Subgroups — #36470475

Order 22784.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151403	C₂	1	2	yes	maximal
3	#26020412	C₂	1	1	yes	maximal, characteristic
4	#26018344	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	—	2	1	no	—
8	C₄×Dic₃₅₆	—	2	2	no	—
9	C₄×Dic₃₅₆	C₄	1	4	yes	complemented
10	C₂×#15667946	—	2	1	no	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	C₂×#15667946	C₂²	1	1	yes	characteristic
13	C₂×#15667937	—	2	1	no	—
14	#19151468	—	2	1	no	—
15	#19151403	—	2	4	no	—
16	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₉	—	2	2	no	—
19	C₂²×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₃₅₆	—	2	1	no	—
21	C₂×Dic₃₅₆	—	2	1	no	—
22	C₂×Dic₃₅₆	—	2	1	no	—
23	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
24	C₂×Dic₃₅₆	D₄	1	2	yes	—
25	C₂×Dic₃₅₆	Q₈	1	2	yes	—
26	C₄×Dic₁₇₈	—	2	4	no	—
27	C₄×Dic₁₇₈	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	—	4	1	no	—
31	C₂×C₄⋊Dic₈₉	—	4	1	no	—
32	C₂×#13083553	—	4	1	no	—
33	C₂×C₄×Dic₈₉	—	4	1	no	—
34	#15667946	—	2	1	no	—
35	#15667946	—	2	2	no	—
36	#15667946	—	2	2	no	—
37	#15667946	—	4	2	no	—
38	#15667946	C₂×C₄	1	2	yes	—
39	#15667937	—	2	4	no	—
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₈₉	—	2	2	no	—
44	C₂×C₁₆×C₈₉	—	2	2	no	—
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₉	D₄	1	2	yes	—
48	C₂×Dic₁₇₈	—	2	2	no	—
49	C₂×Dic₁₇₈	—	2	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	—	4	1	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂×Dic₁₇₈	—	4	1	no	—
54	C₂²×Dic₈₉	—	4	1	no	—
55	C₂²×Dic₈₉	—	4	1	no	—
56	C₂²×Dic₈₉	—	4	2	no	—
57	Dic₃₅₆	—	2	2	no	—
58	Dic₃₅₆	—	2	2	no	—
59	Dic₃₅₆	—	2	2	no	—
60	Dic₃₅₆	C₄⋊C₄	1	4	yes	complemented
61	C₄⋊Dic₈₉	—	2	2	no	—
62	C₄⋊Dic₈₉	—	4	1	no	—
63	C₄⋊Dic₈₉	—	4	2	no	—
64	C₄⋊Dic₈₉	—	4	2	no	—
65	C₄⋊Dic₈₉	—	8	2	no	—
66	C₄⋊Dic₈₉	—	8	2	no	—
67	#13083553	—	4	4	no	—
68	C₄×Dic₈₉	—	4	4	no	—
69	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	—	2	1	no	—
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
81	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₈₉	SD₁₆	1	2	yes	—
83	C₁₆×C₈₉	—	4	2	no	—
84	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	#102	1	2	yes	—
88	C₂×C₄×C₈₉	#102	1	2	yes	—
89	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
90	C₂×C₄×C₈₉	Q₃₂	1	2	yes	—
91	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
97	C₂×Dic₈₉	—	4	2	no	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	2	no	—
101	C₂×Dic₈₉	—	8	1	no	—
102	C₂×Dic₈₉	—	8	1	no	—
103	C₂×Dic₈₉	—	8	2	no	—
104	C₂×Dic₈₉	—	8	2	no	—
105	C₂×Dic₈₉	—	8	2	no	—
106	Dic₁₇₈	—	2	4	no	—
107	Dic₁₇₈	—	4	2	no	—
108	Dic₁₇₈	—	4	2	no	—
109	Dic₁₇₈	—	4	2	no	—
110	C₈×C₈₉	—	2	2	no	—
111	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₈₉	#117	1	1	yes	characteristic
113	C₈×C₈₉	#121	1	1	yes	characteristic
114	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
116	C₂²×C₈₉	#459	1	1	yes	characteristic
117	C₂²×C₈₉	#474	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
120	C₂²×C₈₉	#509	1	1	yes	characteristic
121	C₂²×C₈₉	#510	1	1	yes	characteristic
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	#358	1	4	yes	complemented
124	C₄×C₈₉	#380	1	1	yes	characteristic
125	C₄×C₈₉	#417	1	1	yes	characteristic
126	C₄×C₈₉	#461	1	1	yes	characteristic
127	C₄×C₈₉	#476	1	1	yes	characteristic
128	Dic₈₉	—	4	4	no	—
129	Dic₈₉	—	8	2	no	—
130	Dic₈₉	—	16	2	no	—
131	#12812	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	#1802	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#358	1	1	yes	characteristic
134	C₂×C₈₉	#2139	1	1	yes	characteristic
135	C₂×C₈₉	#2207	1	1	yes	characteristic
136	C₂×C₈₉	#2209	1	1	yes	characteristic
137	C₂×C₈₉	#2225	1	1	yes	characteristic
138	C₂×C₈₉	#2227	1	1	yes	characteristic
139	C₂×C₄×Q₁₆	—	89	1	no	—
140	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
141	C₂×#358	—	89	2	no	—
142	#1845	—	89	1	no	—
143	#1380	—	89	2	no	—
144	C₈₉	#12812	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₁₆	—	89	1	no	—
146	C₂×C₄×Q₈	—	178	1	no	—
147	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
148	C₄×Q₁₆	—	89	4	no	—
149	C₄×Q₁₆	—	178	2	no	—
