Subgroups — #36447673

Order 22784.

430 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028965	C₂	1	1	yes	maximal, characteristic
2	#26020424	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018927	C₂	1	2	yes	maximal
4	#26018127	C₂	1	2	yes	maximal, complemented
5	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668018	—	2	1	no	—
7	#19152148	C₂²	1	1	yes	characteristic
8	#19151677	—	2	1	no	—
9	#19151677	C₄	1	2	yes	—
10	C₈×D₃₅₆	—	2	1	no	—
11	C₈×D₃₅₆	C₄	1	2	yes	—
12	C₂×#15667948	—	2	1	no	—
13	C₂×#15667948	—	2	1	no	—
14	#19151622	C₂²	1	2	yes	—
15	#19151622	C₂²	1	1	yes	characteristic
16	#19151564	—	2	1	no	—
17	#19151343	—	2	2	no	—
18	#19151341	—	2	2	no	—
19	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#334×C₈₉	—	2	1	no	—
22	#334×C₈₉	—	2	1	no	—
23	C₂×#13083574	—	2	1	no	—
24	#15668018	—	4	2	no	—
25	#15668018	—	4	2	no	—
26	C₂×#13083558	—	2	1	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083557	—	2	2	no	—
29	C₂×C₈×D₈₉	—	2	2	no	—
30	#15668012	D₄	1	1	yes	characteristic
31	#15668009	D₄	1	1	yes	characteristic
32	#15668000	—	2	1	no	—
33	#15667986	—	2	1	no	—
34	#15667986	Q₈	1	2	yes	—
35	C₄×D₃₅₆	—	2	1	no	—
36	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
37	#15667948	—	4	2	no	—
38	#15667948	—	4	2	no	—
39	C₄⋊#10623494	—	2	1	no	—
40	C₄⋊#10623494	—	2	2	no	—
41	C₄⋊#10623494	—	2	2	no	—
42	C₄⋊#10623494	C₂×C₄	1	2	yes	—
43	#15667924	—	2	2	no	—
44	#15667924	—	2	2	no	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₈₉	—	2	2	no	—
50	C₄⋊C₈×C₈₉	—	2	2	no	—
51	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
55	#13083574	—	4	1	no	—
56	#13083574	—	4	1	no	—
57	#13083574	—	4	1	no	—
58	#13083574	—	4	1	no	—
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
63	C₂×C₂²⋊D₈₉	—	2	2	no	—
64	C₂×C₄×D₈₉	—	2	2	no	—
65	#13083558	—	4	1	no	—
66	#13083558	—	4	1	no	—
67	#13083558	—	4	2	no	—
68	#13083558	—	4	2	no	—
69	#13083557	—	4	4	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	4	2	no	—
72	C₈×D₈₉	—	4	4	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
75	#13083551	—	2	2	no	—
76	#13083551	—	2	2	no	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	4	2	no	—
83	C₂×C₈×C₈₉	—	8	2	no	—
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	D₈	1	2	yes	—
89	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
90	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	#41	1	1	yes	characteristic
94	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂²×D₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
108	C₂²⋊D₈₉	—	4	2	no	—
109	C₂²⋊D₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	2	2	no	—
111	D₃₅₆	—	4	2	no	—
112	D₃₅₆	—	4	1	no	—
113	D₃₅₆	—	4	1	no	—
114	D₃₅₆	—	4	2	no	—
115	C₄×D₈₉	—	4	2	no	—
116	C₄×D₈₉	—	4	2	no	—
117	Dic₁₇₈	—	4	1	no	—
118	Dic₁₇₈	—	4	1	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	8	2	no	—
122	C₈×C₈₉	#102	1	2	yes	—
123	C₈×C₈₉	#102	1	2	yes	—
124	#10623494	—	4	2	no	—
125	#10623494	—	8	2	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	#380	1	1	yes	characteristic
129	C₂²×C₈₉	#474	1	1	yes	characteristic
130	C₂²×C₈₉	#476	1	1	yes	characteristic
131	C₂×D₈₉	—	4	2	no	—
132	C₂×D₈₉	—	4	2	no	—
133	C₂×D₈₉	—	4	2	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
138	C₄×C₈₉	#417	1	1	yes	characteristic
139	C₄×C₈₉	#459	1	1	yes	characteristic
140	C₄×C₈₉	#461	1	1	yes	characteristic
141	C₄×C₈₉	#471	1	2	yes	—
142	Dic₈₉	—	4	2	no	—
143	#8682	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	#1802	1	1	yes	characteristic
147	C₂×C₈₉	#1813	1	1	yes	characteristic
148	C₂×C₈₉	#1815	1	1	yes	characteristic
149	D₈₉	—	8	2	no	—
150	#2931	—	89	1	no	—
151	#1848	—	89	1	no	—
152	C₂×#334	D₈₉	1	1	yes	characteristic, complemented
153	#1547	—	89	2	no	—
154	#1292	—	89	2	no	—
155	C₈₉	#8682	1	1	yes	characteristic, complemented, Sylow
156	C₂×#131	—	178	1	no	—
157	#532	—	89	1	no	—
158	#436	—	89	2	no	—
159	#436	—	178	1	no	—
160	C₈×D₄	—	89	2	no	—
161	C₈×D₄	—	178	1	no	—
162	C₂×#108	—	178	1	no	—
163	C₂×#108	—	178	1	no	—
164	#423	—	89	1	no	—
165	#423	—	89	2	no	—
166	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
167	#408	—	178	1	no	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#334	—	2	1	no	—
