	—	2	2	no	—
64	C₄⋊Dic₈₉	—	2	2	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	#13083553	—	4	2	no	—
68	#13083553	—	4	2	no	—
69	C₂²⋊Dic₈₉	—	4	2	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	—	4	2	no	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	4	1	no	—
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₈₉	—	2	4	no	—
82	C₄⋊C₄×C₈₉	—	4	2	no	—
83	C₄²×C₈₉	—	4	4	no	—
84	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₈₉	#41	1	1	yes	characteristic
87	C₄²×C₈₉	#41	1	1	yes	characteristic
88	C₂³×C₈₉	#127	1	1	yes	characteristic
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	4	1	no	—
92	C₂×C₄×C₈₉	—	4	4	no	—
93	C₂×C₄×C₈₉	—	8	2	no	—
94	C₂×C₄×C₈₉	—	8	2	no	—
95	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₈₉	#135	1	2	yes	—
103	C₂×Dic₈₉	—	4	2	no	—
104	C₂×Dic₈₉	—	4	2	no	—
105	Dic₁₇₈	—	4	2	no	—
106	Dic₁₇₈	—	4	2	no	—
107	Dic₁₇₈	—	8	2	no	—
108	C₈×C₈₉	—	2	1	no	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	#104	1	2	yes	—
112	#10623494	—	2	4	no	—
113	#10623494	—	4	2	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	#390	1	1	yes	characteristic
117	C₂²×C₈₉	#494	1	1	yes	characteristic
118	C₂²×C₈₉	#495	1	1	yes	characteristic
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	8	4	no	—
123	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
124	C₄×C₈₉	#421	1	1	yes	characteristic
125	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
126	C₄×C₈₉	#467	1	2	yes	—
127	Dic₈₉	—	8	2	no	—
128	Dic₈₉	—	8	2	no	—
129	#9718	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	1	no	—
131	C₂×C₈₉	—	2	1	no	—
132	C₂×C₈₉	#1969	1	1	yes	characteristic
133	C₂×C₈₉	#1976	1	1	yes	characteristic
134	C₂×C₈₉	#1980	1	1	yes	characteristic
135	#3170	—	89	1	no	—
136	#1986	—	89	2	no	—
137	C₄×#107	D₈₉	1	1	yes	characteristic
138	#1503	—	89	2	no	—
139	C₈×M₁₆	—	89	1	no	—
140	C₈₉	#9718	1	1	yes	characteristic, complemented, Sylow
141	#533	—	89	2	no	—
142	#500	—	89	1	no	—
143	#494	—	89	1	no	—
144	#487	—	89	2	no	—
145	#467	—	178	1	no	—
146	#467	—	178	1	no	—
147	C₄×Q₁₆	—	178	1	no	—
148	C₄×Q₁₆	—	178	1	no	—
149	#431	—	89	1	no	—
150	C₂×#107	—	2	1	no	—
151	C₄×M₁₆	—	89	1	no	—
152	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
154	#340	—	2	1	no	—
155	#339	—	178	4	no	—
156	#328	—	178	4	no	—
157	#322	—	178	1	no	—
158	C₈²	—	178	1	no	—
159	C₂×Q₁₆	—	178	1	no	—
160	C₂×Q₁₆	—	178	1	no	—
161	C₂×M₁₆	—	89	2	no	—
162	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
164	C₄⋊Q₈	—	89	2	no	—
165	C₄⋊Q₈	—	89	2	no	—
166	#125	—	178	2	no	—
167	C₂²⋊Q₈	—	178	2	no	—
168	C₂²⋊Q₈	—	178	2	no	—
169	C₄×Q₈	—	178	2	no	—
170	C₄×Q₈	—	178	2	no	—
171	#117	—	178	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	—	178	1	no	—
177	#107	—	178	1	no	—
178	C₄⋊C₈	—	178	1	no	—
179	#103	—	178	2	no	—
180	#103	—	178	2	no	—
181	C₂²⋊C₈	—	89	2	no	—
182	#97	—	178	1	no	—
183	C₄×C₈	—	178	1	no	—
184	C₄×C₈	—	178	1	no	—
185	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₄×C₈	D₃₅₆	1	1	yes	characteristic
188	C₄×C₈	D₃₅₆	1	1	yes	characteristic
189	#95	—	2	2	no	—
190	C₂×Q₈	—	178	2	no	—
191	C₂×Q₈	—	178	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
194	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
196	Q₁₆	—	356	2	no	—
197	Q₁₆	—	356	2	no	—
198	M₁₆	—	89	4	no	—
199	M₁₆	—	178	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	178	2	no	—
203	C₂×C₈	—	178	2	no	—
204	C₂×C₈	#13083551	1	1	yes	characteristic
205	C₂×C₈	#13083551	1	1	yes	characteristic
206	C₂×C₈	#13083551	1	1	yes	characteristic
207	C₂×C₈	#13083551	1	1	yes	characteristic
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	356	2	no	—
211	C₄⋊C₄	—	356	2	no	—
212	C₄⋊C₄	—	178	2	no	—
213	C₄⋊C₄	—	178	2	no	—
214	C₄⋊C₄	—	178	2	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₂²⋊C₄	—	356	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	—	356	2	no	—
219	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
221	C₄²	#13083575	1	1	yes	characteristic
222	C₄²	#13083575	1	1	yes	characteristic
223	C₂³	#15668008	1	1	yes	characteristic
224	Q₈	—	356	2	no	—
225	Q₈	—	356	2	no	—
226	Q₈	—	712	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	356	2	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
236	C₂×C₄	#15667974	1	1	yes	characteristic
237	C₂×C₄	#15667974	1	1	yes	characteristic
238	C₂×C₄	#15667987	1	1	yes	characteristic
239	C₂×C₄	#15667987	1	1	yes	characteristic
240	C₂×C₄	#15668001	1	1	yes	characteristic
241	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
242	C₂×C₄	#15668031	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	356	2	no	—
247	C₈	—	178	4	no	—
248	C₈	#15667940	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#19151499	1	1	yes	characteristic
252	C₂²	#19151976	1	1	yes	characteristic
253	C₂²	#19151981	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	4	no	—
258	C₄	—	712	2	no	—
259	C₄	—	712	2	no	—
260	C₄	C₂×#15667940	1	1	yes	characteristic
261	C₄	#19151616	1	1	yes	characteristic
262	C₄	#19151692	1	2	yes	—
263	C₄	#19151840	1	2	yes	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#26021111	1	1	yes	characteristic
267	C₂	#26021148	1	1	yes	characteristic
268	C₂	#26021170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#9718
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. #19152003	C₄	semidirect
8. #19151978	C₄	semidirect
9. #19151949	C₄	semidirect
60. #13083558	Q₁₆	semidirect
140. C₈₉	#9718	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26021195	2
#26018750	2
C₈×#13083558	2
C₄×#107×C₈₉	2
#9718	89