Subgroups — #36454439

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668004	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	#19151496	C₄	1	2	yes	—
8	#19151482	—	2	2	no	—
9	#19151464	C₄	1	1	yes	characteristic
10	C₄×#13083553	—	2	2	no	—
11	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
12	#19151353	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	2	2	no	—
18	#15668004	C₈	1	4	yes	complemented
19	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
21	C₂×#13083553	—	2	2	no	—
22	C₂×#13083553	—	2	2	no	—
23	C₂×C₄×Dic₈₉	—	2	2	no	—
24	C₄⋊#10623494	—	2	4	no	—
25	#15667919	—	2	2	no	—
26	#15667919	—	2	2	no	—
27	#15667919	—	2	2	no	—
28	C₂²×C₈×C₈₉	—	2	2	no	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	D₄	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₈₉	—	2	8	no	—
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	Q₈	1	4	yes	—
37	C₂²×Dic₈₉	—	2	2	no	—
38	C₂²×Dic₈₉	—	2	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	—	4	4	no	—
41	C₄⋊Dic₈₉	—	2	2	no	—
42	C₄⋊Dic₈₉	—	2	4	no	—
43	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
44	#13083553	—	2	8	no	—
45	#13083553	—	4	4	no	—
46	C₄×Dic₈₉	—	4	4	no	—
47	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	8	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₄²×C₈₉	—	2	4	no	—
55	C₄²×C₈₉	—	2	2	no	—
56	C₄²×C₈₉	—	2	4	no	—
57	C₄²×C₈₉	C₂×C₈	1	2	yes	—
58	C₄²×C₈₉	M₁₆	1	2	yes	—
59	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	C₂²⋊C₈	1	4	yes	—
68	C₂×C₄×C₈₉	#104	1	4	yes	—
69	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#131	1	2	yes	—
76	C₂×C₄×C₈₉	#135	1	4	yes	—
77	C₂×Dic₈₉	—	2	4	no	—
78	C₂×Dic₈₉	—	4	2	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	—	4	2	no	—
82	C₂×Dic₈₉	—	4	2	no	—
83	C₈×C₈₉	—	2	8	no	—
84	#10623494	—	8	2	no	—
85	C₂²×C₈₉	#391	1	1	yes	characteristic
86	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₈₉	#408	1	1	yes	characteristic
88	C₂²×C₈₉	#416	1	2	yes	—
89	C₂²×C₈₉	C₂×#104	1	2	yes	—
90	C₄×C₈₉	—	2	4	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	2	4	no	—
93	C₄×C₈₉	C₈×Q₈	1	2	yes	—
94	C₄×C₈₉	#446	1	2	yes	—
95	C₄×C₈₉	#488	1	2	yes	—
96	C₄×C₈₉	#499	1	2	yes	—
97	Dic₈₉	—	4	4	no	—
98	Dic₈₉	—	8	2	no	—
99	#9871	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	#1476	1	2	yes	—
101	C₂×C₈₉	#1494	1	2	yes	—
102	C₂×C₈₉	#1984	1	1	yes	characteristic
103	C₂×C₈₉	#1991	1	1	yes	characteristic
104	C₂×C₈₉	#1994	1	1	yes	characteristic
105	C₄×#125	—	89	1	no	—
106	#1831	—	89	1	no	—
107	C₄²×C₈	D₈₉	1	1	yes	characteristic
108	#1279	—	89	4	no	—
109	C₈₉	#9871	1	1	yes	characteristic, complemented, Sylow
110	C₂×#125	—	89	1	no	—
111	C₂×C₄⋊C₈	—	89	2	no	—
112	C₂×C₄×C₈	—	2	2	no	—
113	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
114	#389	—	89	2	no	—
115	#387	—	178	2	no	—
116	#382	—	89	1	no	—
117	C₄×C₄⋊C₄	—	89	2	no	—
118	C₄×C₄⋊C₄	—	178	2	no	—
119	C₄³	C₂×D₈₉	1	1	yes	characteristic
120	#336	—	178	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	178	2	no	—
123	#125	—	89	4	no	—
124	#125	—	178	2	no	—
125	C₂×C₄⋊C₄	—	89	2	no	—
126	C₂×C₄⋊C₄	—	89	2	no	—
127	C₂×C₄⋊C₄	—	178	2	no	—
128	C₂×C₄⋊C₄	—	178	2	no	—
129	C₂×C₄²	—	178	2	no	—
130	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
132	C₂×C₄²	D₃₅₆	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
134	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
135	C₄⋊C₈	—	178	4	no	—
136	C₄×C₈	—	2	8	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	Dic₁₇₈	1	4	yes	—
139	#95	—	178	2	no	—
140	#95	—	178	2	no	—
141	#95	—	178	2	no	—
142	C₂²×C₄	—	178	2	no	—
143	C₂²×C₄	—	178	2	no	—
144	C₂²×C₄	#13083551	1	2	yes	—
145	C₂²×C₄	#13083551	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	356	2	no	—
152	C₂×C₈	—	356	4	no	—
153	C₄⋊C₄	—	89	4	no	—
154	C₄⋊C₄	—	356	4	no	—
155	C₄⋊C₄	—	178	4	no	—
156	C₄⋊C₄	—	178	2	no	—
157	C₄⋊C₄	—	178	8	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	356	4	no	—
162	C₄²	C₈×D₈₉	1	2	yes	—
163	C₄²	#13083557	1	2	yes	—
164	C₄²	C₂×Dic₁₇₈	1	2	yes	—
165	C₄²	#13083574	1	2	yes	—
166	C₂³	C₂×#13083551	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	356	2	no	—
173	C₂×C₄	—	356	2	no	—
174	C₂×C₄	—	356	2	no	—
175	C₂×C₄	—	356	2	no	—
176	C₂×C₄	—	356	2	no	—
177	C₂×C₄	—	178	4	no	—
178	C₂×C₄	#15667924	1	4	yes	—
179	C₂×C₄	#15667940	1	4	yes	—
180	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
181	C₂×C₄	#15667991	1	2	yes	—
182	C₂×C₄	#15668000	1	1	yes	characteristic
183	C₂×C₄	#15668012	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
185	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
186	C₂×C₄	#15668018	1	2	yes	—
187	C₂×C₄	#15668034	1	4	yes	—
188	C₈	—	2	8	no	—
189	C₈	—	712	2	no	—
190	C₂²	#19151504	1	1	yes	characteristic
191	C₂²	C₂×#15667924	1	1	yes	characteristic
192	C₂²	#19151564	1	1	yes	characteristic
193	C₂²	#19151595	1	2	yes	—
194	C₂²	C₂×#15667940	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	356	4	no	—
199	C₄	—	712	2	no	—
200	C₄	C₈×Dic₁₇₈	1	2	yes	—
201	C₄	#19151728	1	2	yes	—
202	C₄	#19151953	1	2	yes	—
203	C₄	#19152000	1	2	yes	—
204	C₂	#26018635	1	2	yes	—
205	C₂	#26018713	1	2	yes	—
206	C₂	#26021188	1	1	yes	characteristic
207	C₂	#26021217	1	1	yes	characteristic
208	C₂	#26021229	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	#9871
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
18. #15668004	C₈	semidirect
109. C₈₉	#9871	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668004	2
#26020369	2
#26018092	2
C₄²×C₈×C₈₉	2
#9871	89