Subgroups — #36524961

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#26029186	C₂	1	2	yes	maximal
3	#26019150	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₃₅₆	C₂	1	4	yes	maximal, complemented
5	#26019011	C₂	1	1	yes	maximal, characteristic
6	#26018708	C₂	1	2	yes	maximal
7	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19152155	C₂²	1	2	yes	—
9	#19152003	C₄	1	1	yes	characteristic
10	#19151978	C₄	1	1	yes	characteristic
11	#19151949	C₄	1	2	yes	—
12	#19151839	C₄	1	4	yes	—
13	#19151728	—	2	2	no	—
14	#19151728	C₂²	1	4	yes	—
15	C₈×Dic₁₇₈	—	2	2	no	—
16	C₈×Dic₁₇₈	C₂²	1	4	yes	—
17	C₄×Dic₃₅₆	C₂²	1	4	yes	—
18	#19151624	—	2	2	no	—
19	#19151622	C₂²	1	2	yes	—
20	#19151553	—	2	2	no	—
21	#19151351	C₂²	1	4	yes	—
22	#19151330	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₈²×C₈₉	C₂²	1	2	yes	—
27	C₈²×C₈₉	C₂²	1	2	yes	—
28	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
29	C₂×#13083558	—	2	2	no	—
30	#15668012	C₂×C₄	1	2	yes	—
31	#15668012	C₂×C₄	1	2	yes	—
32	#15668004	—	2	2	no	—
33	#15667991	—	2	2	no	—
34	#15667991	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₇₈	—	2	2	no	—
36	C₄×Dic₁₇₈	C₂³	1	4	yes	—
37	#15667971	—	2	2	no	—
38	#15667946	C₂×C₄	1	4	yes	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	C₂³	1	4	yes	—
41	#15667941	—	2	4	no	—
42	#15667941	—	2	4	no	—
43	#15667937	C₂×C₄	1	8	yes	—
44	#15667922	—	2	4	no	—
45	C₈×Dic₈₉	—	2	4	no	—
46	C₂²×C₈×C₈₉	—	2	1	no	—
47	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₈₉	—	2	1	no	—
52	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂³	1	2	yes	—
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₇₈	—	2	2	no	—
61	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
62	Dic₃₅₆	—	2	8	no	—
63	#13083558	—	4	4	no	—
64	C₂×#10623494	—	2	4	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	—	2	2	no	—
68	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
69	#13083553	—	2	4	no	—
70	#13083553	—	2	4	no	—
71	C₂²⋊Dic₈₉	—	2	4	no	—
72	C₄×Dic₈₉	—	2	4	no	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	#41	1	1	yes	characteristic
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₈₉	#135	1	2	yes	—
99	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₈₉	—	2	4	no	—
103	C₂×Dic₈₉	—	2	4	no	—
104	Dic₁₇₈	—	2	8	no	—
105	Dic₁₇₈	—	4	4	no	—
106	C₈×C₈₉	—	2	4	no	—
107	C₈×C₈₉	—	2	2	no	—
108	C₈×C₈₉	—	2	1	no	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	#131	1	4	yes	—
111	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
112	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
113	#10623494	—	4	4	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
117	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
121	C₄×C₈₉	#443	1	2	yes	—
122	C₄×C₈₉	C₂×#131	1	2	yes	—
123	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
124	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
125	Dic₈₉	—	4	4	no	—
126	Dic₈₉	—	4	4	no	—
127	#17958	—	89	1	no	maximal, Sylow
128	C₂×C₈₉	—	2	2	no	—
129	C₂×C₈₉	#2931	1	1	yes	characteristic
130	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
131	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
132	#3170	—	89	1	no	—
133	#2960	—	89	2	no	—
134	#1591	—	89	4	no	—
135	C₈×Q₁₆	—	89	4	no	—
136	#1567	—	89	1	no	—
137	#1493	—	89	2	no	—
138	C₂×C₈²	D₈₉	1	1	yes	characteristic
139	C₈₉	#17958	1	1	yes	characteristic, complemented, Sylow
140	#533	—	89	2	no	—
141	#500	—	89	1	no	—
142	#494	—	89	1	no	—
143	#487	—	89	2	no	—
144	#467	—	89	4	no	—
145	#446	—	89	4	no	—
146	#446	—	178	2	no	—
147	C₈×Q₈	—	89	4	no	—
148	C₈×Q₈	—	178	2	no	—
149	C₄×Q₁₆	—	89	4	no	—
150	#424	—	178	2	no	—
