Finite Group Databasebeta

← #37510047

Subgroups — #37510047

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×#19153143C212yesmaximal, complemented
2C2×#19153040C212yesmaximal, complemented
3#26038344C211yesmaximal, characteristic, complemented
4#26038208C211yesmaximal, characteristic, complemented
5#537×C179C211yesmaximal, characteristic
6C22×Dic358C2211yescharacteristic
7#19153143—24no—
8#19153143—24no—
9C2×#15668249—22no—
10C2×#15668247—22no—
11C2×#15668243C2212yes—
12C2×C4×Dic179C2211yescharacteristic
13#19153040—24no—
14C179⋊#95—22no—
15C2×#41×C179C2212yes—
16C22⋊Q8×C179—22no—
17#121×C179—22no—
18#121×C179—22no—
19C2×C4⋊C4×C179C2211yescharacteristic
20C2×Dic358—22no—
21C2×Dic358—22no—
22C2×Dic358—21no—
23C2×Dic358D412yescomplemented
24C22×Dic179—21no—
25C22×Dic179—21no—
26#15668249—28no—
27#15668247—28no—
28#15668243—22no—
29#15668243D414yescomplemented
30C2×C179⋊C8—22no—
31C4×Dic179—22no—
32#41×C179—22no—
33#41×C179—22no—
34#41×C179—22no—
35#41×C179—22no—
36C2×Q8×C179D412yes—
37C2×D4×C179—22no—
38C2×D4×C179—22no—
39C2×D4×C179D412yes—
40C22×C4×C179—22no—
41C22×C4×C179D412yes—
42C22×C4×C179C2311yescharacteristic
43C4⋊C4×C179—22no—
44C4⋊C4×C179—22no—
45C22⋊C4×C179—24no—
46Dic358—24no—
47Dic358—42no—
48C2×Dic179—22no—
49C2×Dic179—22no—
50C2×Dic179—41no—
51C2×Dic179—41no—
52C2×Dic179—41no—
53C2×Dic179—41no—
54C179⋊C8—24no—
55C23×C179—22no—
56C23×C179C2×D411yescharacteristic
57Q8×C179—22no—
58Q8×C179—22no—
59D4×C179—22no—
60D4×C179—22no—
61D4×C179—42no—
62D4×C179—42no—
63D4×C179—82no—
64C2×C4×C179—22no—
65C2×C4×C179—22no—
66C2×C4×C179—42no—
67C2×C4×C179—42no—
68C2×C4×C179—42no—
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×D412yes—
71C2×C4×C179C2×D412yes—
72C2×C4×C179#4111yescharacteristic
73C22×C179—21no—
74C22×C179—21no—
75C22×C179—42no—
76C22×C179—42no—
77C22×C179—42no—
78C22×C179#12011yescharacteristic
79C22×C179#12111yescharacteristic
80C22×C179#12711yescharacteristic
81C4×C179—42no—
82C4×C179—42no—
83C4×C179#12012yes—
84C4×C179#12112yes—
85Dic179—42no—
86Dic179—42no—
87C2×C179—21no—
88C2×C179—21no—
89C2×C179—82no—
90C2×C179#39211yescharacteristic
91C2×C179#45511yescharacteristic
92C2×C179#45611yescharacteristic
93C179#200711yescharacteristic, complemented, Sylow
94#2007—1791nomaximal, Sylow
95C2×#137—1792no—
96#537D17911yescharacteristic
97C2×#104—1792no—
98#391—1791no—
99#343—1791no—
100C2×#41C2×D17912yes—
101C22×Q8—1791no—
102#137—3584no—
103#137—3584no—
104C2×Q16—3582no—
105C2×SD16—3582no—
106C2×M16—1792no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D17911yescharacteristic
111C2×C42—1791no—
112#104—3584no—
113#95—3582no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—3581no—
119C2×Q8—3582no—
120C2×Q8—3582no—
121C2×Q8—1792no—
122C2×Q8C22⋊D17912yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D17912yes—
126C22×C4—22no—
127C22×C4—3581no—
128C22×C4—3581no—
129C22×C4C22⋊D17912yes—
130C22×C4C22×D17911yescharacteristic
131Q16—3588no—
132SD16—3588no—
133M16—3582no—
134M16—1794no—
135C2×C8—3582no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—3582no—
140C23—22no—
141C23C2×C22⋊D17911yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—3584no—
145Q8—7162no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—3582no—
157C2×C4—3582no—
158C2×C4—7161no—
159C2×C4—7161no—
160C2×C4—7161no—
161C2×C4—7161no—
162C2×C4D4×D17911yescharacteristic
163C2×C4D4×D17912yes—
164C2×C4C2×C22⋊D17912yes—
165C2×C4#1566825811yescharacteristic
166C8—3584no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1915308211yescharacteristic
173C22#1915308411yescharacteristic
174C22#1915310911yescharacteristic
175C4—42no—
176C4—42no—
177C4—7162no—
178C4—7162no—
179C4#1915308112yes—
180C4#1915308512yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2603834711yescharacteristic
185C2#2603861711yescharacteristic
186C2#2603862211yescharacteristic
subgroupG/H
centerC22#19153082
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting, prime radical#537×C179C2
socleC22×C179#120
hypercenter#537D179
hypocenterC179#2007
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#19153143C2semidirect
2. C2×#19153040C2semidirect
3. #26038344C2semidirect
4. #26038208C2semidirect
23. C2×Dic358D4semidirect
29. #15668243D4semidirect
93. C179#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#191531432
C2×#191530402
#260383442
#260382082
#537×C1792
#2007179