Subgroups — #36454518

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028630	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
3	#26018134	C₂	1	4	yes	maximal
4	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	1	yes	characteristic
6	C₂×#15668012	C₄	1	1	yes	characteristic
7	C₂×#15667923	—	2	1	no	—
8	C₂×#15667923	C₂²	1	2	yes	—
9	C₂×C₄×#10623494	—	2	1	no	—
10	#19151504	—	2	1	no	—
11	#19151504	—	2	1	no	—
12	#19151504	—	2	1	no	—
13	#19151504	—	2	1	no	—
14	#19151464	C₄	1	2	yes	—
15	#19151464	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×#97×C₈₉	C₂²	1	2	yes	—
18	C₄³×C₈₉	C₂²	1	1	yes	characteristic
19	#336×C₈₉	—	2	1	no	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂²×Dic₁₇₈	—	2	1	no	—
22	C₂²×#10623494	—	2	2	no	—
23	#15668012	—	4	2	no	—
24	#15668012	—	4	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	#15667923	—	2	4	no	—
30	#15667923	—	2	2	no	—
31	#15667923	Q₈	1	4	yes	complemented
32	C₄×#10623494	—	2	4	no	—
33	#15667919	—	2	2	no	—
34	#15667919	—	2	2	no	—
35	#15667919	—	2	2	no	—
36	#15667919	—	2	2	no	—
37	C₂²×C₈×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₈₉	—	2	4	no	—
46	#97×C₈₉	—	2	4	no	—
47	C₂×Dic₁₇₈	—	4	2	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂×Dic₁₇₈	—	4	2	no	—
50	C₂×Dic₁₇₈	—	4	2	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂²×Dic₈₉	—	2	2	no	—
54	C₂²×Dic₈₉	—	2	2	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	—	2	4	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₂×C₈×C₈₉	—	4	2	no	—
72	C₄²×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	4	1	no	—
78	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
79	C₄²×C₈₉	#41	1	2	yes	—
80	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	#100	1	2	yes	—
95	C₂×C₄×C₈₉	#101	1	2	yes	—
96	C₂×C₄×C₈₉	#104	1	2	yes	—
97	C₂×C₄×C₈₉	#104	1	2	yes	—
98	C₂×C₄×C₈₉	#104	1	2	yes	—
99	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×Dic₈₉	—	4	2	no	—
104	C₂×Dic₈₉	—	4	2	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	Dic₁₇₈	—	8	4	no	—
110	Dic₁₇₈	—	8	4	no	—
111	C₈×C₈₉	—	4	4	no	—
112	#10623494	—	2	8	no	—
113	C₂²×C₈₉	#391	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
116	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
117	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
119	C₂²×C₈₉	#421	1	1	yes	characteristic
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	—	2	1	no	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	—	2	2	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	4	1	no	—
127	C₄×C₈₉	#329	1	4	yes	—
128	Dic₈₉	—	8	2	no	—
129	Dic₈₉	—	8	2	no	—
130	#9890	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	C₂×#329	1	1	yes	characteristic
132	C₂×C₈₉	#1526	1	1	yes	characteristic
133	C₂×C₈₉	#1527	1	1	yes	characteristic
134	C₂×C₈₉	#1531	1	1	yes	characteristic
135	C₂×C₈₉	#1986	1	1	yes	characteristic
136	C₂×C₈₉	#1994	1	1	yes	characteristic
137	C₂×C₈₉	#1997	1	1	yes	characteristic
138	#2862	—	89	1	no	—
139	#1767	D₈₉	1	1	yes	characteristic
140	C₄×#97	—	89	1	no	—
141	#1295	—	89	4	no	—
142	C₈₉	#9890	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊Q₈	—	89	1	no	—
144	C₂×C₄⋊Q₈	—	89	1	no	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×#97	—	89	2	no	—
