Finite Group Databasebeta

← #27141706

Subgroups — #27141706

Order 12416.

588 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 121 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#20260752C212yesmaximal, complemented
2#20260740C212yesmaximal, complemented
3#20260719C211yesmaximal, characteristic, complemented
4#20260709C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C97C211yesmaximal, characteristic
6C2×#13095495C2212yescomplemented
7#15690769C2212yescomplemented
8#15690737—24no—
9#15690731—24no—
10#15690729—21no—
11#15690727—21no—
12#15690725—22no—
13H(5,𝔽2)×C97—24no—
14H(5,𝔽2)×C97—24no—
15C2×#41×C97—21no—
16C2×#41×C97C412yescomplemented
17C2×#41×C97C2211yescharacteristic
18C2×#41×C97C2211yescharacteristic
19C22×D4×C97—22no—
20C22×D4×C97—21no—
21C22×D4×C97C412yescomplemented
22C22×D4×C97C2211yescharacteristic
23#13095495—22no—
24#13095495—22no—
25C2×#11715895—22no—
26C4⋊Dic97—22no—
27C22⋊Dic97—22no—
28C4×Dic97—22no—
29C24×C97—22no—
30C24×C97—21no—
31#41×C97—28no—
32#41×C97—24no—
33#41×C97—24no—
34#41×C97—22no—
35#41×C97—22no—
36#41×C97—22no—
37#41×C97D414yes—
38C2×Q8×C97C2×C411yescharacteristic
39C2×Q8×C97D411yescharacteristic
40C2×D4×C97—216no—
41C2×D4×C97—28no—
42C2×D4×C97—22no—
43C2×D4×C97—21no—
44C2×D4×C97—24no—
45C2×D4×C97—24no—
46C2×D4×C97—22no—
47C2×D4×C97—21no—
48C2×D4×C97—22no—
49C2×D4×C97C2×C411yescharacteristic
50C2×D4×C97C2×C412yes—
51C2×D4×C97D412yes—
52C2×D4×C97D411yescharacteristic
53C2×D4×C97D414yes—
54C22×C4×C97—22no—
55C22×C4×C97—21no—
56C22×C4×C97C2×C412yes—
57C22×C4×C97C2311yescharacteristic
58C23×C97—28no—
59C23×C97—22no—
60C23×C97—21no—
61C23×C97—24no—
62C23×C97—82no—
63C23×C97—84no—
64C23×C97C22⋊C412yes—
65C23×C97C2×D411yescharacteristic
66Q8×C97—21no—
67Q8×C97—22no—
68Q8×C97C22⋊C412yes—
69D4×C97—28no—
70D4×C97—28no—
71D4×C97—24no—
72D4×C97—24no—
73D4×C97—22no—
74D4×C97—21no—
75D4×C97—22no—
76D4×C97—24no—
77D4×C97C22⋊C412yes—
78D4×C97C22⋊C414yes—
79C2×C4×C97—28no—
80C2×C4×C97—21no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—24no—
84C2×C4×C97C22⋊C412yes—
85C2×C4×C97C22×C411yescharacteristic
86C2×C4×C97C2×D412yes—
87C2×C4×C97C2×D412yes—
88C2×C4×C97C2×D411yescharacteristic
89C2×Dic97—42no—
90#11715895—42no—
91C22×C97—22no—
92C22×C97—24no—
93C22×C97—22no—
94C22×C97—41no—
95C22×C97—42no—
96C22×C97—44no—
97C22×C97—42no—
98C22×C97—84no—
99C22×C97—88no—
100C22×C97C2×C22⋊C411yescharacteristic
101C22×C97#12012yes—
102C4×C97—22no—
103C4×C97—22no—
104C4×C97C2×C22⋊C411yescharacteristic
105C4×C97C2×C22⋊C411yescharacteristic
106C4×C97#12012yes—
107Dic97—82no—
108C2×C97—22no—
109C2×C97—21no—
110C2×C97—44no—
111C2×C97—42no—
112C2×C97—42no—
113C2×C97#37911yescharacteristic
114#1795—971nomaximal, Sylow
115C97#179511yescharacteristic, complemented, Sylow
116C2×H(5,𝔽2)D9711yescharacteristic
117#421—972no—
118#418—972no—
119#413—971no—
120#410—971no—
121H(5,𝔽2)—24no—
122H(5,𝔽2)—24no—
123C2×#41—21no—
124C2×#41Dic9712yescomplemented
125C2×#41C2×D9711yescharacteristic
126C2×#41C2×D9711yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4Dic9712yescomplemented
130C22×D4C2×D9711yescharacteristic
131C2×M16—972no—
132#117—972no—
133#104—1944no—
134#102—1944no—
135#101—1941no—
136#100—1941no—
137#99—1942no—
138C24—22no—
139C24—21no—
140#41—28no—
141#41—24no—
142#41—24no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41C22⋊D9714yes—
147C2×Q8C2×Dic9711yescharacteristic
148C2×Q8C22⋊D9711yescharacteristic
149C2×D4—216no—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4C2×Dic9711yescharacteristic
159C2×D4C2×Dic9712yes—
160C2×D4C22⋊D9712yes—
161C2×D4C22⋊D9711yescharacteristic
162C2×D4C22⋊D9714yes—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4C2×Dic9712yes—
166C22×C4C22×D9711yescharacteristic
167M16—1942no—
168M16—1942no—
169C2×C8—1942no—
170C4⋊C4—1942no—
171C22⋊C4—1942no—
172C42—1942no—
173C23—28no—
174C23—22no—
175C23—21no—
176C23—24no—
177C23—82no—
178C23—84no—
179C23C22⋊Dic9712yes—
180C23C2×C22⋊D9711yescharacteristic
181Q8—21no—
182Q8—22no—
183Q8C22⋊Dic9712yes—
184D4—28no—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—21no—
190D4—22no—
191D4—24no—
192D4C22⋊Dic9712yes—
193D4C22⋊Dic9714yes—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—3882no—
200C2×C4C22⋊Dic9712yes—
201C2×C4C22×Dic9711yescharacteristic
202C2×C4C2×C22⋊D9712yes—
203C2×C4C2×C22⋊D9712yes—
204C2×C4C2×C22⋊D9711yescharacteristic
205C8—3882no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—41no—
210C22—42no—
211C22—44no—
212C22—42no—
213C22—84no—
214C22—88no—
215C22C2×C22⋊Dic9711yescharacteristic
216C22#1569077912yes—
217C4—22no—
218C4—22no—
219C4—7762no—
220C4C2×C22⋊Dic9711yescharacteristic
221C4C2×C22⋊Dic9711yescharacteristic
222C4#1569077912yes—
223C2—22no—
224C2—21no—
225C2—44no—
226C2—42no—
227C2—42no—
228C2#2026059111yescharacteristic
subgroupG/H
centerC2#20260591
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×H(5,𝔽2)×C97C2
socleC2×C97#379
cosocleC22×C4×C97C23
hypercenterC2×H(5,𝔽2)D97
hypocenterC97#1795
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20260752C2semidirect
2. #20260740C2semidirect
3. #20260719C2semidirect
4. #20260709C2semidirect
6. C2×#13095495C22semidirect
7. #15690769C22semidirect
16. C2×#41×C97C4semidirect
21. C22×D4×C97C4semidirect
115. C97#1795semidirect
124. C2×#41Dic97semidirect
129. C22×D4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607522
#202607402
#202607192
#202607092
C2×H(5,𝔽2)×C972
#179597