Subgroups — #36707740

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668034	C₂	1	2	yes	maximal, complemented
3	#26022129	C₂	1	2	yes	maximal, complemented
4	#26022124	C₂	1	2	yes	maximal, complemented
5	#26022118	C₂	1	4	yes	maximal, complemented
6	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021960	C₂	1	2	yes	maximal, complemented
8	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	2	yes	—
10	C₄×#13083574	—	2	2	no	—
11	#19152030	C₄	1	8	yes	complemented
12	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
13	C₂×#15667957	C₂²	1	2	yes	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₃₅₆	C₂²	1	4	yes	—
16	C₄×#13083563	—	2	4	no	—
17	C₄×D₇₁₂	C₂²	1	4	yes	—
18	#19151636	C₂²	1	2	yes	—
19	#19151595	—	2	2	no	—
20	#19151426	C₂²	1	2	yes	—
21	#19151424	C₂²	1	2	yes	—
22	#19151403	C₂²	1	4	yes	—
23	#19151400	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	#346×C₈₉	C₂²	1	2	yes	—
28	#346×C₈₉	C₂²	1	2	yes	—
29	C₂×#13083574	—	2	2	no	—
30	#15668034	—	2	4	no	—
31	#15668034	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₃₅₆	C₂³	1	2	yes	—
33	C₂×#13083563	—	2	2	no	—
34	C₂×D₇₁₂	C₂³	1	2	yes	—
35	C₄×Dic₁₇₈	—	2	4	no	—
36	C₄×D₃₅₆	—	2	4	no	—
37	C₄×C₂²⋊D₈₉	—	4	4	no	—
38	#15667971	—	2	2	no	—
39	#15667969	—	4	2	no	—
40	C₄²×D₈₉	—	4	2	no	—
41	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
42	#15667957	C₂×C₄	1	8	yes	complemented
43	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
44	#15667946	C₂³	1	4	yes	—
45	#15667944	—	2	2	no	—
46	#15667937	—	2	4	no	—
47	#15667934	—	2	4	no	—
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₈₉	—	2	2	no	—
52	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₈₉	D₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	#13083574	—	2	8	no	—
59	#13083574	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×C₂²⋊D₈₉	—	4	2	no	—
63	C₂×C₄×D₈₉	—	4	2	no	—
64	C₂×C₄×D₈₉	—	4	4	no	—
65	Dic₃₅₆	—	2	2	no	—
66	Dic₃₅₆	C₂²×C₄	1	4	yes	—
67	#13083563	—	2	8	no	—
68	D₇₁₂	—	2	2	no	—
69	D₇₁₂	C₂²×C₄	1	4	yes	—
70	C₄⋊Dic₈₉	—	2	4	no	—
71	#13083553	—	4	4	no	—
72	C₂²⋊Dic₈₉	—	4	2	no	—
73	#13083551	—	4	4	no	—
74	C₄×Dic₈₉	—	4	2	no	—
75	C₄×Dic₈₉	—	4	4	no	—
76	C₂²×C₄×C₈₉	D₈	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	#41	1	2	yes	—
83	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₄²×C₈₉	D₈	1	2	yes	—
85	C₄²×C₈₉	C₂×D₄	1	2	yes	—
86	C₁₆×C₈₉	—	2	2	no	—
87	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
88	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
90	C₂²×D₈₉	—	4	2	no	—
91	C₂×C₄×C₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#135	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₈₉	—	4	8	no	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	4	no	—
100	D₃₅₆	—	2	4	no	—
101	D₃₅₆	—	4	2	no	—
102	C₄×D₈₉	—	4	8	no	—
103	C₄×D₈₉	—	8	8	no	—
104	Dic₁₇₈	—	2	4	no	—
105	Dic₁₇₈	—	4	2	no	—
106	C₈×C₈₉	—	2	2	no	—
107	C₈×C₈₉	C₄×D₄	1	2	yes	—
108	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	C₄×D₈	1	2	yes	—
112	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₈₉	—	4	4	no	—
114	C₂×D₈₉	—	8	4	no	—
115	C₄×C₈₉	—	2	4	no	—
116	C₄×C₈₉	C₄×D₈	1	2	yes	—
117	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
119	Dic₈₉	—	4	4	no	—
120	Dic₈₉	—	8	4	no	—
121	#33451	—	89	1	no	maximal, Sylow
122	C₂×C₈₉	—	2	2	no	—
123	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
124	C₂×C₈₉	#3414	1	2	yes	—
125	D₈₉	—	8	4	no	—
126	C₂×#508	—	89	1	no	—
127	C₄×#135	—	89	2	no	—
128	#2177	—	89	2	no	—
129	#2176	—	89	2	no	—
130	#2175	—	89	4	no	—
131	#2158	—	89	1	no	—
132	#2138	—	89	2	no	—
133	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
134	C₈₉	#33451	1	1	yes	characteristic, complemented, Sylow
135	C₂×#135	—	89	2	no	—
136	C₄×#41	—	178	2	no	—
137	#508	—	89	8	no	—
138	C₂×Q₃₂	—	89	1	no	—
139	C₂×SD₃₂	—	89	2	no	—
140	C₂×D₁₆	—	89	1	no	—
141	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
142	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
