Subgroups — #36454553

Order 22784.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028650	C₂	1	1	yes	maximal, characteristic
2	#26020404	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	C₂×#15668012	—	2	1	no	—
7	C₂×#15668012	C₄	1	2	yes	—
8	C₂×C₄⋊#10623494	—	2	1	no	—
9	C₂×C₄⋊#10623494	—	2	1	no	—
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	#19151468	—	2	1	no	—
12	#19151468	—	2	1	no	—
13	#19151468	C₄	1	2	yes	—
14	#19151468	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	—	2	1	no	—
17	C₄³×C₈₉	C₂²	1	1	yes	characteristic
18	#336×C₈₉	—	2	1	no	—
19	C₂²×Dic₁₇₈	—	2	2	no	—
20	C₂²×#10623494	—	2	2	no	—
21	#15668012	—	4	4	no	—
22	#15668012	—	4	2	no	—
23	#15668012	—	4	2	no	—
24	#15668012	—	4	2	no	—
25	#15668012	—	4	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	—	2	2	no	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
32	C₄⋊#10623494	—	2	4	no	—
33	C₄⋊#10623494	—	4	2	no	—
34	C₄⋊#10623494	—	4	2	no	—
35	C₂²×C₈×C₈₉	—	2	2	no	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	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	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	—	2	2	no	—
44	C₄⋊C₈×C₈₉	Q₈	1	4	yes	complemented
45	C₄×C₈×C₈₉	—	4	2	no	—
46	C₂×Dic₁₇₈	—	4	4	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂×Dic₁₇₈	—	4	4	no	—
49	C₂²×Dic₈₉	—	2	4	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	4	no	—
54	C₄⋊Dic₈₉	—	4	4	no	—
55	C₄⋊Dic₈₉	—	4	4	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	—	4	4	no	—
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₂×C₈×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	—	4	2	no	—
71	C₄²×C₈₉	—	2	2	no	—
72	C₄²×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	D₈	1	2	yes	—
75	C₄²×C₈₉	D₈	1	2	yes	—
76	C₄²×C₈₉	SD₁₆	1	2	yes	—
77	C₄²×C₈₉	SD₁₆	1	2	yes	—
78	C₄²×C₈₉	SD₁₆	1	2	yes	—
79	C₄²×C₈₉	Q₁₆	1	2	yes	—
80	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
81	C₄²×C₈₉	#41	1	2	yes	—
82	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#100	1	2	yes	—
90	C₂×C₄×C₈₉	#101	1	2	yes	—
91	C₂×C₄×C₈₉	#102	1	2	yes	—
92	C₂×C₄×C₈₉	#102	1	2	yes	—
93	C₂×C₄×C₈₉	#102	1	2	yes	—
94	C₂×C₄×C₈₉	#102	1	2	yes	—
95	C₂×C₄×C₈₉	#103	1	2	yes	—
96	C₂×C₄×C₈₉	#103	1	2	yes	—
97	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#137	1	2	yes	—
108	C₂×Dic₈₉	—	4	4	no	—
109	C₂×Dic₈₉	—	4	4	no	—
110	C₂×Dic₈₉	—	4	4	no	—
111	Dic₁₇₈	—	8	8	no	—
112	C₈×C₈₉	—	4	4	no	—
113	#10623494	—	8	2	no	—
114	C₂²×C₈₉	#391	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
116	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
117	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
120	C₂²×C₈₉	#419	1	1	yes	characteristic
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	#331	1	4	yes	—
125	C₄×C₈₉	#332	1	4	yes	—
126	C₄×C₈₉	#474	1	2	yes	—
127	C₄×C₈₉	#475	1	2	yes	—
128	C₄×C₈₉	#486	1	1	yes	characteristic
129	C₄×C₈₉	#487	1	1	yes	characteristic
130	C₄×C₈₉	#498	1	1	yes	characteristic
131	C₄×C₈₉	#500	1	1	yes	characteristic
132	Dic₈₉	—	8	4	no	—
133	#9898	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	C₂×#331	1	1	yes	characteristic
135	C₂×C₈₉	C₂×#332	1	1	yes	characteristic
136	C₂×C₈₉	#1546	1	1	yes	characteristic
137	C₂×C₈₉	#1548	1	1	yes	characteristic
138	C₂×C₈₉	#1988	1	1	yes	characteristic
139	C₂×C₈₉	#1990	1	1	yes	characteristic
140	C₂×C₈₉	#1997	1	1	yes	characteristic
141	#2868	—	89	1	no	—
142	#1842	—	89	1	no	—
143	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
144	#1296	—	89	4	no	—
145	C₈₉	#9898	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄⋊Q₈	—	89	2	no	—
147	C₂×C₄⋊Q₈	—	178	1	no	—
148	C₂×C₄⋊Q₈	—	178	1	no	—
149	C₂×C₄⋊C₈	—	89	2	no	—
150	C₂×C₄⋊C₈	—	178	1	no	—
151	C₂×C₄⋊C₈	—	178	1	no	—
