Subgroups — #36760541

Order 22784.

2502 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152155	C₂	1	2	yes	maximal
2	#26028650	C₂	1	2	yes	maximal, complemented
3	#26028630	C₂	1	2	yes	maximal, complemented
4	#26028615	C₂	1	4	yes	maximal, complemented
5	#26028604	C₂	1	8	yes	maximal
6	C₄×#15668012	C₂	1	8	yes	maximal, complemented
7	#26022659	C₂	1	4	yes	maximal
8	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152155	—	2	16	no	—
10	C₂×#15668012	C₂²	1	8	yes	—
11	C₂×#15668012	C₂²	1	4	yes	—
12	C₂×#15668004	C₂²	1	4	yes	—
13	C₂×#15667991	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₇₈	C₂²	1	8	yes	—
15	C₂×#15667971	C₂²	1	4	yes	—
16	#19151504	C₂²	1	16	yes	—
17	#19151496	C₂²	1	32	yes	—
18	#19151468	C₂²	1	8	yes	—
19	#19151468	C₂²	1	8	yes	—
20	#19151464	C₂²	1	8	yes	—
21	#19151458	C₂²	1	16	yes	—
22	C₄×C₄⋊Dic₈₉	C₂²	1	16	yes	—
23	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
24	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
26	C₄³×C₈₉	C₂²	1	2	yes	—
27	C₄³×C₈₉	C₂²	1	1	yes	characteristic
28	C₄³×C₈₉	C₂²	1	1	yes	characteristic
29	C₄³×C₈₉	C₂²	1	4	yes	—
30	C₂²×Dic₁₇₈	C₂³	1	8	yes	—
31	#15668012	—	2	16	no	—
32	#15668012	—	2	32	no	—
33	#15668004	—	2	16	no	—
34	#15667991	—	2	32	no	—
35	C₄×Dic₁₇₈	—	2	32	no	—
36	#15667971	—	2	16	no	—
37	C₂×C₄⋊Dic₈₉	C₂³	1	32	yes	—
38	C₂×C₄⋊Dic₈₉	C₂³	1	8	yes	—
39	C₂×C₄⋊Dic₈₉	C₂³	1	8	yes	—
40	C₂×#13083553	C₂³	1	16	yes	—
41	C₂×C₂²⋊Dic₈₉	C₂³	1	8	yes	—
42	C₂×C₄×Dic₈₉	C₂³	1	8	yes	—
43	#15667919	C₂³	1	32	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
45	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
46	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	—	2	16	no	—
48	C₂×C₄²×C₈₉	D₄	1	8	yes	—
49	C₂×C₄²×C₈₉	Q₈	1	16	yes	complemented
50	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
57	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
58	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
59	C₂×Dic₁₇₈	—	2	16	no	—
60	C₂×Dic₁₇₈	—	2	32	no	—
61	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
62	C₄⋊Dic₈₉	—	2	16	no	—
63	C₄⋊Dic₈₉	—	2	16	no	—
64	C₄⋊Dic₈₉	—	2	64	no	—
65	#13083553	—	2	32	no	—
66	C₂²⋊Dic₈₉	—	2	16	no	—
67	C₄×Dic₈₉	—	2	16	no	—
68	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	8	no	—
70	C₂²×C₄×C₈₉	—	2	16	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₈₉	C₂×Q₈	1	16	yes	—
74	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
75	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
81	C₄²×C₈₉	—	2	8	no	—
82	C₄²×C₈₉	—	2	16	no	—
83	C₄²×C₈₉	C₂×D₄	1	8	yes	—
84	C₄²×C₈₉	C₂×D₄	1	4	yes	—
85	C₄²×C₈₉	C₂×D₄	1	4	yes	—
86	C₄²×C₈₉	C₂×Q₈	1	8	yes	—
87	C₄²×C₈₉	C₂×Q₈	1	16	yes	—
88	C₄²×C₈₉	C₂×Q₈	1	8	yes	—
89	C₄²×C₈₉	#41	1	16	yes	—
90	C₂³×C₈₉	—	2	4	no	—
91	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
92	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
93	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₈₉	—	2	4	no	—
95	C₂×C₄×C₈₉	—	2	8	no	—
96	C₂×C₄×C₈₉	—	2	16	no	—
97	C₂×C₄×C₈₉	—	2	32	no	—
