Finite Group Databasebeta

← #27172178

Subgroups — #27172178

Order 12928.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#15692286C212yesmaximal, complemented
2#20265758C211yesmaximal, characteristic
3C2×#15692183C212yesmaximal, complemented
4#20264946C211yesmaximal, characteristic, complemented
5#391×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2212yes—
7#15692286—24no—
8#15692286—24no—
9C2×#13108064—22no—
10C2×#13108061—22no—
11C2×#13108058C2212yes—
12#15692235—22no—
13#15692230—22no—
14#15692229—22no—
15#15692183—24no—
16C22×Q8×C101C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C2×C42×C101C2211yescharacteristic
19#95×C101—22no—
20#13108074—22no—
21#13108074—22no—
22#13108074—22no—
23#13108074—22no—
24C2×Dic202D412yes—
25C2×D404—22no—
26C2×D404D412yes—
27C2×C22⋊D101—22no—
28C2×C4×D101—22no—
29#13108064—28no—
30#13108061—28no—
31#13108058—22no—
32#13108058D414yescomplemented
33C2×#11717579—22no—
34C4⋊Dic101—22no—
35#13108051—24no—
36C2×Q8×C101—21no—
37C2×Q8×C101—22no—
38C2×Q8×C101—22no—
39C2×Q8×C101D412yescomplemented
40C22×C4×C101—21no—
41C22×C4×C101—21no—
42C22×C4×C101D412yes—
43C22×C4×C101C2311yescharacteristic
44C4⋊C4×C101—22no—
45C42×C101—22no—
46C23×C101C2×D411yescharacteristic
47C22×D101—22no—
48Q8×C101—24no—
49Q8×C101—42no—
50C2×C4×C101—22no—
51C2×C4×C101—22no—
52C2×C4×C101—22no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101—41no—
57C2×C4×C101—42no—
58C2×C4×C101C2×D412yes—
59C2×C4×C101C2×D412yes—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101#4111yescharacteristic
62C22⋊D101—42no—
63C22⋊D101—42no—
64C2×Dic101—22no—
65D404—22no—
66D404—22no—
67D404—82no—
68C4×D101—42no—
69C4×D101—42no—
70Dic202—22no—
71Dic202—22no—
72#11717579—24no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101#12011yescharacteristic
76C22×C101#12111yescharacteristic
77C22×C101#12711yescharacteristic
78C2×D101—42no—
79C2×D101—42no—
80C2×D101—42no—
81C4×C101—42no—
82C4×C101—42no—
83C4×C101—42no—
84C4×C101#12012yes—
85C4×C101#12112yes—
86Dic101—42no—
87C2×C101—21no—
88C2×C101—21no—
89C2×C101#39211yescharacteristic
90C2×C101#45511yescharacteristic
91C2×C101#45611yescharacteristic
92D101—82no—
93#2007—1011nomaximal, Sylow
94C101#200711yescharacteristic, complemented, Sylow
95C2×#137—1012no—
96#537—1011no—
97C2×#104—1012no—
98#391D10111yescharacteristic, complemented
99#343—1011no—
100C2×#41—1012no—
101C22×Q8C2×D10111yescharacteristic
102#137—2024no—
103#137—2024no—
104C2×Q16—2022no—
105C2×SD16—2022no—
106C2×M16—1012no—
107C22⋊Q8—2022no—
108#121—2022no—
109#121—2022no—
110C2×C4⋊C4C2×D10111yescharacteristic
111C2×C42C2×D10111yescharacteristic
112#104—2024no—
113#95—22no—
114#41—2022no—
115#41—2022no—
116#41—2022no—
117#41—2022no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—1012no—
122C2×Q8D40412yescomplemented
123C2×D4—1012no—
124C2×D4—2022no—
125C2×D4—2022no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—2022no—
129C22×C4C22⋊D10112yes—
130C22×C4C22×D10111yescharacteristic
131Q16—2028no—
132SD16—2028no—
133M16—1014no—
134M16—2022no—
135C2×C8—2022no—
136C4⋊C4—22no—
137C4⋊C4—2022no—
138C22⋊C4—2024no—
139C42—22no—
140C23—2022no—
141C23C2×C22⋊D10111yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—2022no—
145Q8—2022no—
146D4—8082no—
147D4—4042no—
148D4—4042no—
149D4—2022no—
150D4—2022no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—4042no—
160C2×C4—4042no—
161C2×C4—2022no—
162C2×C4D4×D10112yes—
163C2×C4D4×D10112yes—
164C2×C4C2×D40411yescharacteristic
165C2×C4#1310807411yescharacteristic
166C8—2024no—
167C22—21no—
168C22—21no—
169C22—4042no—
170C22—4042no—
171C22—4042no—
172C22#1569222311yescharacteristic
173C22#1569222911yescharacteristic
174C22#1569225111yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—4042no—
179C4#1569222512yes—
180C4#1569222812yes—
181C2—21no—
182C2—21no—
183C2—8082no—
184C2#2026508611yescharacteristic
185C2#2026535411yescharacteristic
186C2#2026535911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#391×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#391D101
hypocenterC101#2007
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
3. C2×#15692183C2semidirect
4. #20264946C2semidirect
5. #391×C101C2semidirect
32. #13108058D4semidirect
39. C2×Q8×C101D4semidirect
94. C101#2007semidirect
98. #391D101semidirect
122. C2×Q8D404semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
#202657582
C2×#156921832
#202649462
#391×C1012
#2007101