Subgroups — #36707039

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 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×#19152000	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151996	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	#26020409	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	#19152000	—	2	4	no	—
13	#19151996	—	2	4	no	—
14	C₂×#15667946	C₂²	1	4	yes	—
15	C₂×#15667946	C₂²	1	2	yes	—
16	C₂×#15667944	C₂²	1	1	yes	characteristic
17	C₂×#15667944	C₂²	1	2	yes	—
18	C₂×#15667944	C₂²	1	4	yes	—
19	C₂×#15667944	C₂²	1	1	yes	characteristic
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	4	yes	—
38	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
40	C₂×#13083553	—	2	2	no	—
41	C₂×#13083553	—	2	2	no	—
42	#15667946	—	2	4	no	—
43	#15667946	—	2	8	no	—
44	#15667944	—	2	4	no	—
45	#15667944	—	2	4	no	—
46	#15667944	—	2	2	no	—
47	#15667944	—	2	2	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	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
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	4	yes	—
60	C₂×Dic₁₇₈	—	4	4	no	—
61	C₂×Dic₁₇₈	—	8	4	no	—
62	C₂²×Dic₈₉	—	2	4	no	—
63	C₂²×Dic₈₉	—	2	2	no	—
64	C₂²×Dic₈₉	—	2	2	no	—
65	C₄⋊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	4	no	—
70	C₄⋊Dic₈₉	—	2	4	no	—
71	C₄⋊Dic₈₉	—	4	4	no	—
72	C₄⋊Dic₈₉	—	8	8	no	—
73	#13083553	—	4	4	no	—
74	#13083553	—	4	4	no	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
83	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
85	C₄²×C₈₉	C₂×D₄	1	2	yes	—
86	C₄²×C₈₉	C₂×D₄	1	2	yes	—
87	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
88	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
89	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	—	2	8	no	—
91	C₂×C₄×C₈₉	—	2	4	no	—
92	C₂×C₄×C₈₉	#127	1	4	yes	—
93	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
94	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	4	4	no	—
102	C₂×Dic₈₉	—	4	8	no	—
103	C₂×Dic₈₉	—	4	4	no	—
104	Dic₁₇₈	—	8	8	no	—
105	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
106	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
107	C₈×C₈₉	C₄⋊Q₈	1	8	yes	—
108	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
109	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
110	C₂²×C₈₉	#577	1	4	yes	—
111	C₄×C₈₉	—	2	4	no	—
112	C₄×C₈₉	—	2	8	no	—
113	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
116	Dic₈₉	—	8	4	no	—
117	Dic₈₉	—	8	2	no	—
118	Dic₈₉	—	8	2	no	—
119	#33318	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	#2868	1	1	yes	characteristic
121	C₂×C₈₉	#3154	1	2	yes	—
122	C₂×C₈₉	#3163	1	4	yes	—
123	C₂×#501	—	89	2	no	—
124	C₂×#499	—	89	1	no	—
125	C₂×#498	—	89	1	no	—
126	#2660	—	89	2	no	—
127	#1847	—	89	4	no	—
128	#1846	—	89	2	no	—
129	#1844	—	89	2	no	—
130	#1726	D₈₉	1	1	yes	characteristic
131	C₈₉	#33318	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊Q₈	—	89	2	no	—
133	C₂×#125	—	89	2	no	—
134	#501	—	178	8	no	—
135	#499	—	178	4	no	—
136	#498	—	178	4	no	—
137	C₂×#107	—	89	4	no	—
138	C₂×#107	—	89	2	no	—
139	C₂×#106	—	89	1	no	—
140	C₂×#106	—	89	2	no	—
141	C₂×#106	—	89	4	no	—
142	C₂×#106	—	89	1	no	—
143	C₂×#106	—	89	2	no	—
144	C₂×#97	C₂×D₈₉	1	2	yes	—
