Subgroups — #36707047

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152007	C₂	1	2	yes	maximal
2	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
3	C₂×#19152000	C₂	1	1	yes	maximal, characteristic
4	#26026371	C₂	1	2	yes	maximal
5	#26020420	C₂	1	4	yes	maximal
6	#26020416	C₂	1	2	yes	maximal
7	#26020412	C₂	1	2	yes	maximal
8	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×#15668012	C₂²	1	2	yes	—
10	C₂×#15668004	C₂²	1	2	yes	—
11	#19152007	—	2	8	no	—
12	#19152003	—	2	4	no	—
13	#19152000	—	2	4	no	—
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667946	C₂²	1	2	yes	—
16	C₂×#15667946	C₂²	1	4	yes	—
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667946	C₂²	1	2	yes	—
19	C₂×#15667944	C₂²	1	4	yes	—
20	C₂×#15667944	C₂²	1	2	yes	—
21	#19151530	—	2	4	no	—
22	#19151519	—	2	4	no	—
23	#19151468	C₂²	1	4	yes	—
24	#19151464	C₂²	1	4	yes	—
25	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#97×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
29	#376×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₇₈	—	2	2	no	—
31	#15668012	—	2	4	no	—
32	#15668012	—	2	4	no	—
33	#15668004	—	2	4	no	—
34	#15668004	—	2	4	no	—
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	C₂×C₄⋊Dic₈₉	—	2	8	no	—
37	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
40	C₂×#13083553	—	2	2	no	—
41	C₂×#13083553	—	2	2	no	—
42	#15667946	—	2	2	no	—
43	#15667946	—	2	2	no	—
44	#15667946	—	2	4	no	—
45	#15667946	—	2	4	no	—
46	#15667946	—	2	8	no	—
47	#15667944	—	2	4	no	—
48	#15667944	—	2	8	no	—
49	#15667919	—	2	8	no	—
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
55	#97×C₈₉	D₄	1	4	yes	—
56	#97×C₈₉	Q₈	1	8	yes	complemented
57	C₄×C₈×C₈₉	D₄	1	4	yes	—
58	C₄×C₈×C₈₉	Q₈	1	8	yes	complemented
59	#95×C₈₉	D₄	1	2	yes	—
60	#95×C₈₉	D₄	1	2	yes	—
61	C₂×Dic₁₇₈	—	4	4	no	—
62	C₂×Dic₁₇₈	—	8	4	no	—
63	C₂²×Dic₈₉	—	2	4	no	—
64	C₂²×Dic₈₉	—	2	2	no	—
65	C₂²×Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	—	2	2	no	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	4	no	—
71	C₄⋊Dic₈₉	—	2	4	no	—
72	C₄⋊Dic₈₉	—	4	4	no	—
73	C₄⋊Dic₈₉	—	8	8	no	—
74	#13083553	—	4	4	no	—
75	#13083553	—	4	4	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
87	C₄²×C₈₉	C₂×D₄	1	2	yes	—
88	C₄²×C₈₉	C₂×D₄	1	2	yes	—
89	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
90	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
91	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	2	4	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	#127	1	2	yes	—
97	C₂×C₄×C₈₉	#127	1	2	yes	—
98	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
99	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
103	C₂×Dic₈₉	—	4	2	no	—
104	C₂×Dic₈₉	—	4	2	no	—
105	C₂×Dic₈₉	—	4	4	no	—
106	C₂×Dic₈₉	—	4	4	no	—
107	C₂×Dic₈₉	—	4	8	no	—
108	C₂×Dic₈₉	—	4	4	no	—
109	Dic₁₇₈	—	8	8	no	—
110	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
111	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
112	C₈×C₈₉	C₄⋊Q₈	1	8	yes	—
113	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
115	C₂²×C₈₉	#576	1	2	yes	—
116	C₂²×C₈₉	#578	1	2	yes	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	8	no	—
120	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
123	Dic₈₉	—	8	4	no	—
124	Dic₈₉	—	8	2	no	—
125	Dic₈₉	—	8	2	no	—
126	#33319	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	#2868	1	1	yes	characteristic
128	C₂×C₈₉	#3155	1	1	yes	characteristic
129	C₂×C₈₉	#3156	1	1	yes	characteristic
130	C₂×C₈₉	#3164	1	4	yes	—
131	C₂×#501	—	89	2	no	—
132	C₂×#500	—	89	1	no	—
133	C₂×#499	—	89	1	no	—
134	#2660	—	89	2	no	—
135	#1847	—	89	4	no	—
136	#1846	—	89	2	no	—
137	#1845	—	89	2	no	—
138	#1726	D₈₉	1	1	yes	characteristic
139	C₈₉	#33319	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊Q₈	—	89	2	no	—
141	C₂×#125	—	89	2	no	—
