Subgroups — #36460218

Order 22784.

950 conjugacy classes of subgroups.

The subgroup lattice has 208 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	2	yes	maximal
2	C₂×#19151353	C₂	1	4	yes	maximal
3	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10623494	C₂²	1	4	yes	—
5	C₂×C₄⋊#10623494	—	2	4	no	—
6	C₂×C₄⋊#10623494	—	2	4	no	—
7	C₂×C₄⋊#10623494	C₄	1	8	yes	complemented
8	#19151353	—	2	8	no	—
9	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	#376×C₈₉	—	2	4	no	—
12	C₂×#95×C₈₉	C₄	1	4	yes	—
13	C₂²×#10623494	—	2	4	no	—
14	C₂²×#10623494	—	2	4	no	—
15	C₂²×#10623494	—	2	8	no	—
16	C₂²×#10623494	—	2	8	no	—
17	C₂²×#10623494	C₂×C₄	1	8	yes	—
18	C₄⋊#10623494	—	2	16	no	—
19	C₂³×C₄×C₈₉	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	—	2	4	no	—
23	C₂×C₄²×C₈₉	—	2	4	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₈₉	D₄	1	4	yes	—
26	C₂×C₄²×C₈₉	D₄	1	2	yes	—
27	C₂×C₄²×C₈₉	D₄	1	4	yes	—
28	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
29	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
30	#95×C₈₉	—	2	8	no	—
31	C₂×#10623494	—	2	8	no	—
32	C₂×#10623494	—	2	16	no	—
33	C₂×#10623494	—	4	8	no	—
34	C₂×#10623494	—	4	4	no	—
35	C₂×#10623494	—	4	4	no	—
36	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	—	2	8	no	—
38	C₂²×C₄×C₈₉	—	2	4	no	—
39	C₂²×C₄×C₈₉	—	2	4	no	—
40	C₂²×C₄×C₈₉	—	2	16	no	—
41	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
42	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₈₉	#41	1	4	yes	—
52	C₄²×C₈₉	—	2	8	no	—
53	C₄²×C₈₉	—	2	8	no	—
54	C₄²×C₈₉	C₂²⋊C₄	1	8	yes	—
55	C₄²×C₈₉	C₄⋊C₄	1	8	yes	—
56	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
57	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₈₉	#117	1	2	yes	—
61	C₂³×C₈₉	#131	1	8	yes	—
62	C₂×C₄×C₈₉	—	2	8	no	—
63	C₂×C₄×C₈₉	—	2	8	no	—
64	C₂×C₄×C₈₉	—	2	8	no	—
65	C₂×C₄×C₈₉	—	2	8	no	—
66	C₂×C₄×C₈₉	—	2	16	no	—
67	C₂×C₄×C₈₉	—	4	8	no	—
68	C₂×C₄×C₈₉	—	4	8	no	—
69	C₂×C₄×C₈₉	#95	1	8	yes	—
70	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
74	C₂×C₄×C₈₉	#121	1	4	yes	—
75	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
76	C₂×C₄×C₈₉	#124	1	2	yes	—
77	C₂×C₄×C₈₉	#125	1	2	yes	—
78	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
81	#10623494	—	4	8	no	—
82	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
83	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
84	C₂²×C₈₉	#380	1	2	yes	—
85	C₂²×C₈₉	#382	1	1	yes	characteristic
86	C₂²×C₈₉	#384	1	1	yes	characteristic
87	C₂²×C₈₉	#405	1	4	yes	—
88	C₂²×C₈₉	#408	1	4	yes	—
89	C₂²×C₈₉	#424	1	8	yes	—
90	C₂²×C₈₉	#424	1	4	yes	—
91	C₂²×C₈₉	#433	1	8	yes	—
92	C₄×C₈₉	—	2	8	no	—
93	C₄×C₈₉	—	4	8	no	—
94	C₄×C₈₉	#389	1	4	yes	—
95	C₄×C₈₉	#390	1	2	yes	—
96	C₄×C₈₉	#391	1	2	yes	—
97	#10833	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
99	C₂×C₈₉	#1729	1	2	yes	—
100	C₂×C₈₉	#1768	1	4	yes	—
101	C₂×C₈₉	#1798	1	4	yes	—
102	C₂×C₈₉	#1832	1	2	yes	—
103	C₂×C₈₉	#1843	1	2	yes	—
104	C₂²×C₄⋊C₈	—	89	2	no	—
105	C₂×#376	D₈₉	1	1	yes	characteristic
106	C₂×#336	—	89	4	no	—
107	C₈₉	#10833	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₈	—	89	4	no	—
109	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
