Subgroups — #36470491

Order 22784.

446 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	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
2	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151407	C₂	1	2	yes	maximal
4	#26018359	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668034	C₂²	1	1	yes	characteristic
7	C₄×#13083574	—	2	1	no	—
8	C₂×#15667952	C₂²	1	2	yes	—
9	C₄×Dic₃₅₆	C₄	1	2	yes	complemented
10	C₄×#13083563	—	2	2	no	—
11	C₄×D₇₁₂	C₄	1	2	yes	complemented
12	#19151636	C₂²	1	1	yes	characteristic
13	#19151595	—	2	1	no	—
14	C₄⋊#13083549	—	2	2	no	—
15	#19151407	—	2	4	no	—
16	C₂×#106×C₈₉	—	2	1	no	—
17	C₂×#106×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#384×C₈₉	—	2	1	no	—
20	C₂×#13083574	—	2	1	no	—
21	#15668034	—	2	1	no	—
22	#15668034	—	2	1	no	—
23	#15668034	D₄	1	2	yes	—
24	#15668034	Q₈	1	2	yes	—
25	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
26	C₂×#13083563	—	2	1	no	—
27	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₇₈	—	2	2	no	—
29	C₄×D₃₅₆	—	2	2	no	—
30	C₄×C₂²⋊D₈₉	—	4	2	no	—
31	#15667971	—	2	1	no	—
32	#15667969	—	4	1	no	—
33	C₄²×D₈₉	—	4	1	no	—
34	#15667952	—	2	2	no	—
35	#15667952	—	2	2	no	—
36	C₂×#13083549	—	2	2	no	—
37	#15667946	C₂×C₄	1	2	yes	—
38	#15667944	—	2	1	no	—
39	#15667937	—	2	2	no	—
40	#15667934	—	2	2	no	—
41	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	#106×C₈₉	—	2	2	no	—
47	#106×C₈₉	—	2	2	no	—
48	#106×C₈₉	—	4	2	no	—
49	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₈₉	D₄	1	2	yes	—
51	#13083574	—	2	2	no	—
52	#13083574	—	2	2	no	—
53	#13083574	—	4	1	no	—
54	#13083574	—	4	1	no	—
55	C₂×Dic₁₇₈	—	2	1	no	—
56	C₂×D₃₅₆	—	2	1	no	—
57	C₂×C₂²⋊D₈₉	—	4	1	no	—
58	C₂×C₄×D₈₉	—	4	1	no	—
59	C₂×C₄×D₈₉	—	4	2	no	—
60	Dic₃₅₆	—	2	1	no	—
61	Dic₃₅₆	C₄⋊C₄	1	2	yes	complemented
62	#13083563	—	2	2	no	—
63	#13083563	—	2	2	no	—
64	D₇₁₂	—	2	1	no	—
65	D₇₁₂	C₄⋊C₄	1	2	yes	complemented
66	C₄⋊Dic₈₉	—	2	2	no	—
67	#13083553	—	4	2	no	—
68	C₂²⋊Dic₈₉	—	4	1	no	—
69	#13083551	—	4	2	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	C₄×Dic₈₉	—	4	1	no	—
72	#13083549	—	4	2	no	—
73	C₂²×C₄×C₈₉	—	4	2	no	—
74	C₂²×C₄×C₈₉	D₈	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	—	2	1	no	—
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₈₉	—	4	2	no	—
86	C₄⋊C₄×C₈₉	—	4	2	no	—
87	C₄⋊C₄×C₈₉	—	8	4	no	—
88	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
90	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
91	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
92	C₂²×D₈₉	—	4	1	no	—
93	C₂×C₄×C₈₉	—	2	4	no	—
94	C₂×C₄×C₈₉	—	8	4	no	—
95	C₂×C₄×C₈₉	—	8	2	no	—
96	C₂×C₄×C₈₉	#102	1	2	yes	—
97	C₂×C₄×C₈₉	#102	1	2	yes	—
98	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
104	C₂²⋊D₈₉	—	4	2	no	—
105	C₂²⋊D₈₉	—	4	2	no	—