98	C₂×C₄×C₈₉	C₄⋊Q₈	1	16	yes	complemented
99	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
100	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
103	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
104	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
105	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
106	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
107	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
108	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
109	C₂×Dic₈₉	—	2	16	no	—
110	C₂×Dic₈₉	—	2	32	no	—
111	Dic₁₇₈	—	4	32	no	—
112	C₂²×C₈₉	—	2	4	no	—
113	C₂²×C₈₉	—	2	8	no	—
114	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
115	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
116	C₂²×C₈₉	C₂³×Q₈	1	2	yes	—
117	C₂²×C₈₉	C₂²×#41	1	4	yes	—
118	C₄×C₈₉	—	2	4	no	—
119	C₄×C₈₉	—	2	8	no	—
120	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
121	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
122	C₄×C₈₉	#532	1	8	yes	—
123	C₄×C₈₉	#533	1	16	yes	complemented
124	Dic₈₉	—	4	16	no	—
125	#36645	—	89	1	no	maximal, Sylow
126	C₂×C₈₉	—	2	4	no	—
127	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#532	1	2	yes	—
129	C₂×C₈₉	C₂×#533	1	4	yes	—
130	C₂×#533	—	89	2	no	—
131	#2868	—	89	2	no	—
132	#2862	—	89	2	no	—
133	#2859	—	89	4	no	—
134	#2857	—	89	8	no	—
135	C₄×C₄⋊Q₈	—	89	8	no	—
136	#2285	—	89	4	no	—
137	C₂×C₄³	D₈₉	1	1	yes	characteristic
138	C₈₉	#36645	1	1	yes	characteristic, complemented, Sylow
139	#533	—	178	16	no	—
140	C₂×C₄⋊Q₈	—	89	8	no	—
141	C₂×C₄⋊Q₈	—	89	4	no	—
142	C₂×#125	—	89	4	no	—
143	C₂×C₂²⋊Q₈	—	89	8	no	—
144	C₂×C₄×Q₈	—	89	8	no	—
145	C₂×#117	—	89	4	no	—
146	C₂²×C₄²	C₂×D₈₉	1	4	yes	—
147	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
148	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
149	#391	—	89	16	no	—
150	#389	—	89	32	no	—
151	#384	—	89	8	no	—
152	#384	—	89	8	no	—
153	#382	—	89	8	no	—
154	#380	—	89	16	no	—
155	C₄×C₄⋊C₄	—	89	16	no	—
156	C₄³	C₂×D₈₉	1	4	yes	—
157	C₄³	C₂×D₈₉	1	2	yes	—
158	C₄³	C₂×D₈₉	1	1	yes	characteristic
159	C₄³	C₂×D₈₉	1	1	yes	characteristic
160	C₂²×Q₈	—	89	8	no	—
161	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
162	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
163	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
164	C₄⋊Q₈	—	178	16	no	—
165	C₄⋊Q₈	—	178	32	no	—
166	#125	—	178	16	no	—
167	C₂²⋊Q₈	—	178	32	no	—
168	C₄×Q₈	—	178	32	no	—
169	#117	—	178	16	no	—
170	C₂×C₄⋊C₄	—	89	8	no	—
171	C₂×C₄⋊C₄	—	89	16	no	—
172	C₂×C₄⋊C₄	—	89	8	no	—
173	C₂×C₄⋊C₄	—	89	32	no	—
174	C₂×C₂²⋊C₄	—	89	8	no	—
175	C₂×C₄²	—	2	16	no	—
176	C₂×C₄²	—	89	8	no	—
177	C₂×C₄²	Dic₁₇₈	1	16	yes	complemented
178	C₂×C₄²	D₃₅₆	1	8	yes	—
179	C₂×C₄²	C₂²×D₈₉	1	8	yes	—
180	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
181	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
184	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
185	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
186	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