147	C₂×#97	—	178	1	no	—
148	C₂×#97	C₂×D₈₉	1	2	yes	—
149	C₂×C₄×C₈	—	178	1	no	—
150	#391	—	178	1	no	—
151	#391	—	178	1	no	—
152	#391	—	178	1	no	—
153	#391	—	178	1	no	—
154	#382	—	89	2	no	—
155	#382	—	89	2	no	—
156	C₄³	C₂×D₈₉	1	1	yes	characteristic
157	#336	—	2	1	no	—
158	C₂²×Q₈	—	178	1	no	—
159	C₂²×Q₈	—	178	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	178	2	no	—
162	C₄⋊Q₈	—	356	2	no	—
163	C₄⋊Q₈	—	356	2	no	—
164	C₂×C₄⋊C₄	—	89	2	no	—
165	C₂×C₄⋊C₄	—	89	2	no	—
166	C₂×C₄⋊C₄	—	178	1	no	—
167	C₂×C₄⋊C₄	—	178	1	no	—
168	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
170	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
171	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	#97	—	2	4	no	—
177	#97	—	89	4	no	—
178	#97	—	178	4	no	—
179	#97	—	178	2	no	—
180	C₄×C₈	—	178	4	no	—
181	#95	—	178	2	no	—
182	#95	—	178	2	no	—
183	#95	—	178	2	no	—
184	#95	—	178	2	no	—
185	C₂×Q₈	—	356	2	no	—
186	C₂×Q₈	—	356	2	no	—
187	C₂×Q₈	—	356	2	no	—
188	C₂×Q₈	—	356	2	no	—
189	C₂×Q₈	—	356	2	no	—
190	C₂×Q₈	—	356	2	no	—
191	C₂²×C₄	—	178	2	no	—
192	C₂²×C₄	—	178	2	no	—
193	C₂²×C₄	#13083551	1	1	yes	characteristic
194	C₂²×C₄	#13083551	1	1	yes	characteristic
195	C₂²×C₄	#13083551	1	1	yes	characteristic
196	C₂²×C₄	#13083551	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	178	4	no	—
204	C₂×C₈	—	178	4	no	—
205	C₂×C₈	—	178	4	no	—
206	C₄⋊C₄	—	356	2	no	—
207	C₄⋊C₄	—	356	2	no	—
208	C₄⋊C₄	—	356	2	no	—
209	C₄⋊C₄	—	356	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	1	no	—
216	C₄²	Q₈×D₈₉	1	2	yes	—
217	C₄²	#13083573	1	2	yes	—
218	C₂³	C₂×#13083551	1	1	yes	characteristic
219	Q₈	—	712	4	no	—
220	Q₈	—	712	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	—	356	2	no	—
236	C₂×C₄	—	356	2	no	—
237	C₂×C₄	—	356	2	no	—
238	C₂×C₄	—	356	2	no	—
239	C₂×C₄	—	356	2	no	—
240	C₂×C₄	#15667928	1	2	yes	—
241	C₂×C₄	#15667930	1	2	yes	—
242	C₂×C₄	#15667938	1	2	yes	—
243	C₂×C₄	#15667938	1	2	yes	—
244	C₂×C₄	#15667940	1	2	yes	—
245	C₂×C₄	#15667977	1	2	yes	—
246	C₂×C₄	#15667992	1	2	yes	—
247	C₂×C₄	#15667999	1	1	yes	characteristic
248	C₂×C₄	#15668010	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	178	8	no	—
251	C₂²	#19151503	1	1	yes	characteristic
252	C₂²	C₂×#15667928	1	1	yes	characteristic
253	C₂²	C₂×#15667930	1	1	yes	characteristic
254	C₂²	C₂×#15667938	1	1	yes	characteristic
255	C₂²	C₂×#15667938	1	1	yes	characteristic
256	C₂²	C₂×#15667940	1	1	yes	characteristic
257	C₂²	#19151616	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	712	2	no	—
266	C₄	—	712	2	no	—
267	C₄	#19151337	1	4	yes	—
268	C₂	C₂×#19151337	1	1	yes	characteristic
269	C₂	#26018846	1	1	yes	characteristic
270	C₂	#26018851	1	1	yes	characteristic
271	C₂	#26018868	1	1	yes	characteristic
272	C₂	#26021195	1	1	yes	characteristic
273	C₂	#26021228	1	1	yes	characteristic
274	C₂	#26021240	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#9890
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
31. #15667923	Q₈	semidirect
142. C₈₉	#9890	semidirect

Maximal subgroups

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

subgroup	index
#26028630	2
C₄×#15667923	2
#26018134	2
C₈₉×#1767	2
#9890	89