Finite Group Databasebeta

← #37510044

Subgroups — #37510044

Order 22912.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153141C212yesmaximal, complemented
2#26039020C211yesmaximal, characteristic, complemented
3C2×#19153039C212yesmaximal, complemented
4#26038340C211yesmaximal, characteristic, complemented
5#343×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2212yes—
7C22×D716C2211yescharacteristic
8#19153141—24no—
9#19153141—24no—
10C2×#15668248—22no—
11C2×D1432—22no—
12#19153093—22no—
13#19153088—22no—
14#19153087—22no—
15C2×(C2×D179)⋊C4—22no—
16C2×C4×Dic179C2211yescharacteristic
17#19153039—24no—
18C2×M16×C179C2212yes—
19C2×C4⋊C4×C179C2211yescharacteristic
20C23×D179—21no—
21#15668259—22no—
22#15668259—22no—
23#15668259—22no—
24#15668259—22no—
25C2×Dic358D412yes—
26C2×D716—22no—
27C2×D716—22no—
28C2×D716—22no—
29C2×D716—21no—
30C2×D716D412yes—
31C2×D716D412yescomplemented
32C2×C22⋊D179—22no—
33C22×Dic179—21no—
34C2×C4×D179—22no—
35#15668248—28no—
36D1432—28no—
37C4⋊Dic179—22no—
38(C2×D179)⋊C4—24no—
39(C2×D179)⋊C4—44no—
40C4×Dic179—22no—
41C22×C4×C179D412yes—
42C22×C4×C179C2311yescharacteristic
43M16×C179—22no—
44M16×C179D414yescomplemented
45C2×C8×C179—22no—
46C4⋊C4×C179—22no—
47C22×D179—22no—
48C22×D179—22no—
49C22×D179—42no—
50C22×D179—42no—
51C22×D179—41no—
52C22×D179—41no—
53Dic358—22no—
54Dic358—22no—
55D716—22no—
56D716—24no—
57D716—22no—
58D716—42no—
59D716—82no—
60C22⋊D179—42no—
61C22⋊D179—42no—
62C2×Dic179—22no—
63C2×Dic179—22no—
64C2×Dic179—41no—
65C2×Dic179—41no—
66C4×D179—42no—
67C4×D179—42no—
68C23×C179C2×D411yescharacteristic
69C2×C4×C179—22no—
70C2×C4×C179—42no—
71C2×C4×C179C2×D411yescharacteristic
72C2×C4×C179C2×D412yes—
73C2×C4×C179C2×D412yes—
74C2×C4×C179#4111yescharacteristic
75C8×C179—24no—
76C22×C179—21no—
77C22×C179—21no—
78C22×C179#12011yescharacteristic
79C22×C179#12111yescharacteristic
80C22×C179#12711yescharacteristic
81C2×D179—44no—
82C2×D179—42no—
83C2×D179—42no—
84C2×D179—42no—
85C2×D179—42no—
86C2×D179—82no—
87C2×D179—81no—
88C2×D179—81no—
89C4×C179—42no—
90C4×C179#12012yes—
91C4×C179#12112yes—
92Dic179—42no—
93Dic179—42no—
94C2×C179—21no—
95C2×C179—21no—
96C2×C179#39211yescharacteristic
97C2×C179#45311yescharacteristic
98C2×C179#45411yescharacteristic
99D179—82no—
100D179—82no—
101C179#200611yescharacteristic, complemented, Sylow
102#2006—1791nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1792no—
104#537—1791no—
105C2×#104—1792no—
106#390—1791no—
107#343D17911yescharacteristic, complemented
108C2×#41—1792no—
109C22×D4—1791no—
110AGL(1,ℤ/8)—3584no—
111AGL(1,ℤ/8)—3584no—
112C2×SD16—3582no—
113C2×D8—3582no—
114C2×M16C2×D17912yes—
115C22⋊Q8—3582no—
116#121—3582no—
117#121—3582no—
118C2×C4⋊C4C2×D17911yescharacteristic
119C2×C22⋊C4—3582no—
120C2×C42—1791no—
121#104—3584no—
122C24—3581no—
123#41—3582no—
124#41—3582no—
125#41—3582no—
126#41—3582no—
127C2×Q8—1792no—
128C2×D4—3582no—
129C2×D4—3582no—
130C2×D4—3582no—
131C2×D4—3582no—
132C2×D4—3581no—
133C2×D4—1792no—
134C2×D4—1792no—
135C22×C4—3581no—
136C22×C4—3582no—
137C22×C4D71612yes—
138C22×C4C22×D17911yescharacteristic
139SD16—3588no—
140D8—3588no—
141M16—22no—
142M16D71614yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3582no—
146C22⋊C4—3584no—
147C22⋊C4—7164no—
148C42—3582no—
149C23—3582no—
150C23—3582no—
151C23—7162no—
152C23—7162no—
153C23—7161no—
154C23—7161no—
155C23C2×D71611yescharacteristic
156Q8—3582no—
157Q8—3582no—
158D4—3582no—
159D4—3584no—
160D4—3582no—
161D4—14322no—
162D4—7162no—
163D4—7162no—
164D4—7162no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—3582no—
168C2×C4—3582no—
169C2×C4—7161no—
170C2×C4—7161no—
171C2×C4—7162no—
172C2×C4—7162no—
173C2×C4D4×D17911yescharacteristic
174C2×C4D4×D17912yes—
175C2×C4C2×D71612yes—
176C2×C4#1566826011yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—14322no—
181C22—14321no—
182C22—14321no—
183C22—7164no—
184C22—7162no—
185C22—7162no—
186C22—7162no—
187C22—7162no—
188C22#1915308311yescharacteristic
189C22#1915308811yescharacteristic
190C22#1915311011yescharacteristic
191C4—42no—
192C4—7162no—
193C4—7162no—
194C4#1915308312yes—
195C4#1915308812yes—
196C2—21no—
197C2—21no—
198C2—14322no—
199C2—14322no—
200C2#2603834911yescharacteristic
201C2#2603860911yescharacteristic
202C2#2603861411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#343×C179C2
socleC22×C179#120
hypercenter#343D179
hypocenterC179#2006
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#19153141C2semidirect
2. #26039020C2semidirect
3. C2×#19153039C2semidirect
4. #26038340C2semidirect
5. #343×C179C2semidirect
31. C2×D716D4semidirect
44. M16×C179D4semidirect
101. C179#2006semidirect
107. #343D179semidirect
142. M16D716semidirect

Maximal subgroups

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

subgroupindex
C2×#191531412
#260390202
C2×#191530392
#260383402
#343×C1792
#2006179