Subgroups — #36463032

Order 22784.

958 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#10623494	C₂	1	3	yes	maximal
2	C₂×#19151353	C₂	1	3	yes	maximal
3	C₄×#95×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10623494	C₂²	1	3	yes	—
5	C₂³×#10623494	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊#10623494	—	2	6	no	—
7	C₂×C₄⋊#10623494	—	2	12	no	—
8	#19151353	—	2	6	no	—
9	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
10	C₂×#95×C₈₉	C₄	1	3	yes	—
11	C₂×#95×C₈₉	C₄	1	1	yes	characteristic
12	C₂²×#10623494	—	2	6	no	—
13	C₂²×#10623494	—	2	6	no	—
14	C₂²×#10623494	—	2	3	no	—
15	C₂²×#10623494	—	2	3	no	—
16	C₂²×#10623494	—	2	3	no	—
17	C₂²×#10623494	—	2	3	no	—
18	C₂²×#10623494	D₄	1	6	yes	—
19	C₂²×#10623494	Q₈	1	2	yes	—
20	C₄⋊#10623494	—	2	24	no	—
21	C₂³×C₄×C₈₉	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₈₉	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	—	2	6	no	—
25	C₂×C₄²×C₈₉	—	2	6	no	—
26	C₂×C₄²×C₈₉	D₄	1	6	yes	—
27	C₂×C₄²×C₈₉	Q₈	1	6	yes	—
28	#95×C₈₉	—	2	6	no	—
29	#95×C₈₉	C₈	1	4	yes	complemented
30	C₂×#10623494	—	2	12	no	—
31	C₂×#10623494	—	2	6	no	—
32	C₂×#10623494	—	2	6	no	—
33	C₂×#10623494	—	4	3	no	—
34	C₂×#10623494	—	4	6	no	—
35	C₂×#10623494	—	4	3	no	—
36	C₂×#10623494	—	4	3	no	—
37	C₂×#10623494	—	4	1	no	—
38	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	—	2	6	no	—
40	C₂²×C₄×C₈₉	—	2	6	no	—
41	C₂²×C₄×C₈₉	—	2	6	no	—
42	C₂²×C₄×C₈₉	—	2	6	no	—
43	C₂²×C₄×C₈₉	—	2	3	no	—
44	C₂²×C₄×C₈₉	—	2	3	no	—
45	C₂²×C₄×C₈₉	C₄⋊C₄	1	6	yes	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	6	yes	—
47	C₂²×C₄×C₈₉	C₂×C₈	1	6	yes	—
48	C₂²×C₄×C₈₉	M₁₆	1	6	yes	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×Q₈	1	3	yes	—
53	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	#41	1	3	yes	—
55	C₄²×C₈₉	—	2	6	no	—
56	C₄²×C₈₉	—	2	12	no	—
57	C₄²×C₈₉	—	2	6	no	—
58	C₂³×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
59	C₂³×C₈₉	C₄×D₄	1	3	yes	—
60	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
61	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
62	C₂³×C₈₉	C₂×M₁₆	1	3	yes	—
63	C₂³×C₈₉	#131	1	3	yes	—
64	C₂³×C₈₉	#131	1	1	yes	characteristic
65	C₂×C₄×C₈₉	—	2	6	no	—
66	C₂×C₄×C₈₉	—	2	6	no	—
67	C₂×C₄×C₈₉	—	2	12	no	—
68	C₂×C₄×C₈₉	—	2	12	no	—
69	C₂×C₄×C₈₉	—	2	6	no	—
70	C₂×C₄×C₈₉	—	2	6	no	—
71	C₂×C₄×C₈₉	—	2	6	no	—
72	C₂×C₄×C₈₉	—	4	6	no	—
73	C₂×C₄×C₈₉	—	4	6	no	—
74	C₂×C₄×C₈₉	C₄⋊C₈	1	12	yes	complemented
75	C₂×C₄×C₈₉	#120	1	3	yes	—
