Subgroups — #36453200

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
2	#26020404	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151330	C₂	1	4	yes	maximal
4	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×C₄⋊#10623494	—	2	2	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151504	C₄	1	2	yes	—
10	#19151496	—	2	2	no	—
11	#19151468	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₈₉	—	2	2	no	—
13	#19151330	—	2	8	no	—
14	C₂×C₄×C₈×C₈₉	—	2	2	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
16	C₄³×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
18	C₂²×#10623494	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15668012	C₈	1	4	yes	complemented
21	C₄×Dic₁₇₈	—	2	4	no	—
22	C₄×Dic₁₇₈	—	2	4	no	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
26	C₂×#13083553	—	2	2	no	—
27	C₂×C₄×Dic₈₉	—	2	2	no	—
28	C₄⋊#10623494	—	2	4	no	—
29	C₄⋊#10623494	—	4	4	no	—
30	#15667919	—	2	4	no	—
31	C₂²×C₈×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	2	yes	—
35	C₂×C₄²×C₈₉	D₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₈₉	—	2	8	no	—
38	C₄×C₈×C₈₉	—	2	2	no	—
39	C₄×C₈×C₈₉	D₄	1	4	yes	—
40	C₂×Dic₁₇₈	—	2	4	no	—
41	C₂×Dic₁₇₈	—	2	4	no	—
42	C₂×Dic₁₇₈	—	2	2	no	—
43	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
44	C₂²×Dic₈₉	—	2	2	no	—
45	C₂²×Dic₈₉	—	2	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	4	no	—
48	C₄⋊Dic₈₉	—	2	8	no	—
49	C₄⋊Dic₈₉	—	2	4	no	—
50	C₄⋊Dic₈₉	—	4	4	no	—
51	#13083553	—	4	4	no	—
52	C₄×Dic₈₉	—	4	2	no	—
53	C₄×Dic₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	—	2	8	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₄²×C₈₉	—	2	4	no	—
62	C₄²×C₈₉	—	2	2	no	—
63	C₄²×C₈₉	C₂×C₈	1	2	yes	—
64	C₄²×C₈₉	M₁₆	1	2	yes	—
65	C₄²×C₈₉	SD₁₆	1	4	yes	—
66	C₄²×C₈₉	Q₁₆	1	4	yes	—
67	C₄²×C₈₉	C₂×D₄	1	2	yes	—
68	C₄²×C₈₉	#41	1	2	yes	—
69	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₈₉	#103	1	4	yes	—
75	C₂×C₄×C₈₉	#103	1	4	yes	—
76	C₂×C₄×C₈₉	#104	1	4	yes	—
77	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₈₉	#121	1	2	yes	—
79	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#131	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
86	C₂×Dic₈₉	—	2	4	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	C₂×Dic₈₉	—	4	2	no	—
90	C₂×Dic₈₉	—	4	4	no	—
91	Dic₁₇₈	—	2	8	no	—
92	Dic₁₇₈	—	4	4	no	—
93	C₈×C₈₉	—	2	8	no	—
94	#10623494	—	8	2	no	—
95	C₂²×C₈₉	#390	1	1	yes	characteristic
96	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₈₉	#408	1	1	yes	characteristic
98	C₂²×C₈₉	C₂×#103	1	2	yes	—
99	C₂²×C₈₉	C₂×#104	1	2	yes	—
100	C₄×C₈₉	—	2	2	no	—
101	C₄×C₈₉	—	2	4	no	—
102	C₄×C₈₉	Q₈⋊C₈	1	8	yes	complemented
103	C₄×C₈₉	C₈×D₄	1	2	yes	—
104	C₄×C₈₉	#436	1	2	yes	—
105	C₄×C₈₉	#487	1	2	yes	—
106	C₄×C₈₉	#492	1	1	yes	characteristic
107	C₄×C₈₉	#494	1	1	yes	characteristic
108	Dic₈₉	—	4	4	no	—
109	Dic₈₉	—	8	2	no	—
110	#9637	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	C₂×Q₈⋊C₈	1	2	yes	—
112	C₂×C₈₉	#1493	1	2	yes	—
113	C₂×C₈₉	#1954	1	1	yes	characteristic
114	C₂×C₈₉	#1956	1	1	yes	characteristic
115	C₂×C₈₉	#1962	1	1	yes	characteristic
116	C₄×C₄⋊Q₈	—	89	1	no	—
117	#1842	—	89	1	no	—
118	C₄²×C₈	D₈₉	1	1	yes	characteristic
119	C₂×Q₈⋊C₈	—	89	4	no	—
120	C₈₉	#9637	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊Q₈	—	89	1	no	—
122	C₂×C₄×Q₈	—	89	2	no	—
123	C₂×C₄⋊C₈	—	89	2	no	—
