Finite Group Databasebeta

← #27142180

Subgroups — #27142180

Order 12416.

442 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C4×Dic97C211yesmaximal, characteristic, complemented
2C2×#15690737C212yesmaximal, complemented
3#20260699C211yesmaximal, characteristic, complemented
4#20260494C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C23×Dic97—21no—
7C2×#13095495C2212yes—
8C2×C4×Dic97—24no—
9C2×C4×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—24no—
22C22×Dic97—28no—
23C22×Dic97—22no—
24#13095495—22no—
25#13095495—22no—
26C2×#11715895—22no—
27C4×Dic97—24no—
28C4×Dic97—22no—
29C4×Dic97D414yescomplemented
30C24×C97D411yescharacteristic
31#41×C97—22no—
32#41×C97—22no—
33C2×Q8×C97C2×C411yescharacteristic, complemented
34C2×D4×C97—22no—
35C2×D4×C97—21no—
36C2×D4×C97C2×C411yescharacteristic, complemented
37C22×C4×C97—22no—
38C22×C4×C97—22no—
39C22×C4×C97—21no—
40C22×C4×C97C2×C412yes—
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C4⋊C4×C97C2×C412yescomplemented
46C22⋊C4×C97—22no—
47C22⋊C4×C97—21no—
48C22⋊C4×C97—22no—
49C42×C97—21no—
50C23×C97—24no—
51C23×C97—21no—
52C23×C97—21no—
53C23×C97—21no—
54C23×C97C22⋊C412yes—
55C23×C97C2×D411yescharacteristic
56Q8×C97—22no—
57D4×C97—22no—
58D4×C97—44no—
59D4×C97—41no—
60D4×C97—41no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—24no—
64C2×C4×C97—24no—
65C2×C4×C97—22no—
66C2×C4×C97—21no—
67C2×C4×C97—42no—
68C2×C4×C97C22⋊C412yes—
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D412yes—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97#4112yes—
73C2×Dic97—216no—
74C2×Dic97—28no—
75C2×Dic97—24no—
76#11715895—42no—
77C22×C97—24no—
78C22×C97—24no—
79C22×C97—22no—
80C22×C97—22no—
81C22×C97—22no—
82C22×C97—41no—
83C22×C97—42no—
84C22×C97#10414yescomplemented
85C22×C97C2×C22⋊C411yescharacteristic
86C22×C97#12011yescharacteristic
87C22×C97#12311yescharacteristic
88C4×C97—22no—
89C4×C97—41no—
90C4×C97—42no—
91C4×C97C4×D412yes—
92C4×C97#12112yes—
93Dic97—28no—
94C2×C97—21no—
95C2×C97—24no—
96C2×C97—21no—
97C2×C97—81no—
98C2×C97#38611yescharacteristic
99C2×C97C2×#10412yes—
100#1896—971nomaximal, Sylow
101C97#189611yescharacteristic, complemented, Sylow
102#525D9711yescharacteristic
103C22×C42—971no—
104C2×#104—972no—
105#407—971no—
106#339—972no—
107C2×#41C2×D9711yescharacteristic
108C23×C4—1941no—
109C23×C4C2×D9711yescharacteristic
110C2×M16—972no—
111#123—21no—
112C22⋊Q8Dic9712yescomplemented
113#121Dic9712yescomplemented
114#120—21no—
115C4×D4—22no—
116#117C2×D9711yescharacteristic
117C2×C42—972no—
118C2×C42—1944no—
119C2×C42—1941no—
120#104—1944no—
121C22⋊C8—1942no—
122C24C22⋊D9711yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic9711yescharacteristic, complemented
126C2×D4—22no—
127C2×D4—21no—
128C2×D4C2×Dic9711yescharacteristic, complemented
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—1944no—
133C22×C4—1948no—
134C22×C4—1942no—
135C22×C4C2×Dic9712yes—
136C22×C4C22⋊D9711yescharacteristic
137C22×C4C22⋊D9712yes—
138C22×C4C22×D9711yescharacteristic
139M16—1942no—
140M16—1942no—
141C2×C8—1942no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic9712yescomplemented
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C42—21no—
148C42—974no—
149C42—1944no—
150C42—1942no—
151C23—21no—
152C23—21no—
153C23—24no—
154C23—21no—
155C23C22⋊Dic9712yes—
156C23C2×C22⋊D9711yescharacteristic
157Q8—22no—
158D4—22no—
159D4—41no—
160D4—41no—
161D4—44no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—19416no—
170C2×C4—1948no—
171C2×C4—1944no—
172C2×C4C22⋊Dic9712yes—
173C2×C4C22×Dic9711yescharacteristic
174C2×C4D4×D9712yes—
175C2×C4C2×C22⋊D9711yescharacteristic
176C2×C4#1309551012yes—
177C8—3882no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—41no—
184C22—42no—
185C22#1569073714yescomplemented
186C22C2×C22⋊Dic9711yescharacteristic
187C22#1569077811yescharacteristic
188C22#1569079111yescharacteristic
189C4—22no—
190C4—41no—
191C4—42no—
192C4—1948no—
193C4D4×Dic9712yes—
194C4#1569078212yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—81no—
199C2#2026061511yescharacteristic
200C2C2×#1569073712yes—
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#1896
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. C22×C4×Dic97C2semidirect
2. C2×#15690737C2semidirect
3. #20260699C2semidirect
4. #20260494C2semidirect
16. C22⋊Q8×C97C4semidirect
17. #121×C97C4semidirect
29. C4×Dic97D4semidirect
33. C2×Q8×C97C2×C4semidirect
36. C2×D4×C97C2×C4semidirect
45. C4⋊C4×C97C2×C4semidirect
84. C22×C97#104semidirect
101. C97#1896semidirect
112. C22⋊Q8Dic97semidirect
113. #121Dic97semidirect
125. C2×Q8C2×Dic97semidirect
128. C2×D4C2×Dic97semidirect
143. C4⋊C4C2×Dic97semidirect
185. C22#15690737semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic972
C2×#156907372
#202606992
#202604942
#525×C972
#189697