76	C₂×C₄×C₈₉	#121	1	3	yes	—
77	C₂×C₄×C₈₉	C₂²⋊Q₈	1	6	yes	—
78	C₂×C₄×C₈₉	C₂²⋊Q₈	1	3	yes	—
79	C₂×C₄×C₈₉	#123	1	3	yes	—
80	C₂×C₄×C₈₉	#125	1	3	yes	—
81	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
83	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
84	#10623494	—	4	6	no	—
85	#10623494	—	4	2	no	—
86	C₂²×C₈₉	#384	1	1	yes	characteristic
87	C₂²×C₈₉	#385	1	3	yes	—
88	C₂²×C₈₉	#389	1	3	yes	—
89	C₂²×C₈₉	C₂×C₄⋊C₈	1	3	yes	—
90	C₂²×C₈₉	#423	1	3	yes	—
91	C₂²×C₈₉	#424	1	3	yes	—
92	C₂²×C₈₉	#424	1	3	yes	—
93	C₂²×C₈₉	C₈×D₄	1	3	yes	—
94	C₂²×C₈₉	#435	1	6	yes	—
95	C₂²×C₈₉	#436	1	3	yes	—
96	C₂²×C₈₉	C₈×Q₈	1	1	yes	characteristic
97	C₂²×C₈₉	#446	1	3	yes	—
98	C₄×C₈₉	—	2	12	no	—
99	C₄×C₈₉	—	4	6	no	—
100	C₄×C₈₉	#392	1	1	yes	characteristic
101	C₄×C₈₉	#395	1	3	yes	—
102	C₄×C₈₉	#398	1	3	yes	—
103	C₄×C₈₉	#399	1	1	yes	characteristic
104	#11322	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	#1444	1	1	yes	characteristic
106	C₂×C₈₉	#1842	1	1	yes	characteristic
107	C₂×C₈₉	#1843	1	1	yes	characteristic
108	C₂×C₈₉	#1850	1	3	yes	—
109	C₂×C₈₉	#1851	1	3	yes	—
110	C₂×C₈₉	#1939	1	3	yes	—
111	C₂×C₈₉	#1940	1	3	yes	—
112	C₂²×C₄⋊C₈	—	89	3	no	—
113	C₂×#336	—	89	3	no	—
114	C₄×#95	D₈₉	1	1	yes	characteristic
115	C₈₉	#11322	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₈	—	89	3	no	—
117	C₂³×C₈	—	89	1	no	—
118	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
119	C₂×C₄⋊C₈	—	178	12	no	—
120	C₂×C₄⋊C₈	—	178	6	no	—
121	C₂×#95	Dic₈₉	1	3	yes	—
122	C₂×#95	Dic₈₉	1	1	yes	characteristic
123	#336	—	178	6	no	—
124	C₂³×C₄	C₂×Dic₈₉	1	3	yes	—
125	C₂³×C₄	C₂×Dic₈₉	1	3	yes	—
126	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	C₂²×C₈	—	89	6	no	—
128	C₂²×C₈	—	89	2	no	—
129	C₂²×C₈	—	178	6	no	—
130	C₂²×C₈	—	178	3	no	—
131	C₂²×C₈	—	178	3	no	—
132	C₂²×C₈	—	178	6	no	—
133	C₂²×C₈	—	178	3	no	—
134	C₂²×C₈	—	178	3	no	—
135	C₂×C₄²	—	2	6	no	—
136	C₂×C₄²	—	2	6	no	—
137	C₂×C₄²	Dic₁₇₈	1	6	yes	—
138	C₂×C₄²	D₃₅₆	1	6	yes	—
139	C₄⋊C₈	—	178	24	no	—
140	#95	—	2	6	no	—
141	#95	#10623494	1	4	yes	complemented
142	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
143	C₂²×C₄	—	2	6	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	3	no	—
148	C₂²×C₄	—	2	3	no	—
149	C₂²×C₄	C₄⋊Dic₈₉	1	6	yes	—
150	C₂²×C₄	C₄⋊Dic₈₉	1	6	yes	—
151	C₂²×C₄	C₂×#10623494	1	6	yes	—
152	C₂²×C₄	#13083558	1	6	yes	—
153	C₂²×C₄	D₄×D₈₉	1	3	yes	—
154	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
155	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₇₈	1	3	yes	—