170	#334	—	2	1	no	—
171	#332	—	178	2	no	—
172	#331	—	178	2	no	—
173	C₂×#41	—	178	1	no	—
174	#131	—	356	2	no	—
175	#131	—	356	2	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	1	no	—
178	C₂×M₁₆	—	178	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	178	2	no	—
181	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
182	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
183	C₄⋊Q₈	—	89	1	no	—
184	#127	—	89	1	no	—
185	#124	—	178	1	no	—
186	#121	—	89	2	no	—
187	#121	—	178	1	no	—
188	C₄×D₄	—	89	2	no	—
189	C₄×D₄	—	178	1	no	—
190	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
191	#108	—	356	2	no	—
192	#108	—	356	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	89	2	no	—
196	C₄⋊C₈	—	178	2	no	—
197	C₄⋊C₈	—	178	1	no	—
198	C₄⋊C₈	—	178	2	no	—
199	C₂²⋊C₈	—	178	2	no	—
200	C₂²⋊C₈	—	178	2	no	—
201	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
202	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
203	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	#41	—	356	1	no	—
206	#41	—	356	1	no	—
207	#41	—	356	1	no	—
208	#41	—	356	1	no	—
209	C₂×Q₈	—	178	1	no	—
210	C₂×D₄	—	89	2	no	—
211	C₂×D₄	—	178	2	no	—
212	C₂×D₄	—	178	1	no	—
213	C₂×D₄	—	178	1	no	—
214	C₂²×C₄	—	178	2	no	—
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	#13083573	1	1	yes	characteristic
218	M₁₆	—	356	1	no	—
219	M₁₆	—	356	1	no	—
220	M₁₆	—	356	4	no	—
221	M₁₆	—	356	2	no	—
222	M₁₆	—	356	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	8	2	no	—
227	C₂×C₈	—	356	4	no	—
228	C₂×C₈	—	356	2	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	#13083551	1	1	yes	characteristic
231	C₂×C₈	#13083551	1	1	yes	characteristic
232	C₂×C₈	#13083551	1	1	yes	characteristic
233	C₂×C₈	#13083551	1	1	yes	characteristic
234	C₂×C₈	D₇₁₂	1	2	yes	—
235	C₂×C₈	#13083563	1	2	yes	—
236	C₄⋊C₄	—	89	2	no	—
237	C₄⋊C₄	—	178	1	no	—
238	C₂²⋊C₄	—	178	2	no	—
239	C₂²⋊C₄	—	178	2	no	—
240	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₄²	D₄×D₈₉	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄²	#13083575	1	1	yes	characteristic
244	C₂³	—	178	2	no	—
245	C₂³	#15667992	1	1	yes	characteristic
246	Q₈	—	356	1	no	—
247	Q₈	—	356	1	no	—
248	D₄	—	356	2	no	—
249	D₄	—	356	2	no	—
250	D₄	—	356	2	no	—
251	D₄	—	356	1	no	—
252	D₄	—	356	1	no	—
253	D₄	—	356	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	356	2	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	178	2	no	—
259	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
260	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
261	C₂×C₄	#15667970	1	1	yes	characteristic
262	C₂×C₄	#15667983	1	1	yes	characteristic
263	C₂×C₄	#15667987	1	1	yes	characteristic
264	C₂×C₄	#15667988	1	1	yes	characteristic
265	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
266	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
267	C₂×C₄	#15668040	1	1	yes	characteristic
268	C₂×C₄	#15668045	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	8	2	no	—
272	C₈	—	356	2	no	—
273	C₈	—	712	2	no	—
274	C₈	#15667934	1	2	yes	—
275	C₈	#15667934	1	2	yes	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	356	2	no	—
279	C₂²	—	356	2	no	—
280	C₂²	—	356	2	no	—
281	C₂²	#19151455	1	1	yes	characteristic
282	C₂²	#19151879	1	1	yes	characteristic
283	C₂²	#19151889	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	356	2	no	—
288	C₄	C₂×#15667934	1	1	yes	characteristic
289	C₄	#19151600	1	1	yes	characteristic
290	C₄	#19151794	1	1	yes	characteristic
291	C₄	#19151804	1	1	yes	characteristic
292	C₄	#19151858	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	712	2	no	—
296	C₂	#26020253	1	1	yes	characteristic
297	C₂	#26020299	1	1	yes	characteristic
298	C₂	#26020312	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151455
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#8682
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. #26020424	C₂	semidirect
4. #26018127	C₂	semidirect
5. C₂×#334×C₈₉	C₂	semidirect
152. C₂×#334	D₈₉	semidirect
155. C₈₉	#8682	semidirect

Maximal subgroups

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

subgroup	index
#26028965	2
#26020424	2
#26018927	2
#26018127	2
C₂×#334×C₈₉	2
#8682	89