Subgroups — #36441040

Order 22784.

332 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020404	C₂	1	2	yes	maximal
2	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10623494	—	2	2	no	—
4	C₂×C₄⋊#10623494	—	2	2	no	—
5	C₂×C₄⋊#10623494	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
7	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
8	C₄³×C₈₉	C₂²	1	1	yes	characteristic
9	C₂²×#10623494	—	2	4	no	—
10	C₄⋊#10623494	—	4	4	no	—
11	C₄⋊#10623494	—	4	2	no	—
12	C₄⋊#10623494	—	4	2	no	—
13	C₄⋊#10623494	—	4	2	no	—
14	C₄⋊#10623494	—	4	2	no	—
15	C₂²×C₈×C₈₉	—	2	2	no	—
16	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₈₉	D₄	1	2	yes	—
20	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	—	4	2	no	—
22	C₄⋊C₈×C₈₉	—	4	2	no	—
23	C₄⋊C₈×C₈₉	—	4	2	no	—
24	C₂×#10623494	—	4	4	no	—
25	C₂×#10623494	—	4	4	no	—
26	C₂×#10623494	—	4	4	no	—
27	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	—	4	2	no	—
33	C₂×C₈×C₈₉	—	4	4	no	—
34	C₄²×C₈₉	—	2	1	no	—
35	C₄²×C₈₉	—	2	2	no	—
36	C₄²×C₈₉	—	2	2	no	—
37	C₄²×C₈₉	—	2	1	no	—
38	C₄²×C₈₉	—	4	1	no	—
39	C₄²×C₈₉	D₈	1	2	yes	—
40	C₄²×C₈₉	D₈	1	1	yes	characteristic
41	C₄²×C₈₉	SD₁₆	1	2	yes	—
42	C₄²×C₈₉	SD₁₆	1	2	yes	—
43	C₄²×C₈₉	SD₁₆	1	2	yes	—
44	C₄²×C₈₉	Q₁₆	1	2	yes	—
45	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
46	C₂³×C₈₉	#95	1	1	yes	characteristic
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	#100	1	2	yes	—
54	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#101	1	2	yes	—
56	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#102	1	2	yes	—
58	C₂×C₄×C₈₉	#102	1	2	yes	—
59	C₂×C₄×C₈₉	#102	1	2	yes	—
60	C₂×C₄×C₈₉	#103	1	2	yes	—
61	C₂×C₄×C₈₉	#103	1	2	yes	—
62	C₂×C₄×C₈₉	#103	1	2	yes	—
63	C₂×C₄×C₈₉	#106	1	2	yes	—
64	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#107	1	2	yes	—
66	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
67	C₈×C₈₉	—	8	2	no	—
68	#10623494	—	8	4	no	—
69	C₂²×C₈₉	#338	1	1	yes	characteristic
70	C₂²×C₈₉	#340	1	2	yes	—
71	C₂²×C₈₉	#340	1	1	yes	characteristic
72	C₂²×C₈₉	#343	1	2	yes	—
73	C₂²×C₈₉	#343	1	1	yes	characteristic
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	—	2	1	no	—
78	C₄×C₈₉	—	4	1	no	—
79	C₄×C₈₉	#331	1	2	yes	—
80	C₄×C₈₉	#331	1	1	yes	characteristic
81	C₄×C₈₉	#332	1	2	yes	—
82	C₄×C₈₉	#332	1	2	yes	—
83	C₄×C₈₉	#332	1	2	yes	—
84	C₄×C₈₉	#333	1	2	yes	—
85	C₄×C₈₉	#333	1	1	yes	characteristic
86	#7050	—	89	1	no	maximal, Sylow
87	C₂×C₈₉	#1292	1	2	yes	—
88	C₂×C₈₉	#1292	1	1	yes	characteristic
89	C₂×C₈₉	#1296	1	1	yes	characteristic
90	C₂×C₈₉	#1296	1	2	yes	—
91	C₂×C₈₉	#1309	1	1	yes	characteristic
92	#1842	—	89	2	no	—
93	#1842	D₈₉	1	1	yes	characteristic
94	C₈₉	#7050	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄⋊C₈	—	2	2	no	—
96	C₂×C₄⋊C₈	—	89	4	no	—
97	C₂×C₄⋊C₈	—	178	2	no	—
98	C₂×C₄⋊C₈	—	178	2	no	—
99	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
100	C₄³	C₂×D₈₉	1	1	yes	characteristic
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	178	4	no	—
103	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
104	C₂×C₄²	C₄×D₈₉	1	2	yes	—
105	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
106	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
107	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	—	4	2	no	—
110	C₄⋊C₈	—	4	2	no	—
111	C₄⋊C₈	—	356	4	no	—
112	C₄⋊C₈	—	356	2	no	—
113	C₄⋊C₈	—	356	2	no	—
114	C₄⋊C₈	—	356	2	no	—
115	C₄⋊C₈	—	356	2	no	—
116	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
117	C₂²×C₄	#13083551	1	2	yes	—
118	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
119	C₂²×C₄	#13083553	1	2	yes	—
120	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	4	no	—
123	C₂×C₈	—	356	4	no	—
124	C₂×C₈	—	356	4	no	—
125	C₂×C₈	—	356	4	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	#13083559	1	2	yes	—
132	C₄²	D₇₁₂	1	1	yes	characteristic
133	C₄²	#13083561	1	2	yes	—
134	C₄²	#13083562	1	2	yes	—
135	C₄²	#13083563	1	2	yes	—
136	C₄²	#13083564	1	2	yes	—
137	C₄²	Dic₃₅₆	1	1	yes	characteristic
138	C₂³	#15667919	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	#15667928	1	2	yes	—
146	C₂×C₄	#15667929	1	1	yes	characteristic
147	C₂×C₄	#15667930	1	2	yes	—
148	C₂×C₄	#15667931	1	1	yes	characteristic
149	C₂×C₄	#15667932	1	2	yes	—
150	C₂×C₄	#15667933	1	2	yes	—
151	C₂×C₄	#15667934	1	2	yes	—
152	C₂×C₄	#15667935	1	2	yes	—
153	C₂×C₄	#15667936	1	2	yes	—
154	C₂×C₄	#15667937	1	2	yes	—
155	C₂×C₄	#15667943	1	2	yes	—
156	C₂×C₄	#15667944	1	1	yes	characteristic
157	C₂×C₄	#15667945	1	2	yes	—
158	C₂×C₄	#15667946	1	1	yes	characteristic
159	C₈	—	8	2	no	—
160	C₈	—	712	4	no	—
161	C₂²	#19151356	1	1	yes	characteristic
162	C₂²	#19151360	1	2	yes	—
163	C₂²	#19151361	1	1	yes	characteristic
164	C₂²	#19151366	1	2	yes	—
165	C₂²	#19151367	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	1	no	—
171	C₄	#19151341	1	2	yes	—
172	C₄	#19151342	1	1	yes	characteristic
173	C₄	#19151343	1	2	yes	—
174	C₄	#19151344	1	2	yes	—
175	C₄	#19151345	1	2	yes	—
176	C₄	#19151346	1	2	yes	—
177	C₄	#19151347	1	1	yes	characteristic
178	C₂	#26018126	1	2	yes	—
179	C₂	#26018127	1	1	yes	characteristic
180	C₂	#26018135	1	1	yes	characteristic
181	C₂	#26018136	1	2	yes	—
182	C₂	#26018165	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₄³	C₂×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	#95
cosocle	C₄³×C₈₉	C₂²
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#7050
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
94. C₈₉	#7050	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
C₈₉×#1842	2
#7050	89