157	C₂²×C₄	#13083573	1	3	yes	—
158	C₂²×C₄	#13083575	1	1	yes	characteristic
159	C₂×C₈	—	356	3	no	—
160	C₂×C₈	—	356	6	no	—
161	C₂×C₈	—	356	3	no	—
162	C₂×C₈	—	356	3	no	—
163	C₂×C₈	—	356	1	no	—
164	C₂×C₈	—	178	12	no	—
165	C₂×C₈	—	178	6	no	—
166	C₂×C₈	—	178	6	no	—
167	C₄²	—	2	6	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₂³	C₂×C₄⋊Dic₈₉	1	3	yes	—
171	C₂³	D₄×Dic₈₉	1	3	yes	—
172	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
173	C₂³	C₂²×#10623494	1	1	yes	characteristic
174	C₂³	C₂×#13083558	1	3	yes	—
175	C₂³	#15668019	1	3	yes	—
176	C₂³	#15668019	1	1	yes	characteristic
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	12	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	6	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	4	6	no	—
185	C₂×C₄	—	4	6	no	—
186	C₂×C₄	C₄⋊#10623494	1	12	yes	complemented
187	C₂×C₄	#15667982	1	3	yes	—
188	C₂×C₄	#15667987	1	3	yes	—
189	C₂×C₄	#15667989	1	6	yes	—
190	C₂×C₄	#15667992	1	3	yes	—
191	C₂×C₄	#15667996	1	3	yes	—
192	C₂×C₄	#15668003	1	3	yes	—
193	C₂×C₄	#15668009	1	1	yes	characteristic
194	C₂×C₄	#15668011	1	3	yes	—
195	C₂×C₄	#15668012	1	3	yes	—
196	C₈	—	356	6	no	—
197	C₈	—	356	2	no	—
198	C₂²	#19151468	1	1	yes	characteristic
199	C₂²	#19151473	1	3	yes	—
200	C₂²	#19151497	1	3	yes	—
201	C₂²	C₂×C₄⋊#10623494	1	3	yes	—
202	C₂²	#19151622	1	3	yes	—
203	C₂²	#19151625	1	3	yes	—
204	C₂²	#19151625	1	3	yes	—
205	C₂²	D₄×#10623494	1	3	yes	—
206	C₂²	#19151670	1	6	yes	—
207	C₂²	#19151676	1	3	yes	—
208	C₂²	Q₈×#10623494	1	1	yes	characteristic
209	C₂²	#19151729	1	3	yes	—
210	C₄	—	2	12	no	—
211	C₄	—	4	6	no	—
212	C₄	#19151508	1	1	yes	characteristic
213	C₄	#19151519	1	3	yes	—
214	C₄	#19151530	1	3	yes	—
215	C₄	#19151535	1	1	yes	characteristic
216	C₂	#26018517	1	1	yes	characteristic
217	C₂	#26020404	1	1	yes	characteristic
218	C₂	#26020406	1	1	yes	characteristic
219	C₂	#26020432	1	3	yes	—
220	C₂	#26020437	1	3	yes	—
221	C₂	#26020971	1	3	yes	—
222	C₂	#26020976	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₈₉
commutator	C₂³×C₈₉	C₂²×C₈
Fitting	C₄×#95×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₄×#95	D₈₉
hypocenter	C₈₉	#11322
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
29. #95×C₈₉	C₈	semidirect
74. C₂×C₄×C₈₉	C₄⋊C₈	semidirect
115. C₈₉	#11322	semidirect
141. #95	#10623494	semidirect
186. C₂×C₄	C₄⋊#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#10623494	2
C₂×#19151353	2
C₄×#95×C₈₉	2
#11322	89