Subgroups — #36460341

Order 22784.

766 conjugacy classes of subgroups.

The subgroup lattice has 258 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	2	yes	maximal
2	C₂×#19151363	C₂	1	2	yes	maximal
3	#26019946	C₂	1	2	yes	maximal
4	C₂²×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	4	yes	—
6	C₂×#15667948	—	2	4	no	—
7	C₂×#15667948	—	2	4	no	—
8	C₂×#15667948	C₄	1	8	yes	complemented
9	#19151363	—	2	4	no	—
10	#19151353	—	2	4	no	—
11	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#97×C₈₉	—	2	4	no	—
15	C₂×#97×C₈₉	C₄	1	4	yes	—
16	C₂×#13083558	—	2	4	no	—
17	C₂×#13083558	—	2	4	no	—
18	C₂×#13083558	—	2	4	no	—
19	C₂×#13083558	—	2	4	no	—
20	C₂×#13083558	—	2	4	no	—
21	C₂×#13083558	C₂×C₄	1	8	yes	—
22	C₂²×#10623494	—	2	4	no	—
23	#15667948	—	2	16	no	—
24	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₈₉	D₄	1	2	yes	—
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₂²×C₈×C₈₉	D₄	1	4	yes	—
34	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
35	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
36	C₂×C₄²×C₈₉	—	2	2	no	—
37	C₂×C₄²×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	D₄	1	2	yes	—
40	#97×C₈₉	—	2	8	no	—
41	#13083558	—	2	8	no	—
42	#13083558	—	2	8	no	—
43	#13083558	—	4	4	no	—
44	#13083558	—	4	4	no	—
45	C₂×#10623494	—	2	8	no	—
46	C₂×#10623494	—	4	4	no	—
47	C₂×#10623494	—	4	4	no	—
48	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	—	2	4	no	—
50	C₂²×C₄×C₈₉	—	4	4	no	—
51	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	#41	1	2	yes	—
61	C₂²×C₄×C₈₉	#41	1	2	yes	—
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	C₄²	1	4	yes	—
67	C₂×C₈×C₈₉	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
71	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
72	C₄²×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	—	2	4	no	—
75	C₂³×C₈₉	—	2	4	no	—
76	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
78	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
79	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
80	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	2	4	no	—
82	C₂×C₄×C₈₉	—	2	8	no	—
83	C₂×C₄×C₈₉	—	4	8	no	—
84	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₈₉	#121	1	2	yes	—
96	C₂×C₄×C₈₉	#121	1	2	yes	—
97	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₈₉	#124	1	2	yes	—
99	C₂×C₄×C₈₉	#125	1	2	yes	—
100	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₈×C₈₉	—	2	4	no	—
103	C₈×C₈₉	#95	1	8	yes	—
104	#10623494	—	4	8	no	—
105	C₂²×C₈₉	—	2	4	no	—
106	C₂²×C₈₉	—	2	2	no	—
107	C₂²×C₈₉	—	2	2	no	—
108	C₂²×C₈₉	—	2	4	no	—
109	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₈₉	#380	1	1	yes	characteristic
111	C₂²×C₈₉	#384	1	1	yes	characteristic
112	C₂²×C₈₉	#389	1	2	yes	—
113	C₂²×C₈₉	#391	1	2	yes	—
114	C₂²×C₈₉	#430	1	4	yes	—
115	C₄×C₈₉	—	4	4	no	—
116	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
117	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₈₉	#380	1	1	yes	characteristic
119	C₄×C₈₉	#382	1	1	yes	characteristic
120	C₄×C₈₉	#389	1	2	yes	—
121	C₄×C₈₉	#390	1	2	yes	—
122	#10865	—	89	1	no	maximal, Sylow
123	C₂×C₈₉	—	2	2	no	—
124	C₂×C₈₉	—	2	2	no	—
125	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
126	C₂×C₈₉	#1778	1	2	yes	—
127	C₂×C₈₉	#1811	1	2	yes	—
128	C₂×C₈₉	#1849	1	2	yes	—
129	C₂²×#108	—	89	2	no	—
130	C₂²×#97	D₈₉	1	1	yes	characteristic
131	C₂×#341	—	89	2	no	—
132	#1728	—	89	2	no	—
133	C₈₉	#10865	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	—	89	4	no	—
135	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
