Finite Group Databasebeta

← #27142185

Subgroups — #27142185

Order 12416.

358 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15690769C211yesmaximal, characteristic, complemented
2C2×#15690737C212yesmaximal, complemented
3#20260699C211yesmaximal, characteristic, complemented
4#20260494C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7#15690769—24no—
8C2×C4⋊Dic97—21no—
9C2×C22⋊Dic97—21no—
10C2×C4×Dic97C2212yes—
11#15690737—24no—
12C22⋊#11715895—22no—
13C2×#41×C97C2211yescharacteristic
14C23×C4×C97C2211yescharacteristic
15#123×C97—21no—
16C22⋊Q8×C97C412yescomplemented
17#121×C97C412yescomplemented
18#120×C97—21no—
19C4×D4×C97—22no—
20#117×C97C2211yescharacteristic
21C22×Dic97—22no—
22#13095495—22no—
23#13095495—22no—
24C2×#11715895—22no—
25C4⋊Dic97—24no—
26C22⋊Dic97—24no—
27C4×Dic97—22no—
28C4×Dic97D414yescomplemented
29C24×C97D411yescharacteristic
30#41×C97—22no—
31#41×C97—22no—
32C2×Q8×C97C2×C411yescharacteristic
33C2×D4×C97—22no—
34C2×D4×C97—21no—
35C2×D4×C97C2×C411yescharacteristic
36C22×C4×C97—22no—
37C22×C4×C97—22no—
38C22×C4×C97—21no—
39C22×C4×C97C2×C412yes—
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D412yes—
42C22×C4×C97C2311yescharacteristic
43C4⋊C4×C97—22no—
44C4⋊C4×C97C2×C412yes—
45C22⋊C4×C97—22no—
46C22⋊C4×C97—21no—
47C22⋊C4×C97—22no—
48C42×C97—21no—
49C23×C97—21no—
50C23×C97—21no—
51C23×C97—21no—
52C23×C97—42no—
53C23×C97C22⋊C412yes—
54C23×C97C2×D411yescharacteristic
55Q8×C97—22no—
56D4×C97—22no—
57D4×C97—44no—
58D4×C97—41no—
59D4×C97—41no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—24no—
63C2×C4×C97—24no—
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—42no—
67C2×C4×C97C22⋊C412yes—
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97#4112yes—
72C2×Dic97—24no—
73C2×Dic97—44no—
74#11715895—42no—
75C22×C97—22no—
76C22×C97—22no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97—41no—
82C22×C97—42no—
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97#12011yescharacteristic
85C22×C97#12311yescharacteristic
86C4×C97—22no—
87C4×C97—41no—
88C4×C97—42no—
89C4×C97C4×D412yes—
90C4×C97#12112yes—
91Dic97—44no—
92C2×C97—21no—
93C2×C97—21no—
94C2×C97—42no—
95C2×C97—81no—
96C2×C97#38611yescharacteristic
97C2×C97#42112yes—
98#1897—971nomaximal, Sylow
99C97#189711yescharacteristic, complemented, Sylow
100#525D9711yescharacteristic
101C2×#117—971no—
102C2×#104—972no—
103#407—971no—
104#339—972no—
105C2×#41C2×D9711yescharacteristic
106C23×C4C2×D9711yescharacteristic
107C2×M16—972no—
108#123—21no—
109C22⋊Q8Dic9712yescomplemented
110#121Dic9712yescomplemented
111#120—21no—
112C4×D4—22no—
113#117—1944no—
114#117C2×D9711yescharacteristic
115C2×C4⋊C4—1941no—
116C2×C22⋊C4—1941no—
117C2×C42—972no—
118#104—1944no—
119C22⋊C8—1942no—
120C24C22⋊D9711yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×Dic9711yescharacteristic
124C2×D4—22no—
125C2×D4—21no—
126C2×D4C2×Dic9711yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—1942no—
131C22×C4C2×Dic9712yes—
132C22×C4C22⋊D9711yescharacteristic
133C22×C4C22⋊D9712yes—
134C22×C4C22×D9711yescharacteristic
135M16—1942no—
136M16—1942no—
137C2×C8—1942no—
138C4⋊C4—22no—
139C4⋊C4—1944no—
140C4⋊C4C2×Dic9712yes—
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—22no—
144C22⋊C4—1944no—
145C42—21no—
146C42—974no—
147C42—1942no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23C22⋊Dic9712yes—
153C23C2×C22⋊D9711yescharacteristic
154Q8—22no—
155D4—22no—
156D4—44no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—3884no—
167C2×C4—1944no—
168C2×C4C22⋊Dic9712yes—
169C2×C4C22×Dic9711yescharacteristic
170C2×C4D4×D9712yes—
171C2×C4C2×C22⋊D9711yescharacteristic
172C2×C4#1309551012yes—
173C8—3882no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—42no—
182C22C2×C22⋊Dic9711yescharacteristic
183C22#1569077811yescharacteristic
184C22#1569079111yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—3884no—
189C4D4×Dic9712yes—
190C4#1569078212yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—81no—
195C2#2026061511yescharacteristic
196C2#2026075212yes—
subgroupG/H
centerC2×C4C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#525×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#525D97
hypocenterC97#1897
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#15690769C2semidirect
2. C2×#15690737C2semidirect
3. #20260699C2semidirect
4. #20260494C2semidirect
16. C22⋊Q8×C97C4semidirect
17. #121×C97C4semidirect
28. C4×Dic97D4semidirect
99. C97#1897semidirect
109. C22⋊Q8Dic97semidirect
110. #121Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
C2×#156907372
#202606992
#202604942
#525×C972
#189797