Subgroups — #36453219

Order 22784.

530 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020369	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10623494	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₂²	1	2	yes	—
7	#19151353	—	2	2	no	—
8	#19151353	—	2	2	no	—
9	Q₈⋊#10623494	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
12	#391×C₈₉	C₄	1	2	yes	—
13	#389×C₈₉	—	2	2	no	—
14	#384×C₈₉	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
16	C₄³×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₄⋊#10623494	—	2	4	no	—
20	C₄×#10623494	—	2	2	no	—
21	C₄×#10623494	D₄	1	4	yes	—
22	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
23	C₄⋊Q₈×C₈₉	—	2	2	no	—
24	C₄⋊Q₈×C₈₉	C₈	1	4	yes	complemented
25	C₄×Q₈×C₈₉	—	2	4	no	—
26	C₄×Q₈×C₈₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×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	#95×C₈₉	—	2	4	no	—
38	C₂×#10623494	—	2	4	no	—
39	C₂×#10623494	—	4	4	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	4	no	—
42	C₂×Q₈×C₈₉	—	2	2	no	—
43	C₂×Q₈×C₈₉	—	2	4	no	—
44	C₂×Q₈×C₈₉	—	2	4	no	—
45	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₈₉	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₈₉	—	2	8	no	—
54	C₄⋊C₄×C₈₉	—	2	4	no	—
55	C₄⋊C₄×C₈₉	—	4	4	no	—
56	C₄⋊C₄×C₈₉	—	4	4	no	—
57	C₄²×C₈₉	—	2	2	no	—
58	C₄²×C₈₉	—	2	4	no	—
59	C₄²×C₈₉	—	4	2	no	—
60	C₄²×C₈₉	—	4	2	no	—
61	C₄²×C₈₉	—	4	2	no	—
62	C₄²×C₈₉	C₂×C₈	1	2	yes	—
63	C₄²×C₈₉	M₁₆	1	2	yes	—
64	C₄²×C₈₉	C₂×D₄	1	2	yes	—
65	C₄²×C₈₉	#41	1	2	yes	—
66	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	Q₈×C₈₉	—	2	8	no	—
68	Q₈×C₈₉	—	4	4	no	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	—	4	2	no	—
78	C₂×C₄×C₈₉	—	4	2	no	—
79	C₂×C₄×C₈₉	—	4	4	no	—
80	C₂×C₄×C₈₉	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₈₉	#121	1	2	yes	—
83	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#131	1	2	yes	—
88	C₂×C₄×C₈₉	#137	1	4	yes	—
89	#10623494	—	4	4	no	—
90	#10623494	—	8	2	no	—
91	C₂²×C₈₉	#390	1	1	yes	characteristic
92	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂²×C₈₉	#408	1	1	yes	characteristic
94	C₂²×C₈₉	#419	1	2	yes	—
95	C₂²×C₈₉	#421	1	2	yes	—
96	C₄×C₈₉	—	2	2	no	—
97	C₄×C₈₉	—	2	4	no	—
98	C₄×C₈₉	—	4	4	no	—
99	C₄×C₈₉	—	4	2	no	—
100	C₄×C₈₉	—	8	2	no	—
101	C₄×C₈₉	C₈×D₄	1	2	yes	—
102	C₄×C₈₉	#436	1	2	yes	—
103	C₄×C₈₉	#491	1	2	yes	—
104	C₄×C₈₉	#497	1	2	yes	—
105	#9641	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	#1479	1	2	yes	—
107	C₂×C₈₉	#1497	1	2	yes	—
108	C₂×C₈₉	#1954	1	1	yes	characteristic
109	C₂×C₈₉	#1960	1	1	yes	characteristic
110	C₂×C₈₉	#1963	1	1	yes	characteristic
111	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
112	#1831	—	89	1	no	—
113	#1744	—	89	1	no	—
114	C₂×Q₈⋊C₈	—	89	4	no	—
115	C₈₉	#9641	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
117	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
118	C₂×C₄⋊C₈	—	89	2	no	—
119	C₂×C₄×C₈	—	89	2	no	—
120	#391	Dic₈₉	1	2	yes	—
121	#389	—	2	2	no	—
122	#384	Dic₈₉	1	1	yes	characteristic
123	C₄×C₄⋊C₄	—	2	2	no	—
124	C₄³	C₂×D₈₉	1	1	yes	characteristic
125	#336	—	178	2	no	—
126	#336	—	178	2	no	—
127	Q₈⋊C₈	—	178	8	no	—
128	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
129	C₂²×C₈	—	178	2	no	—
130	C₂²×C₈	—	178	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	#10623494	1	4	yes	complemented
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
141	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
143	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
144	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊C₈	—	178	4	no	—
146	C₄×C₈	—	89	4	no	—
147	C₄×C₈	—	178	2	no	—
148	#95	—	2	4	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	C₂×#10623494	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
160	C₂×C₈	—	356	4	no	—
161	C₂×C₈	—	356	2	no	—
162	C₂×C₈	—	356	4	no	—
163	C₂×C₈	—	178	4	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×#10623494	1	2	yes	—
174	C₄²	#13083558	1	2	yes	—
175	C₄²	D₄×D₈₉	1	2	yes	—
176	C₄²	#13083573	1	2	yes	—
177	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₂²⋊#10623494	1	4	yes	—
192	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
193	C₂×C₄	#15667983	1	2	yes	—
194	C₂×C₄	#15667999	1	1	yes	characteristic
195	C₂×C₄	#15668008	1	1	yes	characteristic
196	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
197	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
198	C₂×C₄	#15668019	1	2	yes	—
199	C₂×C₄	#15668043	1	4	yes	—
200	C₈	—	356	4	no	—
201	C₈	—	712	2	no	—
202	C₂²	#19151501	1	1	yes	characteristic
203	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
204	C₂²	#19151565	1	1	yes	characteristic
205	C₂²	#19151607	1	2	yes	—
206	C₂²	#19151615	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	4	no	—
211	C₄	—	8	2	no	—
212	C₄	D₄×#10623494	1	2	yes	—
213	C₄	#19151676	1	2	yes	—
214	C₄	#19151966	1	2	yes	—
215	C₄	#19151990	1	2	yes	—
216	C₂	#26018647	1	2	yes	—
217	C₂	#26018725	1	2	yes	—
218	C₂	#26021047	1	1	yes	characteristic
219	C₂	#26021073	1	1	yes	characteristic
220	C₂	#26021085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#9641
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊Q₈×C₈₉	C₈	semidirect
115. C₈₉	#9641	semidirect
132. C₄⋊Q₈	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26020369	2
#26019998	2
C₂×Q₈⋊#10623494	2
C₄×C₄⋊Q₈×C₈₉	2
#9641	89