Subgroups — #36453470

Order 22784.

570 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	#26028650	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151343	C₂	1	4	yes	maximal
4	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	C₂×#15668012	—	2	1	no	—
7	C₂×#15668012	C₂²	1	2	yes	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₄×#10623494	—	2	1	no	—
10	#19151468	—	2	2	no	—
11	#19151468	C₄	1	4	yes	—
12	#19151353	—	2	1	no	—
13	#19151343	—	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	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	#15668012	—	2	4	no	—
20	#15668012	—	2	4	no	—
21	#15668012	—	4	4	no	—
22	#15668012	—	4	2	no	—
23	#15668012	—	4	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	4	no	—
25	C₂×C₄⋊Dic₈₉	—	2	4	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
28	C₄⋊#10623494	—	2	2	no	—
29	C₄⋊#10623494	D₄	1	4	yes	—
30	C₄×#10623494	—	4	2	no	—
31	C₂²×C₈×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	—	2	4	no	—
38	C₄⋊C₈×C₈₉	—	4	4	no	—
39	C₂×Dic₁₇₈	—	4	4	no	—
40	C₂×Dic₁₇₈	—	4	2	no	—
41	C₂×Dic₁₇₈	—	4	2	no	—
42	C₂×Dic₁₇₈	—	4	4	no	—
43	C₂²×Dic₈₉	—	2	4	no	—
44	C₂×#10623494	—	2	4	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₄⋊Dic₈₉	—	2	8	no	—
48	C₄⋊Dic₈₉	—	4	8	no	—
49	C₄⋊Dic₈₉	—	4	8	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₄⋊Dic₈₉	—	4	2	no	—
52	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	4	2	no	—
58	C₂×C₈×C₈₉	—	4	4	no	—
59	C₄²×C₈₉	—	2	4	no	—
60	C₄²×C₈₉	—	2	2	no	—
61	C₄²×C₈₉	D₈	1	2	yes	—
62	C₄²×C₈₉	SD₁₆	1	2	yes	—
63	C₄²×C₈₉	SD₁₆	1	4	yes	—
64	C₄²×C₈₉	Q₁₆	1	4	yes	—
65	C₄²×C₈₉	C₂×D₄	1	2	yes	—
66	C₄²×C₈₉	#41	1	2	yes	—
67	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	#101	1	4	yes	—
72	C₂×C₄×C₈₉	#102	1	4	yes	—
73	C₂×C₄×C₈₉	#103	1	4	yes	—
74	C₂×C₄×C₈₉	#103	1	4	yes	—
75	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₈₉	#121	1	2	yes	—
77	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	—	4	4	no	—
86	C₂×Dic₈₉	—	4	4	no	—
87	Dic₁₇₈	—	8	4	no	—
88	Dic₁₇₈	—	8	4	no	—
89	C₈×C₈₉	—	8	2	no	—
90	#10623494	—	4	4	no	—
91	C₂²×C₈₉	#390	1	1	yes	characteristic
92	C₂²×C₈₉	C₂×#101	1	2	yes	—
93	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
94	C₂²×C₈₉	C₂×#103	1	2	yes	—
95	C₂²×C₈₉	#418	1	1	yes	characteristic
96	C₄×C₈₉	—	2	4	no	—
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	#332	1	8	yes	complemented
99	C₄×C₈₉	#459	1	2	yes	—
100	C₄×C₈₉	#460	1	2	yes	—
101	C₄×C₈₉	#487	1	2	yes	—
102	C₄×C₈₉	#492	1	1	yes	characteristic
103	C₄×C₈₉	#494	1	1	yes	characteristic
104	Dic₈₉	—	8	4	no	—
105	#9694	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	C₂×#332	1	2	yes	—
107	C₂×C₈₉	#1549	1	2	yes	—
108	C₂×C₈₉	#1956	1	1	yes	characteristic
109	C₂×C₈₉	#1966	1	1	yes	characteristic
110	C₂×C₈₉	#1982	1	1	yes	characteristic
111	#2868	—	89	1	no	—
112	#1842	D₈₉	1	1	yes	characteristic
113	C₄×C₄⋊C₈	—	89	1	no	—
114	C₂×#332	—	89	4	no	—
115	C₈₉	#9694	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄⋊Q₈	—	89	2	no	—
117	C₂×C₄⋊Q₈	—	178	1	no	—
118	C₂×C₄⋊Q₈	—	178	1	no	—
119	C₂×C₄⋊C₈	—	2	2	no	—
120	C₂×C₄⋊C₈	—	89	2	no	—