152	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#384	—	89	2	no	—
155	#384	—	89	2	no	—
156	#384	—	178	1	no	—
157	#384	—	178	1	no	—
158	C₄³	C₂×D₈₉	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	C₂²×Q₈	—	178	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	178	2	no	—
163	C₄⋊Q₈	—	356	4	no	—
164	C₄⋊Q₈	—	356	2	no	—
165	C₄⋊Q₈	—	356	2	no	—
166	C₄⋊Q₈	—	356	2	no	—
167	C₄⋊Q₈	—	356	2	no	—
168	C₂×C₄⋊C₄	—	89	4	no	—
169	C₂×C₄⋊C₄	—	178	2	no	—
170	C₂×C₄⋊C₄	—	178	2	no	—
171	C₂×C₄⋊C₄	—	178	2	no	—
172	C₂×C₄⋊C₄	—	178	2	no	—
173	C₂×C₄⋊C₄	—	178	2	no	—
174	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
177	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	356	2	no	—
183	C₄⋊C₈	—	356	2	no	—
184	C₄⋊C₈	—	178	4	no	—
185	C₄⋊C₈	Dic₁₇₈	1	4	yes	complemented
186	C₄×C₈	—	4	2	no	—
187	C₂×Q₈	—	356	4	no	—
188	C₂×Q₈	—	356	4	no	—
189	C₂×Q₈	—	356	4	no	—
190	C₂²×C₄	—	178	4	no	—
191	C₂²×C₄	#13083551	1	1	yes	characteristic
192	C₂²×C₄	#13083551	1	1	yes	characteristic
193	C₂²×C₄	#13083551	1	1	yes	characteristic
194	C₂²×C₄	#13083551	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	356	2	no	—
202	C₂×C₈	—	356	2	no	—
203	C₂×C₈	—	356	2	no	—
204	C₄⋊C₄	—	356	4	no	—
205	C₄⋊C₄	—	356	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	C₄⋊C₄	—	356	4	no	—
211	C₄⋊C₄	—	356	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	#13083559	1	2	yes	—
216	C₄²	D₇₁₂	1	2	yes	—
217	C₄²	#13083561	1	2	yes	—
218	C₄²	#13083563	1	2	yes	—
219	C₄²	#13083563	1	2	yes	—
220	C₄²	Dic₃₅₆	1	2	yes	—
221	C₄²	C₂×Dic₁₇₈	1	2	yes	—
222	C₄²	#13083574	1	2	yes	—
223	C₂³	C₂×#13083551	1	1	yes	characteristic
224	Q₈	—	712	8	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	—	356	4	no	—
232	C₂×C₄	—	356	4	no	—
233	C₂×C₄	—	356	4	no	—
234	C₂×C₄	#15667928	1	2	yes	—
235	C₂×C₄	#15667930	1	2	yes	—
236	C₂×C₄	#15667932	1	2	yes	—
237	C₂×C₄	#15667932	1	2	yes	—
238	C₂×C₄	#15667934	1	2	yes	—
239	C₂×C₄	#15667934	1	2	yes	—
240	C₂×C₄	#15667937	1	2	yes	—
241	C₂×C₄	#15667937	1	2	yes	—
242	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
243	C₂×C₄	#15667991	1	2	yes	—
244	C₂×C₄	#15668000	1	1	yes	characteristic
245	C₂×C₄	#15668012	1	1	yes	characteristic
246	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
247	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
248	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
249	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
250	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
252	C₂×C₄	#15668042	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	712	2	no	—
255	C₂²	#19151504	1	1	yes	characteristic
256	C₂²	C₂×#15667928	1	1	yes	characteristic
257	C₂²	C₂×#15667930	1	1	yes	characteristic
258	C₂²	C₂×#15667932	1	1	yes	characteristic
259	C₂²	C₂×#15667934	1	1	yes	characteristic
260	C₂²	C₂×#15667937	1	1	yes	characteristic
261	C₂²	#19151606	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	712	4	no	—
266	C₄	#19151341	1	4	yes	—
267	C₄	#19151345	1	4	yes	—
268	C₄	#19151878	1	2	yes	—
269	C₄	#19151883	1	2	yes	—
270	C₄	#19151945	1	1	yes	characteristic
271	C₄	#19151949	1	1	yes	characteristic
272	C₄	#19151996	1	1	yes	characteristic
273	C₄	#19152003	1	1	yes	characteristic
274	C₂	C₂×#19151341	1	1	yes	characteristic
275	C₂	C₂×#19151345	1	1	yes	characteristic
276	C₂	#26018923	1	1	yes	characteristic
277	C₂	#26018932	1	1	yes	characteristic
278	C₂	#26021204	1	1	yes	characteristic
279	C₂	#26021213	1	1	yes	characteristic
280	C₂	#26021241	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#9898
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
44. C₄⋊C₈×C₈₉	Q₈	semidirect
145. C₈₉	#9898	semidirect
185. C₄⋊C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26028650	2
#26020404	2
#26018136	2
C₄×C₄⋊C₈×C₈₉	2
#9898	89