124	C₂×C₄⋊C₈	—	178	2	no	—
125	C₂×C₄×C₈	—	2	2	no	—
126	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
127	#391	—	89	2	no	—
128	#389	—	178	2	no	—
129	#384	—	89	1	no	—
130	C₄×C₄⋊C₄	—	178	2	no	—
131	C₄³	C₂×D₈₉	1	1	yes	characteristic
132	Q₈⋊C₈	—	178	8	no	—
133	C₂²×Q₈	—	89	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	178	2	no	—
136	C₄⋊Q₈	—	89	4	no	—
137	C₄⋊Q₈	—	178	2	no	—
138	C₄×Q₈	—	178	4	no	—
139	C₄×Q₈	—	178	4	no	—
140	C₂×C₄⋊C₄	—	89	2	no	—
141	C₂×C₄⋊C₄	—	178	2	no	—
142	C₂×C₄⋊C₄	—	178	2	no	—
143	C₂×C₄⋊C₄	—	178	2	no	—
144	C₂×C₄²	—	178	2	no	—
145	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
147	C₂×C₄²	D₃₅₆	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
149	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
150	C₄⋊C₈	—	356	4	no	—
151	C₄⋊C₈	—	178	4	no	—
152	C₄×C₈	—	2	8	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	D₃₅₆	1	4	yes	—
155	#95	—	178	4	no	—
156	C₂×Q₈	—	89	4	no	—
157	C₂×Q₈	—	178	4	no	—
158	C₂×Q₈	—	178	4	no	—
159	C₂×Q₈	—	178	2	no	—
160	C₂²×C₄	—	178	2	no	—
161	C₂²×C₄	—	178	2	no	—
162	C₂²×C₄	#13083551	1	2	yes	—
163	C₂²×C₄	#13083551	1	2	yes	—
164	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	356	2	no	—
170	C₂×C₈	—	356	4	no	—
171	C₄⋊C₄	—	356	4	no	—
172	C₄⋊C₄	—	356	4	no	—
173	C₄⋊C₄	—	178	4	no	—
174	C₄⋊C₄	—	178	8	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	356	2	no	—
178	C₄²	—	356	2	no	—
179	C₄²	C₈×D₈₉	1	2	yes	—
180	C₄²	#13083557	1	2	yes	—
181	C₄²	#13083563	1	4	yes	—
182	C₄²	Dic₃₅₆	1	4	yes	—
183	C₄²	C₂×D₃₅₆	1	2	yes	—
184	C₄²	#13083574	1	2	yes	—
185	C₂³	C₂×#13083551	1	1	yes	characteristic
186	Q₈	—	356	4	no	—
187	Q₈	—	178	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	356	2	no	—
192	C₂×C₄	—	356	2	no	—
193	C₂×C₄	—	356	2	no	—
194	C₂×C₄	—	356	4	no	—
195	C₂×C₄	—	178	4	no	—
196	C₂×C₄	#15667924	1	4	yes	—
197	C₂×C₄	#15667937	1	4	yes	—
198	C₂×C₄	#15667937	1	4	yes	—
199	C₂×C₄	#15667940	1	4	yes	—
200	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
201	C₂×C₄	#15667986	1	2	yes	—
202	C₂×C₄	#15668000	1	1	yes	characteristic
203	C₂×C₄	#15668009	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
205	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
206	C₂×C₄	#15668018	1	2	yes	—
207	C₂×C₄	C₂×#13083563	1	2	yes	—
208	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
209	C₈	—	2	8	no	—
210	C₈	—	712	2	no	—
211	C₂²	#19151500	1	1	yes	characteristic
212	C₂²	C₂×#15667924	1	1	yes	characteristic
213	C₂²	#19151564	1	1	yes	characteristic
214	C₂²	C₂×#15667937	1	2	yes	—
215	C₂²	C₂×#15667940	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	356	4	no	—
219	C₄	—	712	2	no	—
220	C₄	#19151330	1	8	yes	complemented
221	C₄	C₈×D₃₅₆	1	2	yes	—
222	C₄	#19151677	1	2	yes	—
223	C₄	#19151949	1	2	yes	—
224	C₄	#19151971	1	1	yes	characteristic
225	C₄	#19151978	1	1	yes	characteristic
226	C₂	C₂×#19151330	1	2	yes	—
227	C₂	#26018708	1	2	yes	—
228	C₂	#26021046	1	1	yes	characteristic
229	C₂	#26021054	1	1	yes	characteristic
230	C₂	#26021080	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#9637
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
20. #15668012	C₈	semidirect
102. C₄×C₈₉	Q₈⋊C₈	semidirect
120. C₈₉	#9637	semidirect
220. C₄	#19151330	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668012	2
#26020404	2
C₂×#19151330	2
C₄²×C₈×C₈₉	2
#9637	89