Finite Group Databasebeta

← #27142117

Subgroups — #27142117

Order 12416.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#20261502C211yesmaximal, characteristic, complemented
2#20261281C211yesmaximal, characteristic
3#20260753C212yesmaximal, complemented
4#20260494C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7#15690817—24no—
8C2×#13095495—22no—
9C2×#13095495C2212yes—
10#15690808C411yescharacteristic, complemented
11#15690806C411yescharacteristic, complemented
12#15690799C411yescharacteristic, complemented
13#15690797C411yescharacteristic, complemented
14#15690782—22no—
15C4×C22⋊D97—22no—
16C2×C4×Dic97C2211yescharacteristic
17#15690738—24no—
18C2×#41×C97C2211yescharacteristic
19#117×C97C2212yes—
20#99×C97—22no—
21#13095511—22no—
22#13095511—22no—
23C2×Dic194C2×C411yescharacteristic
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97—22no—
26C2×C22⋊D97—21no—
27C22×Dic97—21no—
28C2×C4×D97—21no—
29#13095495—24no—
30#13095495—22no—
31#13095495D414yescomplemented
32C2×#11715895—24no—
33C2×#11715895—22no—
34#13095490—22no—
35C22⋊Dic97—22no—
36#13095488—22no—
37C4×Dic97—21no—
38C4×Dic97C2×C411yescharacteristic
39C4×Dic97C2×C411yescharacteristic
40#41×C97—22no—
41#41×C97—22no—
42C2×Q8×C97C2×C411yescharacteristic
43C2×D4×C97C2×C411yescharacteristic
44C2×D4×C97D412yes—
45C22×C4×C97D412yes—
46C22×C4×C97C2311yescharacteristic
47C4⋊C4×C97—22no—
48C22⋊C4×C97—22no—
49C42×C97—22no—
50C23×C97C22⋊C412yes—
51C23×C97C2×D411yescharacteristic
52C22×D97—21no—
53Q8×C97—22no—
54D4×C97—24no—
55D4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—42no—
59C2×C4×C97C22⋊C412yes—
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D412yes—
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97#4112yes—
64C22⋊D97—44no—
65C22⋊D97—42no—
66C2×Dic97—21no—
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97—42no—
70D388—22no—
71C4×D97—41no—
72C4×D97—41no—
73Dic194—22no—
74#11715895—24no—
75C22×C97—22no—
76C22×C97—41no—
77C22×C97—42no—
78C22×C97C2×C22⋊C411yescharacteristic
79C22×C97#12112yes—
80C2×D97—42no—
81C2×D97—41no—
82C4×C97—22no—
83C4×C97—82no—
84C4×C97C4×D412yes—
85C4×C97#12011yescharacteristic
86C4×C97#12311yescharacteristic
87Dic97—22no—
88Dic97—41no—
89Dic97—41no—
90C2×C97—21no—
91C2×C97—22no—
92C2×C97—42no—
93C2×C97#38611yescharacteristic
94D97—81no—
95#1884—971nomaximal, Sylow
96C97#188411yescharacteristic, complemented, Sylow
97#568—971no—
98#532—971no—
99#421—972no—
100#410D9711yescharacteristic, complemented
101#339—972no—
102C2×#41—971no—
103C2×#41C2×D9711yescharacteristic
104#131—1944no—
105C2×M16—972no—
106C2×M16—1942no—
107C4⋊Q8—971no—
108#127—971no—
109#124—971no—
110#124—971no—
111#121—1942no—
112C4×D4—1942no—
113#117C2×D9712yes—
114C2×C42—971no—
115#104—1944no—
116#99—22no—
117#41—22no—
118#41—22no—
119#41—1942no—
120#41—1942no—
121C2×Q8—971no—
122C2×Q8C4×D9711yescharacteristic
123C2×D4—971no—
124C2×D4—1942no—
125C2×D4—1941no—
126C2×D4C4×D9711yescharacteristic
127C2×D4C22⋊D9712yes—
128C22×C4—1941no—
129C22×C4—1941no—
130C22×C4D38812yes—
131C22×C4C22×D9711yescharacteristic
132M16—974no—
133M16—1944no—
134M16—1942no—
135C2×C8—1944no—
136C2×C8—1942no—
137C4⋊C4—22no—
138C4⋊C4—1942no—
139C22⋊C4—22no—
140C22⋊C4—1942no—
141C22⋊C4—1942no—
142C42—22no—
143C42—971no—
144C42—971no—
145C42—1941no—
146C23—1941no—
147C23#1309548812yes—
148C23C2×C22⋊D9711yescharacteristic
149Q8—22no—
150Q8—1942no—
151D4—24no—
152D4—22no—
153D4—3882no—
154D4—3884no—
155D4—1942no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—3882no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—1941no—
163C2×C4—1941no—
164C2×C4—1941no—
165C2×C4#1309548812yes—
166C2×C4C2×C4×D9711yescharacteristic
167C2×C4D4×D9712yes—
168C2×C4C2×D38811yescharacteristic
169C2×C4#1309551012yes—
170C8—1944no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—3882no—
175C22—3881no—
176C22C2×#1309548811yescharacteristic
177C22#1569078212yes—
178C4—22no—
179C4—82no—
180C4—3881no—
181C4—3881no—
182C4—1942no—
183C4#1569077012yes—
184C4#1569078011yescharacteristic
185C4#1569079411yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—7761no—
190C2#2026061211yescharacteristic
subgroupG/H
centerC4#15690780
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#410×C97C2
socleC2×C97#386
prime radical#20261281C2
cosocleC22×C4×C97C23
hypercenter#410D97
hypocenterC97#1884
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #20261502C2semidirect
3. #20260753C2semidirect
4. #20260494C2semidirect
5. #410×C97C2semidirect
10. #15690808C4semidirect
11. #15690806C4semidirect
12. #15690799C4semidirect
13. #15690797C4semidirect
31. #13095495D4semidirect
96. C97#1884semidirect
100. #410D97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202612812
#202607532
#202604942
#410×C972
#188497