150	C₂×#107	—	89	1	no	—
151	C₂×#107	—	89	2	no	—
152	C₂×#107	—	178	1	no	—
153	C₂×#103	—	178	1	no	—
154	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
155	#384	—	178	1	no	—
156	C₄⋊C₁₆	—	2	2	no	—
157	#358	—	178	4	no	—
158	C₂²×Q₈	—	178	1	no	—
159	C₂×Q₁₆	—	89	2	no	—
160	C₂×Q₁₆	—	89	2	no	—
161	C₂×Q₁₆	—	89	2	no	—
162	C₂×Q₁₆	—	178	1	no	—
163	C₂×Q₁₆	—	178	1	no	—
164	C₂×Q₁₆	—	178	1	no	—
165	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
167	C₄×Q₈	—	356	2	no	—
168	C₄×Q₈	—	178	4	no	—
169	C₂×C₄⋊C₄	—	356	1	no	—
170	C₂×C₄⋊C₄	—	356	1	no	—
171	C₂×C₄⋊C₄	—	356	1	no	—
172	C₂×C₄⋊C₄	—	178	2	no	—
173	C₂×C₄⋊C₄	—	178	1	no	—
174	C₂×C₄²	—	356	1	no	—
175	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	#107	—	89	2	no	—
180	#107	—	356	2	no	—
181	#107	—	178	2	no	—
182	#107	—	178	2	no	—
183	#107	—	178	1	no	—
184	#103	—	178	4	no	—
185	C₄×C₈	C₄×D₈₉	1	2	yes	—
186	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
187	C₂×Q₈	—	356	1	no	—
188	C₂×Q₈	—	356	1	no	—
189	C₂×Q₈	—	356	1	no	—
190	C₂×Q₈	—	178	2	no	—
191	C₂×Q₈	—	178	2	no	—
192	C₂×Q₈	—	178	2	no	—
193	C₂²×C₄	—	356	1	no	—
194	C₂²×C₄	—	356	1	no	—
195	C₂²×C₄	—	356	2	no	—
196	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
197	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
199	Q₁₆	—	89	4	no	—
200	Q₁₆	—	178	2	no	—
201	Q₁₆	—	178	2	no	—
202	Q₁₆	—	178	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	#13083551	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
207	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
210	C₂×C₈	#13083573	1	1	yes	characteristic
211	C₂×C₈	#13083575	1	1	yes	characteristic
212	C₄⋊C₄	—	356	1	no	—
213	C₄⋊C₄	—	356	4	no	—
214	C₄⋊C₄	—	356	2	no	—
215	C₄⋊C₄	—	356	2	no	—
216	C₄⋊C₄	—	178	2	no	—
217	C₄⋊C₄	—	712	2	no	—
218	C₄⋊C₄	—	712	2	no	—
219	C₄²	—	356	4	no	—
220	C₄²	#13083551	1	2	yes	—
221	C₄²	#13083563	1	2	yes	—
222	C₁₆	—	4	2	no	—
223	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
224	Q₈	—	356	2	no	—
225	Q₈	—	356	2	no	—
226	Q₈	—	356	2	no	—
227	Q₈	—	178	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	356	2	no	—
231	C₂×C₄	—	356	2	no	—
232	C₂×C₄	—	356	2	no	—
233	C₂×C₄	—	356	2	no	—
234	C₂×C₄	—	712	2	no	—
235	C₂×C₄	—	712	2	no	—
236	C₂×C₄	—	712	2	no	—
237	C₂×C₄	—	712	1	no	—
238	C₂×C₄	—	712	1	no	—
239	C₂×C₄	#15667934	1	2	yes	—
240	C₂×C₄	#15667934	1	2	yes	—
241	C₂×C₄	#15667957	1	2	yes	—
242	C₂×C₄	Dic₇₁₂	1	2	yes	—
243	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
244	C₂×C₄	#15667987	1	1	yes	characteristic
245	C₂×C₄	#15667992	1	1	yes	characteristic
246	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
247	C₂×C₄	#15668040	1	1	yes	characteristic
248	C₂×C₄	#15668045	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
251	C₈	#15667970	1	1	yes	characteristic
252	C₈	#15667983	1	1	yes	characteristic
253	C₈	#15667988	1	1	yes	characteristic
254	C₂²	C₂×#15667934	1	1	yes	characteristic
255	C₂²	#19151794	1	1	yes	characteristic
256	C₂²	#19151879	1	1	yes	characteristic
257	C₂²	C₂×#15667957	1	1	yes	characteristic
258	C₂²	C₂×Dic₇₁₂	1	1	yes	characteristic
259	C₂²	#19152036	1	1	yes	characteristic
260	C₂²	#19152041	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	356	4	no	—
263	C₄	—	1424	2	no	—
264	C₄	—	712	2	no	—
265	C₄	#19151403	1	4	yes	complemented
266	C₄	#19151455	1	1	yes	characteristic
267	C₄	#19151600	1	1	yes	characteristic
268	C₄	#19151804	1	1	yes	characteristic
269	C₄	#19151889	1	1	yes	characteristic
270	C₂	#26020253	1	1	yes	characteristic
271	C₂	C₂×#19151403	1	1	yes	characteristic
272	C₂	#26021965	1	1	yes	characteristic
273	C₂	#26022293	1	1	yes	characteristic
274	C₂	#26022303	1	1	yes	characteristic
275	C₂	#26022395	1	1	yes	characteristic
276	C₂	#26022405	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#12812
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₃₅₆	C₄	semidirect
60. Dic₃₅₆	C₄⋊C₄	semidirect
123. C₄×C₈₉	#358	semidirect
144. C₈₉	#12812	semidirect
265. C₄	#19151403	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₅₆	2
C₂×#19151403	2
#26020412	2
#26018344	2
C₂×C₄⋊C₁₆×C₈₉	2
#12812	89