187	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
188	#95	—	89	32	no	—
189	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
190	C₂×Q₈	—	178	16	no	—
191	C₂×Q₈	—	178	32	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	16	no	—
194	C₂²×C₄	—	89	16	no	—
195	C₂²×C₄	C₂×D₃₅₆	1	8	yes	—
196	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
197	C₂²×C₄	C₂×Dic₁₇₈	1	16	yes	—
198	C₂²×C₄	C₂×Dic₁₇₈	1	8	yes	—
199	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
200	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
201	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
203	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
204	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
205	C₄⋊C₄	—	178	16	no	—
206	C₄⋊C₄	—	178	32	no	—
207	C₄⋊C₄	—	178	16	no	—
208	C₄⋊C₄	—	178	64	no	—
209	C₂²⋊C₄	—	178	16	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	16	no	—
212	C₄²	—	178	16	no	—
213	C₄²	C₂×D₃₅₆	1	8	yes	—
214	C₄²	C₂×D₃₅₆	1	4	yes	—
215	C₄²	C₂×D₃₅₆	1	4	yes	—
216	C₄²	C₂×Dic₁₇₈	1	8	yes	—
217	C₄²	C₂×Dic₁₇₈	1	16	yes	—
218	C₄²	C₂×Dic₁₇₈	1	8	yes	—
219	C₄²	#13083574	1	16	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	C₂²×D₃₅₆	1	2	yes	—
222	C₂³	C₂²×Dic₁₇₈	1	4	yes	—
223	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
224	Q₈	—	356	32	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	178	16	no	—
230	C₂×C₄	—	178	32	no	—
231	C₂×C₄	#15668012	1	16	yes	complemented
232	C₂×C₄	C₂²×D₃₅₆	1	4	yes	—
233	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
234	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
235	C₂×C₄	C₂²×D₃₅₆	1	4	yes	—
236	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
237	C₂×C₄	C₂²×Dic₁₇₈	1	4	yes	—
238	C₂×C₄	C₂²×Dic₁₇₈	1	4	yes	—
239	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
240	C₂×C₄	C₂×#13083574	1	8	yes	—
241	C₂×C₄	C₂×#13083574	1	16	yes	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	8	no	—
244	C₂²	C₂×#15668012	1	4	yes	—
245	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
246	C₂²	C₂³×Dic₁₇₈	1	2	yes	—
247	C₂²	C₂²×#13083574	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	8	no	—
250	C₄	—	356	16	no	—
251	C₄	C₂×#15668012	1	4	yes	—
252	C₄	C₂×#15668012	1	4	yes	—
253	C₄	#19152148	1	8	yes	—
254	C₄	#19152155	1	16	yes	complemented
255	C₂	—	2	4	no	—
256	C₂	C₂²×#15668012	1	1	yes	characteristic
257	C₂	C₂×#19152148	1	2	yes	—
258	C₂	C₂×#19152155	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	#36645
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #26028650	C₂	semidirect
3. #26028630	C₂	semidirect
4. #26028615	C₂	semidirect
6. C₄×#15668012	C₂	semidirect
49. C₂×C₄²×C₈₉	Q₈	semidirect
98. C₂×C₄×C₈₉	C₄⋊Q₈	semidirect
123. C₄×C₈₉	#533	semidirect
138. C₈₉	#36645	semidirect
177. C₂×C₄²	Dic₁₇₈	semidirect
231. C₂×C₄	#15668012	semidirect
254. C₄	#19152155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152155	2
#26028650	2
#26028630	2
#26028615	2
#26028604	2
C₄×#15668012	2
#26022659	2
C₂×C₄³×C₈₉	2
#36645	89