145	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	#398	—	178	4	no	—
149	#395	—	178	4	no	—
150	#384	—	89	4	no	—
151	#382	—	89	4	no	—
152	#376	C₂×D₈₉	1	1	yes	characteristic
153	C₂²×Q₈	—	178	2	no	—
154	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
157	C₄⋊Q₈	—	178	4	no	—
158	C₄⋊Q₈	—	178	4	no	—
159	#125	—	178	4	no	—
160	#125	—	178	4	no	—
161	C₂×C₄⋊C₄	—	89	2	no	—
162	C₂×C₄⋊C₄	—	89	2	no	—
163	C₂×C₄⋊C₄	—	89	4	no	—
164	C₂×C₄⋊C₄	—	178	2	no	—
165	C₂×C₄⋊C₄	—	178	2	no	—
166	C₂×C₄⋊C₄	—	178	2	no	—
167	C₂×C₄⋊C₄	—	178	8	no	—
168	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
170	#107	—	178	4	no	—
171	#107	—	178	8	no	—
172	#106	—	178	4	no	—
173	#106	—	178	4	no	—
174	#106	—	178	2	no	—
175	#106	—	178	2	no	—
176	#106	—	178	8	no	—
177	#97	Dic₁₇₈	1	8	yes	complemented
178	#97	D₃₅₆	1	4	yes	—
179	C₄×C₈	Dic₁₇₈	1	8	yes	complemented
180	C₄×C₈	D₃₅₆	1	4	yes	—
181	#95	—	178	8	no	—
182	#95	D₃₅₆	1	4	yes	—
183	C₂×Q₈	—	356	4	no	—
184	C₂×Q₈	—	712	4	no	—
185	C₂²×C₄	—	178	2	no	—
186	C₂²×C₄	—	178	4	no	—
187	C₂²×C₄	—	178	2	no	—
188	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
189	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
190	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
191	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
192	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
193	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
194	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
195	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
196	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
197	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
198	C₄⋊C₄	—	356	4	no	—
199	C₄⋊C₄	—	356	4	no	—
200	C₄⋊C₄	—	356	4	no	—
201	C₄⋊C₄	—	178	2	no	—
202	C₄⋊C₄	—	178	4	no	—
203	C₄⋊C₄	—	178	2	no	—
204	C₄⋊C₄	—	178	2	no	—
205	C₄⋊C₄	—	178	4	no	—
206	C₄⋊C₄	—	178	2	no	—
207	C₄⋊C₄	—	712	8	no	—
208	C₄²	C₂×D₃₅₆	1	2	yes	—
209	C₄²	C₂×D₃₅₆	1	2	yes	—
210	C₄²	C₂×Dic₁₇₈	1	4	yes	—
211	C₄²	C₂×Dic₁₇₈	1	4	yes	—
212	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
213	Q₈	—	712	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	356	2	no	—
217	C₂×C₄	—	356	2	no	—
218	C₂×C₄	—	356	4	no	—
219	C₂×C₄	—	356	4	no	—
220	C₂×C₄	—	356	8	no	—
221	C₂×C₄	—	356	4	no	—
222	C₂×C₄	#15668009	1	4	yes	—
223	C₂×C₄	#15668012	1	8	yes	—
224	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₁₇₈	1	2	yes	—
227	C₂×C₄	C₂²×Dic₁₇₈	1	2	yes	—
228	C₈	#15668012	1	4	yes	—
229	C₈	#15668012	1	4	yes	—
230	C₈	#15668012	1	8	yes	—
231	C₂²	C₂×#15668009	1	1	yes	characteristic
232	C₂²	C₂×#15668012	1	2	yes	—
233	C₂²	#19152410	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	712	4	no	—
237	C₄	—	712	2	no	—
238	C₄	—	712	2	no	—
239	C₄	C₂×#15668012	1	1	yes	characteristic
240	C₄	C₂×#15668012	1	1	yes	characteristic
241	C₄	C₂×#15668012	1	2	yes	—
242	C₂	#26028650	1	1	yes	characteristic
243	C₂	#26030728	1	2	yes	—
244	C₂	#26030783	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	#33318
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
131. C₈₉	#33318	semidirect
177. #97	Dic₁₇₈	semidirect
179. 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×#19152000	2
C₂×#19151996	2
#26026371	2
#26020420	2
#26020416	2
#26020409	2
C₈₉×#1726	2
#33318	89