110	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
111	C₂×C₄⋊C₈	—	89	8	no	—
112	C₂×C₄⋊C₈	—	178	4	no	—
113	C₂×C₄⋊C₈	—	178	4	no	—
114	#376	—	2	4	no	—
115	C₂×#95	Dic₈₉	1	4	yes	—
116	#336	—	178	8	no	—
117	C₂³×C₄	C₂×Dic₈₉	1	4	yes	—
118	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
119	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
120	C₂²×C₈	—	89	8	no	—
121	C₂²×C₈	—	178	4	no	—
122	C₂²×C₈	—	178	8	no	—
123	C₂²×C₈	—	178	4	no	—
124	C₂²×C₈	—	178	8	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	—	2	4	no	—
127	C₂×C₄²	Dic₁₇₈	1	4	yes	—
128	C₂×C₄²	Dic₁₇₈	1	2	yes	—
129	C₂×C₄²	C₄×D₈₉	1	4	yes	—
130	C₂×C₄²	D₃₅₆	1	4	yes	—
131	C₂×C₄²	D₃₅₆	1	2	yes	—
132	C₂×C₄²	C₂²⋊D₈₉	1	4	yes	—
133	C₄⋊C₈	—	178	16	no	—
134	#95	—	2	8	no	—
135	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	16	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
141	C₂²×C₄	C₂²⋊Dic₈₉	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₈₉	1	8	yes	—
143	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
144	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
146	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
147	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
149	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
150	C₂²×C₄	#13083574	1	4	yes	—
151	C₂×C₈	—	356	8	no	—
152	C₂×C₈	—	356	4	no	—
153	C₂×C₈	—	356	4	no	—
154	C₂×C₈	—	178	16	no	—
155	C₂×C₈	—	178	8	no	—
156	C₄²	—	2	8	no	—
157	C₄²	—	2	8	no	—
158	C₄²	#13083551	1	8	yes	—
159	C₄²	#13083553	1	8	yes	—
160	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
162	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
163	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
164	C₂³	#15667971	1	2	yes	—
165	C₂³	#15668019	1	8	yes	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#15667919	1	8	yes	—
174	C₂×C₄	C₂×#13083551	1	2	yes	—
175	C₂×C₄	C₂×#13083553	1	2	yes	—
176	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
177	C₂×C₄	C₄×Dic₁₇₈	1	4	yes	—
178	C₂×C₄	#15667986	1	4	yes	—
179	C₂×C₄	#15667991	1	4	yes	—
180	C₂×C₄	#15668000	1	2	yes	—
181	C₂×C₄	#15668004	1	2	yes	—
182	C₂×C₄	#15668009	1	1	yes	characteristic
183	C₂×C₄	#15668012	1	2	yes	—
184	C₂×C₄	#15668012	1	1	yes	characteristic
185	C₈	—	356	8	no	—
186	C₂²	C₂×#15667919	1	1	yes	characteristic
187	C₂²	C₄×C₄⋊Dic₈₉	1	2	yes	—
188	C₂²	#19151458	1	2	yes	—
189	C₂²	#19151464	1	1	yes	characteristic
190	C₂²	#19151468	1	1	yes	characteristic
191	C₂²	#19151554	1	4	yes	—
192	C₂²	#19151565	1	4	yes	—
193	C₂²	#19151625	1	8	yes	—
194	C₂²	#19151625	1	4	yes	—
195	C₂²	#19151659	1	8	yes	—
196	C₄	—	2	8	no	—
197	C₄	—	4	8	no	—
198	C₄	#19151496	1	4	yes	—
199	C₄	#19151500	1	2	yes	—
200	C₄	#19151504	1	2	yes	—
201	C₂	#26018482	1	1	yes	characteristic
202	C₂	#26019949	1	2	yes	—
203	C₂	#26020109	1	4	yes	—
204	C₂	#26020236	1	4	yes	—
205	C₂	#26020372	1	2	yes	—
206	C₂	#26020406	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₈₉
commutator	C₂³×C₈₉	C₂×C₄²
Fitting	C₂×#376×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	#10833
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊#10623494	C₄	semidirect
107. C₈₉	#10833	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
C₂×#376×C₈₉	2
#10833	89