121	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
122	C₂×C₄×C₈	—	178	1	no	—
123	#384	—	89	4	no	—
124	#384	—	178	2	no	—
125	C₄³	C₂×D₈₉	1	1	yes	characteristic
126	#336	—	178	1	no	—
127	#332	—	178	8	no	—
128	C₂²×Q₈	—	178	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	178	2	no	—
131	C₄⋊Q₈	—	356	4	no	—
132	C₄⋊Q₈	—	356	2	no	—
133	C₄⋊Q₈	—	356	2	no	—
134	C₄⋊Q₈	—	178	4	no	—
135	C₄⋊Q₈	—	178	4	no	—
136	C₂×C₄⋊C₄	—	89	4	no	—
137	C₂×C₄⋊C₄	—	178	4	no	—
138	C₂×C₄⋊C₄	—	178	4	no	—
139	C₂×C₄⋊C₄	—	178	2	no	—
140	C₂×C₄²	C₄×D₈₉	1	2	yes	—
141	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
142	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
144	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	4	4	no	—
147	C₄⋊C₈	—	89	4	no	—
148	C₄⋊C₈	—	178	2	no	—
149	C₄×C₈	—	356	2	no	—
150	C₂×Q₈	—	356	4	no	—
151	C₂×Q₈	—	356	2	no	—
152	C₂×Q₈	—	356	2	no	—
153	C₂×Q₈	—	356	4	no	—
154	C₂²×C₄	—	178	4	no	—
155	C₂²×C₄	#13083551	1	2	yes	—
156	C₂²×C₄	#13083551	1	2	yes	—
157	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	—	356	2	no	—
163	C₂×C₈	—	356	2	no	—
164	C₂×C₈	—	178	4	no	—
165	C₄⋊C₄	—	356	8	no	—
166	C₄⋊C₄	—	356	2	no	—
167	C₄⋊C₄	—	356	2	no	—
168	C₄⋊C₄	—	356	8	no	—
169	C₄⋊C₄	—	178	8	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	4	no	—
172	C₄²	D₇₁₂	1	2	yes	—
173	C₄²	#13083562	1	4	yes	—
174	C₄²	#13083563	1	2	yes	—
175	C₄²	#13083564	1	4	yes	—
176	C₄²	D₄×D₈₉	1	2	yes	—
177	C₄²	#13083575	1	2	yes	—
178	C₂³	C₂×#13083551	1	1	yes	characteristic
179	Q₈	—	712	4	no	—
180	Q₈	—	712	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	356	4	no	—
185	C₂×C₄	—	356	4	no	—
186	C₂×C₄	—	356	4	no	—
187	C₂×C₄	#15667930	1	4	yes	—
188	C₂×C₄	#15667934	1	4	yes	—
189	C₂×C₄	#15667935	1	4	yes	—
190	C₂×C₄	#15667935	1	4	yes	—
191	C₂×C₄	#15667974	1	2	yes	—
192	C₂×C₄	#15667987	1	2	yes	—
193	C₂×C₄	#15668001	1	1	yes	characteristic
194	C₂×C₄	#15668008	1	1	yes	characteristic
195	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
196	C₂×C₄	C₂×#13083562	1	2	yes	—
197	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
198	C₂×C₄	C₂×#13083564	1	2	yes	—
199	C₂×C₄	#15668040	1	2	yes	—
200	C₈	—	8	2	no	—
201	C₈	—	356	4	no	—
202	C₂²	#19151499	1	1	yes	characteristic
203	C₂²	C₂×#15667930	1	2	yes	—
204	C₂²	C₂×#15667934	1	1	yes	characteristic
205	C₂²	C₂×#15667935	1	2	yes	—
206	C₂²	#19151604	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	712	4	no	—
210	C₄	#19151343	1	8	yes	complemented
211	C₄	#19151794	1	2	yes	—
212	C₄	#19151799	1	2	yes	—
213	C₄	#19151950	1	2	yes	—
214	C₄	#19151970	1	1	yes	characteristic
215	C₄	#19151977	1	1	yes	characteristic
216	C₂	C₂×#19151343	1	2	yes	—
217	C₂	#26018938	1	2	yes	—
218	C₂	#26021053	1	1	yes	characteristic
219	C₂	#26021095	1	1	yes	characteristic
220	C₂	#26021178	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#9694
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
98. C₄×C₈₉	#332	semidirect
115. C₈₉	#9694	semidirect
210. C₄	#19151343	semidirect

Maximal subgroups

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

subgroup	index
#26028650	2
C₄×C₄⋊#10623494	2
C₂×#19151343	2
C₈₉×#1842	2
#9694	89