143	C₄×Q₁₆	—	89	4	no	—
144	C₄×SD₁₆	—	178	4	no	—
145	C₄×D₈	—	89	4	no	—
146	#427	—	89	2	no	—
147	#416	—	178	2	no	—
148	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
149	#367	—	89	2	no	—
150	#366	—	89	2	no	—
151	#358	—	89	4	no	—
152	#357	—	89	4	no	—
153	#346	C₂×D₈₉	1	2	yes	—
154	#346	C₂×D₈₉	1	2	yes	—
155	C₂×#41	—	178	2	no	—
156	#135	—	89	8	no	—
157	#135	—	178	4	no	—
158	C₂×Q₁₆	—	89	2	no	—
159	C₂×SD₁₆	—	178	2	no	—
160	C₂×D₈	—	89	2	no	—
161	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	C₄×Q₈	—	178	4	no	—
164	C₄×D₄	—	356	4	no	—
165	C₄×D₄	—	178	4	no	—
166	#117	—	356	2	no	—
167	#117	—	178	2	no	—
168	C₂×C₄²	—	356	2	no	—
169	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
170	Q₃₂	—	89	4	no	—
171	SD₃₂	—	89	8	no	—
172	D₁₆	—	89	4	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
175	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
176	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
178	#107	—	89	4	no	—
179	#106	—	178	2	no	—
180	#103	—	178	4	no	—
181	#102	—	178	4	no	—
182	C₄×C₈	D₃₅₆	1	2	yes	—
183	C₄×C₈	C₂²×D₈₉	1	2	yes	—
184	#41	—	356	4	no	—
185	#41	—	178	8	no	—
186	C₂×Q₈	—	178	2	no	—
187	C₂×D₄	—	356	2	no	—
188	C₂×D₄	—	178	2	no	—
189	C₂²×C₄	—	356	2	no	—
190	C₂²×C₄	—	356	4	no	—
191	C₂²×C₄	D₇₁₂	1	2	yes	—
192	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
193	Q₁₆	—	89	4	no	—
194	Q₁₆	—	178	2	no	—
195	SD₁₆	—	178	8	no	—
196	D₈	—	89	4	no	—
197	D₈	—	178	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
200	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
201	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
202	C₂×C₈	#13083574	1	2	yes	—
203	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
204	C₄⋊C₄	—	356	4	no	—
205	C₄⋊C₄	—	178	4	no	—
206	C₂²⋊C₄	—	356	2	no	—
207	C₂²⋊C₄	—	356	4	no	—
208	C₄²	—	356	4	no	—
209	C₄²	—	356	2	no	—
210	C₄²	D₇₁₂	1	2	yes	—
211	C₄²	C₂×D₃₅₆	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
214	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
215	C₂³	—	356	2	no	—
216	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
217	Q₈	—	356	2	no	—
218	Q₈	—	178	4	no	—
219	D₄	—	356	8	no	—
220	D₄	—	356	2	no	—
221	D₄	—	178	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	—	356	8	no	—
225	C₂×C₄	—	356	4	no	—
226	C₂×C₄	—	712	8	no	—
227	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
228	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
229	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
230	C₂×C₄	#15668034	1	2	yes	—
231	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₃₅₆	1	2	yes	—
234	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
235	C₈	C₂×#13083574	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	356	4	no	—
238	C₂²	—	712	4	no	—
239	C₂²	C₄×D₇₁₂	1	2	yes	—
240	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	356	4	no	—
243	C₄	—	712	4	no	—
244	C₄	C₄×D₇₁₂	1	2	yes	—
245	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
246	C₄	C₂×#15668034	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	712	4	no	—
249	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
250	C₂	#26033388	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#19152030	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#33451
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₄×#15668034	C₂	semidirect
3. #26022129	C₂	semidirect
4. #26022124	C₂	semidirect
5. #26022118	C₂	semidirect
6. #26022036	C₂	semidirect
7. #26021960	C₂	semidirect
8. C₂×#346×C₈₉	C₂	semidirect
11. #19152030	C₄	semidirect
31. #15668034	C₂×C₄	semidirect
41. Dic₇₁₂	C₂×C₄	semidirect
42. #15667957	C₂×C₄	semidirect
43. D₁₄₂₄	C₂×C₄	semidirect
52. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
133. C₂×#346	D₈₉	semidirect
134. C₈₉	#33451	semidirect
174. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152030	2
C₄×#15668034	2
#26022129	2
#26022124	2
#26022118	2
#26022036	2
#26021960	2
C₂×#346×C₈₉	2
#33451	89