106	C₂×Dic₈₉	—	4	1	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	D₃₅₆	—	2	2	no	—
109	D₃₅₆	—	4	1	no	—
110	C₄×D₈₉	—	4	2	no	—
111	C₄×D₈₉	—	4	2	no	—
112	C₄×D₈₉	—	8	4	no	—
113	Dic₁₇₈	—	2	2	no	—
114	Dic₁₇₈	—	4	1	no	—
115	C₈×C₈₉	—	2	2	no	—
116	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₈₉	#117	1	1	yes	characteristic
118	C₈×C₈₉	#121	1	1	yes	characteristic
119	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
122	C₂²×C₈₉	#459	1	1	yes	characteristic
123	C₂²×C₈₉	#474	1	1	yes	characteristic
124	C₂×D₈₉	—	4	2	no	—
125	C₂×D₈₉	—	8	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	16	2	no	—
128	C₄×C₈₉	#360	1	4	yes	complemented
129	C₄×C₈₉	#380	1	1	yes	characteristic
130	C₄×C₈₉	#417	1	1	yes	characteristic
131	C₄×C₈₉	#461	1	1	yes	characteristic
132	C₄×C₈₉	#476	1	1	yes	characteristic
133	Dic₈₉	—	4	2	no	—
134	Dic₈₉	—	8	2	no	—
135	#12816	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	2	1	no	—
137	C₂×C₈₉	—	2	1	no	—
138	C₂×C₈₉	#1802	1	1	yes	characteristic
139	C₂×C₈₉	C₂×#360	1	2	yes	—
140	D₈₉	—	8	2	no	—
141	C₄×#135	—	89	1	no	—
142	#2150	—	89	1	no	—
143	C₂×#360	—	89	2	no	—
144	#1844	D₈₉	1	1	yes	characteristic, complemented
145	#1386	—	89	2	no	—
146	C₈₉	#12816	1	1	yes	characteristic, complemented, Sylow
147	C₂×#135	—	89	1	no	—
148	C₄×#41	—	178	1	no	—
149	C₂×M₃₂	—	89	2	no	—
150	C₄×Q₁₆	—	89	2	no	—
151	C₄×SD₁₆	—	178	2	no	—
152	C₄×D₈	—	89	2	no	—
153	#427	—	89	1	no	—
154	C₂×#106	—	2	1	no	—
155	C₂×#106	C₂×D₈₉	1	2	yes	—
156	#416	—	178	1	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	#384	—	2	1	no	—
159	C₄⋊C₁₆	—	178	2	no	—
160	#360	—	178	4	no	—
161	C₂×#41	—	178	1	no	—
162	#135	—	89	2	no	—
163	#135	—	89	2	no	—
164	#135	—	178	1	no	—
165	#135	—	178	1	no	—
166	C₂×Q₁₆	—	89	1	no	—
167	C₂×SD₁₆	—	178	1	no	—
168	C₂×D₈	—	89	1	no	—
169	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
171	C₄×Q₈	—	178	2	no	—
172	C₄×D₄	—	356	2	no	—
173	C₄×D₄	—	178	2	no	—
174	#117	—	356	1	no	—
175	#117	—	178	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	4	2	no	—
178	C₂×C₄²	—	356	1	no	—
179	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
180	M₃₂	—	178	2	no	—
181	M₃₂	—	178	2	no	—
182	C₂×C₁₆	—	178	2	no	—
183	#107	—	89	2	no	—
184	#106	—	2	2	no	—
185	#106	—	2	2	no	—
186	#106	—	4	2	no	—
187	#106	—	178	1	no	—
188	#103	—	178	2	no	—
189	#102	—	178	2	no	—
190	C₄×C₈	C₄×D₈₉	1	2	yes	—
191	C₄×C₈	D₃₅₆	1	2	yes	—
192	#41	—	356	1	no	—
193	#41	—	356	1	no	—
194	#41	—	178	2	no	—
195	#41	—	178	2	no	—
196	C₂×Q₈	—	178	1	no	—
197	C₂×D₄	—	356	1	no	—
198	C₂×D₄	—	178	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	356	1	no	—
201	C₂²×C₄	—	356	2	no	—
202	C₂²×C₄	#13083559	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
204	Q₁₆	—	89	2	no	—
205	Q₁₆	—	178	1	no	—
206	SD₁₆	—	178	2	no	—
207	SD₁₆	—	178	2	no	—