137	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
138	C₂×#108	—	89	8	no	—
139	C₂×#108	—	178	4	no	—
140	C₂×#108	—	178	4	no	—
141	C₂×#97	—	2	4	no	—
142	C₂×#97	Dic₈₉	1	4	yes	—
143	#341	—	178	4	no	—
144	#336	—	178	4	no	—
145	C₂³×C₄	Dic₁₇₈	1	2	yes	—
146	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
147	C₂×M₁₆	—	89	8	no	—
148	C₂×M₁₆	—	178	4	no	—
149	C₂×M₁₆	—	178	4	no	—
150	C₂×M₁₆	—	178	4	no	—
151	C₂×M₁₆	—	178	4	no	—
152	C₂×M₁₆	—	178	4	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	178	4	no	—
156	C₂²×C₈	Dic₁₇₈	1	2	yes	—
157	C₂²×C₈	Dic₁₇₈	1	2	yes	—
158	C₂²×C₈	C₄×D₈₉	1	4	yes	—
159	C₂²×C₈	D₃₅₆	1	2	yes	—
160	C₂²×C₈	D₃₅₆	1	2	yes	—
161	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
162	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	D₃₅₆	1	2	yes	—
167	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
168	#108	—	178	16	no	—
169	#97	—	2	8	no	—
170	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
174	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
177	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
182	C₂²×C₄	#13083574	1	2	yes	—
183	C₂²×C₄	#13083574	1	2	yes	—
184	M₁₆	—	356	4	no	—
185	M₁₆	—	356	4	no	—
186	M₁₆	—	178	8	no	—
187	M₁₆	—	178	8	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	356	4	no	—
193	C₂×C₈	—	356	4	no	—
194	C₂×C₈	—	178	8	no	—
195	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
196	C₂×C₈	#13083551	1	8	yes	—
197	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
198	C₂×C₈	#13083553	1	8	yes	—
199	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
200	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
206	C₂³	C₄×Dic₁₇₈	1	2	yes	—
207	C₂³	#15667991	1	2	yes	—
208	C₂³	#15668012	1	1	yes	characteristic
209	C₂³	#15668012	1	1	yes	characteristic
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
214	C₂×C₄	C₂×#13083551	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
216	C₂×C₄	C₂×#13083553	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
219	C₂×C₄	#15667971	1	1	yes	characteristic
220	C₂×C₄	#15667971	1	1	yes	characteristic
221	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
222	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
223	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
224	C₂×C₄	#15667986	1	2	yes	—
225	C₂×C₄	#15667986	1	2	yes	—
226	C₂×C₄	#15667991	1	2	yes	—
227	C₂×C₄	#15668000	1	2	yes	—
228	C₂×C₄	#15668004	1	2	yes	—
229	C₂×C₄	#15668009	1	1	yes	characteristic
230	C₂×C₄	#15668012	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	356	8	no	—
233	C₈	#15667919	1	8	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
239	C₂²	#19151458	1	1	yes	characteristic
240	C₂²	#19151468	1	1	yes	characteristic
241	C₂²	#19151496	1	2	yes	—
242	C₂²	#19151504	1	2	yes	—
243	C₂²	#19151647	1	4	yes	—
244	C₄	—	4	4	no	—
245	C₄	C₂×#15667919	1	1	yes	characteristic
246	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
247	C₄	#19151458	1	1	yes	characteristic
248	C₄	#19151464	1	1	yes	characteristic
249	C₄	#19151496	1	2	yes	—
250	C₄	#19151500	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	2	2	no	—
253	C₂	#26018482	1	1	yes	characteristic
254	C₂	#26020158	1	2	yes	—
255	C₂	#26020294	1	2	yes	—
256	C₂	#26020427	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂²×#97×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂²×#97	D₈₉
hypocenter	C₈₉	#10865
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
8. C₂×#15667948	C₄	semidirect
133. C₈₉	#10865	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×#19151363	2
#26019946	2
C₂²×#97×C₈₉	2
#10865	89