142	#501	—	178	8	no	—
143	#500	—	178	4	no	—
144	#499	—	178	4	no	—
145	C₂×#107	—	89	1	no	—
146	C₂×#107	—	89	2	no	—
147	C₂×#107	—	89	4	no	—
148	C₂×#107	—	89	1	no	—
149	C₂×#107	—	89	2	no	—
150	C₂×#106	—	89	4	no	—
151	C₂×#106	—	89	2	no	—
152	C₂×#97	C₂×D₈₉	1	2	yes	—
153	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
156	#398	—	178	4	no	—
157	#395	—	178	4	no	—
158	#384	—	89	4	no	—
159	#382	—	89	4	no	—
160	#376	C₂×D₈₉	1	1	yes	characteristic
161	C₂²×Q₈	—	178	2	no	—
162	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
165	C₄⋊Q₈	—	178	4	no	—
166	C₄⋊Q₈	—	178	4	no	—
167	#125	—	178	4	no	—
168	#125	—	178	4	no	—
169	C₂×C₄⋊C₄	—	89	2	no	—
170	C₂×C₄⋊C₄	—	89	2	no	—
171	C₂×C₄⋊C₄	—	89	4	no	—
172	C₂×C₄⋊C₄	—	178	2	no	—
173	C₂×C₄⋊C₄	—	178	2	no	—
174	C₂×C₄⋊C₄	—	178	2	no	—
175	C₂×C₄⋊C₄	—	178	8	no	—
176	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
178	#107	—	178	2	no	—
179	#107	—	178	2	no	—
180	#107	—	178	4	no	—
181	#107	—	178	4	no	—
182	#107	—	178	8	no	—
183	#106	—	178	4	no	—
184	#106	—	178	8	no	—
185	#97	Dic₁₇₈	1	8	yes	complemented
186	#97	D₃₅₆	1	4	yes	—
187	C₄×C₈	Dic₁₇₈	1	8	yes	complemented
188	C₄×C₈	D₃₅₆	1	4	yes	—
189	#95	—	178	8	no	—
190	#95	D₃₅₆	1	2	yes	—
191	#95	D₃₅₆	1	2	yes	—
192	C₂×Q₈	—	356	4	no	—
193	C₂×Q₈	—	712	4	no	—
194	C₂²×C₄	—	178	2	no	—
195	C₂²×C₄	—	178	4	no	—
196	C₂²×C₄	—	178	2	no	—
197	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
200	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
201	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
202	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
203	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
204	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
205	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
206	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
207	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
208	C₄⋊C₄	—	356	4	no	—
209	C₄⋊C₄	—	356	4	no	—
210	C₄⋊C₄	—	356	4	no	—
211	C₄⋊C₄	—	178	2	no	—
212	C₄⋊C₄	—	178	2	no	—
213	C₄⋊C₄	—	178	2	no	—
214	C₄⋊C₄	—	178	4	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₄⋊C₄	—	178	4	no	—
217	C₄⋊C₄	—	712	8	no	—
218	C₄²	C₂×D₃₅₆	1	2	yes	—
219	C₄²	C₂×D₃₅₆	1	2	yes	—
220	C₄²	C₂×Dic₁₇₈	1	4	yes	—
221	C₄²	C₂×Dic₁₇₈	1	4	yes	—
222	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
223	Q₈	—	712	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	356	2	no	—
229	C₂×C₄	—	356	2	no	—
230	C₂×C₄	—	356	4	no	—
231	C₂×C₄	—	356	4	no	—
232	C₂×C₄	—	356	8	no	—
233	C₂×C₄	—	356	4	no	—
234	C₂×C₄	#15668009	1	2	yes	—
235	C₂×C₄	#15668009	1	2	yes	—
236	C₂×C₄	#15668012	1	8	yes	—
237	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₁₇₈	1	2	yes	—
240	C₂×C₄	C₂²×Dic₁₇₈	1	2	yes	—
241	C₈	#15668012	1	4	yes	—
242	C₈	#15668012	1	4	yes	—
243	C₈	#15668012	1	8	yes	—
244	C₂²	C₂×#15668009	1	1	yes	characteristic
245	C₂²	C₂×#15668012	1	2	yes	—
246	C₂²	#19152403	1	2	yes	—
247	C₂²	#19152415	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	8	no	—
251	C₄	—	712	4	no	—
252	C₄	—	712	2	no	—
253	C₄	—	712	2	no	—
254	C₄	C₂×#15668012	1	1	yes	characteristic
255	C₄	C₂×#15668012	1	1	yes	characteristic
256	C₄	C₂×#15668012	1	2	yes	—
257	C₂	#26028650	1	1	yes	characteristic
258	C₂	#26030738	1	1	yes	characteristic
259	C₂	#26030746	1	1	yes	characteristic
260	C₂	#26030791	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1726	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#33319
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
56. #97×C₈₉	Q₈	semidirect
58. C₄×C₈×C₈₉	Q₈	semidirect
139. C₈₉	#33319	semidirect
185. #97	Dic₁₇₈	semidirect
187. 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
C₂×#19152007	2
C₂×#19152003	2
C₂×#19152000	2
#26026371	2
#26020420	2
#26020416	2
#26020412	2
C₈₉×#1726	2
#33319	89