Finite Group Databasebeta

← #37510049

Subgroups — #37510049

Order 22912.

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

normal subgroups only

This group has 370 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×#19153146C212yesmaximal, complemented
2#26039020C211yesmaximal, characteristic
3C2×#19153039C212yesmaximal, complemented
4#26038344C211yesmaximal, characteristic, complemented
5#343×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2212yes—
7C22×Dic358C2211yescharacteristic
8#19153146—24no—
9#19153146—24no—
10C2×Dic716—22no—
11C2×#15668248—22no—
12#19153093—22no—
13#19153088—22no—
14#19153087—22no—
15C2×C4×Dic179C2211yescharacteristic
16#19153039—24no—
17C179⋊#95—22no—
18C2×M16×C179C2212yes—
19C2×C4⋊C4×C179C2211yescharacteristic
20#15668259—22no—
21#15668259—22no—
22#15668259—22no—
23#15668259—22no—
24C2×Dic358—22no—
25C2×Dic358—22no—
26C2×Dic358—21no—
27C2×Dic358D412yes—
28C2×Dic358D412yescomplemented
29C2×D716—22no—
30C2×D716D412yes—
31C2×C22⋊D179—22no—
32C22×Dic179—21no—
33C22×Dic179—21no—
34C2×C4×D179—22no—
35Dic716—28no—
36#15668248—28no—
37C4⋊Dic179—22no—
38(C2×D179)⋊C4—24no—
39C4×Dic179—22no—
40C22×C4×C179D412yes—
41C22×C4×C179C2311yescharacteristic
42M16×C179—22no—
43M16×C179D414yescomplemented
44C2×C8×C179—22no—
45C4⋊C4×C179—22no—
46C22×D179—22no—
47Dic358—24no—
48Dic358—22no—
49Dic358—22no—
50Dic358—42no—
51D716—22no—
52D716—22no—
53D716—82no—
54C22⋊D179—42no—
55C22⋊D179—42no—
56C2×Dic179—22no—
57C2×Dic179—22no—
58C2×Dic179—22no—
59C2×Dic179—41no—
60C2×Dic179—41no—
61C2×Dic179—41no—
62C2×Dic179—41no—
63C4×D179—42no—
64C4×D179—42no—
65C23×C179C2×D411yescharacteristic
66C2×C4×C179—22no—
67C2×C4×C179—42no—
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×D412yes—
70C2×C4×C179C2×D412yes—
71C2×C4×C179#4111yescharacteristic
72C8×C179—24no—
73C22×C179—21no—
74C22×C179—21no—
75C22×C179#12011yescharacteristic
76C22×C179#12111yescharacteristic
77C22×C179#12711yescharacteristic
78C2×D179—42no—
79C2×D179—42no—
80C2×D179—42no—
81C4×C179—42no—
82C4×C179#12012yes—
83C4×C179#12112yes—
84Dic179—42no—
85Dic179—42no—
86Dic179—42no—
87C2×C179—21no—
88C2×C179—21no—
89C2×C179#39211yescharacteristic
90C2×C179#45511yescharacteristic
91C2×C179#45611yescharacteristic
92D179—82no—
93C179#200711yescharacteristic, complemented, Sylow
94#2007—1791nomaximal, Sylow
95C2×#137—1792no—
96#537—1791no—
97C2×#104—1792no—
98#391—1791no—
99#343D17911yescharacteristic, complemented
100C2×#41—1792no—
101C22×Q8—1791no—
102#137—3584no—
103#137—3584no—
104C2×Q16—3582no—
105C2×SD16—3582no—
106C2×M16C2×D17912yes—
107C22⋊Q8—3582no—
108#121—3582no—
109#121—3582no—
110C2×C4⋊C4C2×D17911yescharacteristic
111C2×C42—1791no—
112#104—3584no—
113#95—3582no—
114#41—3582no—
115#41—3582no—
116#41—3582no—
117#41—3582no—
118C2×Q8—3581no—
119C2×Q8—3582no—
120C2×Q8—3582no—
121C2×Q8—1792no—
122C2×Q8—1792no—
123C2×D4—3582no—
124C2×D4—3582no—
125C2×D4—1792no—
126C22×C4—3581no—
127C22×C4—3581no—
128C22×C4—3582no—
129C22×C4D71612yes—
130C22×C4C22×D17911yescharacteristic
131Q16—3588no—
132SD16—3588no—
133M16—22no—
134M16D71614yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—3582no—
138C22⋊C4—3584no—
139C42—3582no—
140C23—3582no—
141C23C2×D71611yescharacteristic
142Q8—3584no—
143Q8—3582no—
144Q8—3582no—
145Q8—7162no—
146D4—3582no—
147D4—3582no—
148D4—14322no—
149D4—7162no—
150D4—7162no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—3582no—
154C2×C4—3582no—
155C2×C4—3582no—
156C2×C4—7161no—
157C2×C4—7161no—
158C2×C4—7161no—
159C2×C4—7161no—
160C2×C4—7162no—
161C2×C4—7162no—
162C2×C4D4×D17911yescharacteristic
163C2×C4D4×D17912yes—
164C2×C4C2×D71612yes—
165C2×C4#1566826011yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—7162no—
170C22—7162no—
171C22—7162no—
172C22#1915308311yescharacteristic
173C22#1915308811yescharacteristic
174C22#1915311011yescharacteristic
175C4—42no—
176C4—7162no—
177C4—7162no—
178C4—7162no—
179C4#1915308312yes—
180C4#1915308812yes—
181C2—21no—
182C2—21no—
183C2—14322no—
184C2#2603834911yescharacteristic
185C2#2603861911yescharacteristic
186C2#2603862411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#343×C179C2
socleC22×C179#120
prime radical#26039020C2
hypercenter#343D179
hypocenterC179#2007
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×#19153146C2semidirect
3. C2×#19153039C2semidirect
4. #26038344C2semidirect
5. #343×C179C2semidirect
28. C2×Dic358D4semidirect
43. M16×C179D4semidirect
93. C179#2007semidirect
99. #343D179semidirect
134. M16D716semidirect

Maximal subgroups

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

subgroupindex
C2×#191531462
#260390202
C2×#191530392
#260383442
#343×C1792
#2007179