151	#423	—	89	2	no	—
152	#405	—	178	2	no	—
153	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
156	#335	—	89	4	no	—
157	Q₈⋊C₈	—	89	8	no	—
158	C₈²	C₂×D₈₉	1	2	yes	—
159	C₈²	C₂×D₈₉	1	2	yes	—
160	C₂×Q₁₆	—	89	4	no	—
161	C₂×M₁₆	—	178	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₄×D₈₉	1	2	yes	—
164	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
165	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
166	C₄⋊Q₈	—	89	2	no	—
167	C₄⋊Q₈	—	89	2	no	—
168	#125	—	178	2	no	—
169	C₂²⋊Q₈	—	89	4	no	—
170	C₂²⋊Q₈	—	178	2	no	—
171	C₄×Q₈	—	89	4	no	—
172	C₄×Q₈	—	178	2	no	—
173	#117	—	178	2	no	—
174	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
175	#107	—	89	4	no	—
176	C₄⋊C₈	—	89	4	no	—
177	C₄⋊C₈	—	178	4	no	—
178	C₄⋊C₈	—	178	4	no	—
179	C₄⋊C₈	—	178	2	no	—
180	#103	—	89	8	no	—
181	#97	—	178	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	178	4	no	—
184	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
185	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₄×C₈	D₃₅₆	1	1	yes	characteristic
187	C₄×C₈	D₃₅₆	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₈₉	1	2	yes	—
189	C₄×C₈	C₂²×D₈₉	1	2	yes	—
190	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
192	C₂×Q₈	—	89	4	no	—
193	C₂×Q₈	—	178	2	no	—
194	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
196	C₂²×C₄	#13083574	1	1	yes	characteristic
197	Q₁₆	—	178	8	no	—
198	M₁₆	—	356	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	178	4	no	—
204	C₂×C₈	Dic₃₅₆	1	4	yes	complemented
205	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
207	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
208	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
209	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
210	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
211	C₄⋊C₄	—	89	4	no	—
212	C₄⋊C₄	—	178	2	no	—
213	C₄⋊C₄	—	178	2	no	—
214	C₄⋊C₄	—	178	4	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₄⋊C₄	—	178	4	no	—
217	C₂²⋊C₄	—	178	4	no	—
218	C₄²	—	178	4	no	—
219	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
221	C₄²	#13083574	1	1	yes	characteristic
222	C₄²	C₂³×D₈₉	1	1	yes	characteristic
223	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
224	Q₈	—	356	4	no	—
225	Q₈	—	178	8	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	178	4	no	—
229	C₂×C₄	—	178	4	no	—
230	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
231	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
232	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
234	C₂×C₄	#15668034	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
237	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	356	4	no	—
243	C₈	#15668018	1	4	yes	—
244	C₈	C₂×Dic₃₅₆	1	2	yes	—
245	C₈	C₂×Dic₃₅₆	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₃₅₆	1	2	yes	—
249	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	356	4	no	—
253	C₄	—	356	4	no	—
254	C₄	C₄×Dic₃₅₆	1	2	yes	—
255	C₄	#19151716	1	2	yes	—
256	C₄	C₂×#15668018	1	2	yes	—
257	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
258	C₄	C₂×#15668034	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#26028965	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₃₅₆	1	1	yes	characteristic
262	C₂	C₂×#19151716	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#17958
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #26019150	C₂	semidirect
4. C₈×Dic₃₅₆	C₂	semidirect
80. C₂×C₈×C₈₉	Q₁₆	semidirect
139. C₈₉	#17958	semidirect
204. C₂×C₈	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26029186	2
#26019150	2
C₈×Dic₃₅₆	2
#26019011	2
#26018708	2
C₂×C₈²×C₈₉	2
#17958	89