208	D₈	—	89	2	no	—
209	D₈	—	178	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	#13083551	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
215	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
217	C₂×C₈	#13083573	1	1	yes	characteristic
218	C₂×C₈	#13083575	1	1	yes	characteristic
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	8	4	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	178	2	no	—
224	C₂²⋊C₄	—	356	1	no	—
225	C₂²⋊C₄	—	356	2	no	—
226	C₄²	—	356	1	no	—
227	C₄²	—	356	2	no	—
228	C₄²	#13083551	1	2	yes	—
229	C₄²	#13083561	1	1	yes	characteristic
230	C₄²	#13083561	1	1	yes	characteristic
231	C₁₆	—	356	2	no	—
232	C₂³	—	356	1	no	—
233	C₂³	C₂×#13083559	1	1	yes	characteristic
234	Q₈	—	356	1	no	—
235	Q₈	—	178	2	no	—
236	D₄	—	356	2	no	—
237	D₄	—	356	2	no	—
238	D₄	—	356	1	no	—
239	D₄	—	178	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	356	1	no	—
244	C₂×C₄	—	356	2	no	—
245	C₂×C₄	—	356	2	no	—
246	C₂×C₄	—	356	2	no	—
247	C₂×C₄	—	712	4	no	—
248	C₂×C₄	#15667932	1	2	yes	—
249	C₂×C₄	#15667932	1	2	yes	—
250	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
251	C₂×C₄	#15667983	1	1	yes	characteristic
252	C₂×C₄	#15667988	1	1	yes	characteristic
253	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
254	C₂×C₄	#15668038	1	1	yes	characteristic
255	C₂×C₄	#15668043	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
258	C₈	#15667970	1	1	yes	characteristic
259	C₈	#15667987	1	1	yes	characteristic
260	C₈	#15667992	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	356	2	no	—
263	C₂²	—	712	2	no	—
264	C₂²	C₂×#15667932	1	1	yes	characteristic
265	C₂²	#19151790	1	1	yes	characteristic
266	C₂²	#19151875	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	16	2	no	—
269	C₄	—	356	2	no	—
270	C₄	—	712	2	no	—
271	C₄	#19151407	1	4	yes	complemented
272	C₄	#19151455	1	1	yes	characteristic
273	C₄	#19151597	1	1	yes	characteristic
274	C₄	#19151800	1	1	yes	characteristic
275	C₄	#19151885	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	712	2	no	—
279	C₂	#26020249	1	1	yes	characteristic
280	C₂	C₂×#19151407	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#15667932
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1844	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1844	D₈₉
hypocenter	C₈₉	#12816
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
2. #26022008	C₂	semidirect
4. #26018359	C₂	semidirect
5. C₈₉×#1844	C₂	semidirect
9. C₄×Dic₃₅₆	C₄	semidirect
11. C₄×D₇₁₂	C₄	semidirect
61. Dic₃₅₆	C₄⋊C₄	semidirect
65. D₇₁₂	C₄⋊C₄	semidirect
128. C₄×C₈₉	#360	semidirect
144. #1844	D₈₉	semidirect
146. C₈₉	#12816	semidirect
271. C₄	#19151407	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668034	2
#26022008	2
C₂×#19151407	2
#26018359	2
C₈₉×#1844	2
#12816	89