Finite Group Databasebeta

← #27172179

Subgroups — #27172179

Order 12928.

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×#15692285C212yesmaximal, complemented
2C2×#15692182C212yesmaximal, complemented
3#20265082C211yesmaximal, characteristic, complemented
4#20264946C211yesmaximal, characteristic, complemented
5#537×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic
7#15692285—24no—
8#15692285—24no—
9C2×#13108064—22no—
10C2×#13108062—22no—
11C2×#13108058C2212yes—
12C2×C4×Dic101C2211yescharacteristic
13#15692182—24no—
14#15692162—22no—
15C2×#41×C101C2212yes—
16C22⋊Q8×C101—22no—
17#121×C101—22no—
18#121×C101—22no—
19C2×C4⋊C4×C101C2211yescharacteristic
20C2×Dic202—22no—
21C2×Dic202—22no—
22C2×Dic202—21no—
23C2×Dic202D412yescomplemented
24C22×Dic101—21no—
25C22×Dic101—21no—
26#13108064—28no—
27#13108062—28no—
28#13108058—22no—
29#13108058D414yescomplemented
30C2×#11717579—22no—
31C4×Dic101—22no—
32#41×C101—22no—
33#41×C101—22no—
34#41×C101—22no—
35#41×C101—22no—
36C2×Q8×C101D412yes—
37C2×D4×C101—22no—
38C2×D4×C101—22no—
39C2×D4×C101D412yes—
40C22×C4×C101—22no—
41C22×C4×C101D412yes—
42C22×C4×C101C2311yescharacteristic
43C4⋊C4×C101—22no—
44C4⋊C4×C101—22no—
45C22⋊C4×C101—24no—
46C23×C101—22no—
47C23×C101C2×D411yescharacteristic
48Q8×C101—22no—
49Q8×C101—22no—
50D4×C101—22no—
51D4×C101—22no—
52D4×C101—42no—
53D4×C101—42no—
54D4×C101—82no—
55C2×C4×C101—22no—
56C2×C4×C101—22no—
57C2×C4×C101—42no—
58C2×C4×C101—42no—
59C2×C4×C101—42no—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D412yes—
62C2×C4×C101C2×D412yes—
63C2×C4×C101#4111yescharacteristic
64C2×Dic101—22no—
65C2×Dic101—22no—
66C2×Dic101—41no—
67C2×Dic101—41no—
68C2×Dic101—41no—
69C2×Dic101—41no—
70Dic202—24no—
71Dic202—42no—
72#11717579—24no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—42no—
76C22×C101—42no—
77C22×C101—42no—
78C22×C101#12011yescharacteristic
79C22×C101#12111yescharacteristic
80C22×C101#12711yescharacteristic
81C4×C101—42no—
82C4×C101—42no—
83C4×C101#12012yes—
84C4×C101#12112yes—
85Dic101—42no—
86Dic101—42no—
87C2×C101—21no—
88C2×C101—21no—
89C2×C101—82no—
90C2×C101#39211yescharacteristic
91C2×C101#45511yescharacteristic
92C2×C101#45611yescharacteristic
93#2007—1011nomaximal, Sylow
94C101#200711yescharacteristic, complemented, Sylow
95C2×#137—1012no—
96#537D10111yescharacteristic
97C2×#104—1012no—
98#391—1011no—
99#343—1011no—
100C2×#41C2×D10112yes—
101C22×Q8—1011no—
102#137—2024no—
103#137—2024no—
104C2×Q16—2022no—
105C2×SD16—2022no—
106C2×M16—1012no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D10111yescharacteristic
111C2×C42—1011no—
112#104—2024no—
113#95—2022no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—1012no—
119C2×Q8—2022no—
120C2×Q8—2022no—
121C2×Q8—2021no—
122C2×Q8C22⋊D10112yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D10112yes—
126C22×C4—22no—
127C22×C4—2021no—
128C22×C4—2021no—
129C22×C4C22⋊D10112yes—
130C22×C4C22×D10111yescharacteristic
131Q16—2028no—
132SD16—2028no—
133M16—1014no—
134M16—2022no—
135C2×C8—2022no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—2022no—
140C23—22no—
141C23C2×C22⋊D10111yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—4042no—
145Q8—2024no—
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—4041no—
157C2×C4—4041no—
158C2×C4—4041no—
159C2×C4—4041no—
160C2×C4—2022no—
161C2×C4—2022no—
162C2×C4D4×D10111yescharacteristic
163C2×C4D4×D10112yes—
164C2×C4C2×C22⋊D10112yes—
165C2×C4#1310807311yescharacteristic
166C8—2024no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1569222411yescharacteristic
173C22#1569222611yescharacteristic
174C22#1569225111yescharacteristic
175C4—42no—
176C4—42no—
177C4—4042no—
178C4—4042no—
179C4#1569222312yes—
180C4#1569222712yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2026508511yescharacteristic
185C2#2026535511yescharacteristic
186C2#2026536011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#537×C101C2
socleC22×C101#120
hypercenter#537D101
hypocenterC101#2007
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#15692285C2semidirect
2. C2×#15692182C2semidirect
3. #20265082C2semidirect
4. #20264946C2semidirect
23. C2×Dic202D4semidirect
29. #13108058D4semidirect
94. C101#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#156922852
C2×#156921822
#202650822
#202649462
#537×C1012
#2007101