Finite Group Databasebeta

← #27141695

Subgroups — #27141695

Order 12416.

552 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15690737C212yesmaximal, complemented
2#20260736C212yesmaximal, complemented
3C2×#15690729C211yesmaximal, characteristic, complemented
4#20260642C211yesmaximal, characteristic, complemented
5C2×2−1+4×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7C2×C4⋊Dic97C2211yescharacteristic
8C2×C4×Dic97C2211yescharacteristic
9#15690737—24no—
10#15690734—22no—
11#15690731—22no—
12#15690729—22no—
13#15690717—22no—
142−1+4×C97—24no—
152−1+4×C97—22no—
162−1+4×C97C414yescomplemented
17C2×#41×C97—22no—
18C2×#41×C97—22no—
19C2×#41×C97C2212yes—
20C22×Q8×C97—22no—
21C22×Q8×C97C2211yescharacteristic
22C22×Dic97—21no—
23C22×Dic97—21no—
24#13095495—22no—
25#13095495—22no—
26C2×#11715895—22no—
27C4⋊Dic97—22no—
28C4×Dic97—22no—
29#41×C97—28no—
30#41×C97—28no—
31#41×C97—28no—
32#41×C97—28no—
33#41×C97—22no—
34#41×C97—22no—
35#41×C97C2×C414yes—
36#41×C97D414yes—
37C2×Q8×C97—22no—
38C2×Q8×C97—28no—
39C2×Q8×C97—28no—
40C2×Q8×C97—22no—
41C2×Q8×C97—22no—
42C2×Q8×C97C2×C412yes—
43C2×Q8×C97D412yes—
44C2×Q8×C97D412yes—
45C2×D4×C97—22no—
46C2×D4×C97—22no—
47C2×D4×C97D412yes—
48C22×C4×C97—24no—
49C22×C4×C97—22no—
50C22×C4×C97D412yes—
51C22×C4×C97C2311yescharacteristic
52C23×C97—22no—
53C23×C97C2×D411yescharacteristic
54Q8×C97—28no—
55Q8×C97—24no—
56Q8×C97—22no—
57Q8×C97—24no—
58Q8×C97C22⋊C414yes—
59D4×C97—28no—
60D4×C97—24no—
61D4×C97—24no—
62D4×C97—22no—
63D4×C97C22⋊C414yes—
64C2×C4×C97—216no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—24no—
69C2×C4×C97—24no—
70C2×C4×C97C22⋊C414yes—
71C2×C4×C97C22×C411yescharacteristic
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D412yes—
74C2×C4×C97C2×D411yescharacteristic
75C2×Dic97—22no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—42no—
79C2×Dic97—41no—
80#11715895—42no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—24no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97C2×C22⋊C411yescharacteristic
87C22×C97#12011yescharacteristic
88C22×C97#12011yescharacteristic
89C4×C97—24no—
90C4×C97—24no—
91C4×C97C2×C22⋊C412yes—
92C4×C97#12012yes—
93Dic97—42no—
94Dic97—81no—
95C2×C97—21no—
96C2×C97—21no—
97C2×C97—44no—
98C2×C97#37911yescharacteristic
99C2×C97#45411yescharacteristic
100C2×C97#45611yescharacteristic
101#1793—971nomaximal, Sylow
102C97#179311yescharacteristic, complemented, Sylow
103C2×2−1+4D9711yescharacteristic
104C2×#104—972no—
105#417—972no—
106C2×#101—971no—
107#391—971no—
1082−1+4—24no—
1092−1+4—22no—
1102−1+4Dic9714yescomplemented
111C2×#41—22no—
112C2×#41—22no—
113C2×#41C2×D9712yes—
114C22×Q8—22no—
115C22×Q8C2×D9711yescharacteristic
116C2×M16—972no—
117C2×C4⋊C4—971no—
118C2×C42—971no—
119#104—1944no—
120#103—1942no—
121#102—1942no—
122#101—1942no—
123#95—1942no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C2×Dic9714yes—
131#41C22⋊D9714yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×Dic9712yes—
138C2×Q8C22⋊D9712yes—
139C2×Q8C22⋊D9712yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22⋊D9712yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—1941no—
146C22×C4—1941no—
147C22×C4C22⋊D9712yes—
148C22×C4C22×D9711yescharacteristic
149M16—1942no—
150M16—1942no—
151C2×C8—1942no—
152C4⋊C4—1942no—
153C42—1942no—
154C23—22no—
155C23C2×C22⋊D9711yescharacteristic
156Q8—28no—
157Q8—24no—
158Q8—22no—
159Q8—24no—
160Q8C22⋊Dic9714yes—
161D4—28no—
162D4—24no—
163D4—24no—
164D4—22no—
165D4C22⋊Dic9714yes—
166C2×C4—216no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—3882no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—1942no—
177C2×C4C22⋊Dic9714yes—
178C2×C4C22×Dic9711yescharacteristic
179C2×C4C2×C22⋊D9712yes—
180C2×C4C2×C22⋊D9712yes—
181C2×C4C2×C22⋊D9711yescharacteristic
182C8—3882no—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—42no—
187C22—42no—
188C22C2×C22⋊Dic9711yescharacteristic
189C22#1569077911yescharacteristic
190C22#1569077911yescharacteristic
191C4—24no—
192C4—24no—
193C4—7761no—
194C4—3882no—
195C4C2×C22⋊Dic9712yes—
196C4#1569077912yes—
197C2—21no—
198C2—21no—
199C2—44no—
200C2#2026059111yescharacteristic
201C2#2026090811yescharacteristic
202C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×2−1+4×C97C2
socleC22×C97#120
cosocleC22×C4×C97C23
hypercenterC2×2−1+4D97
hypocenterC97#1793
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#15690737C2semidirect
2. #20260736C2semidirect
3. C2×#15690729C2semidirect
4. #20260642C2semidirect
16. 2−1+4×C97C4semidirect
102. C97#1793semidirect
110. 2−1+4Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907372
#202607362
C2×#156907292
#202606422
C2×2−1+4×C972
#179397