Finite Group Databasebeta

← #27141690

Subgroups — #27141690

Order 12416.

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

normal subgroups only

This group has 620 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×#15690727C211yesmaximal, characteristic, complemented
4#20260638C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C97C211yesmaximal, characteristic
6C2×#13095495C2212yescomplemented
7C2×C4⋊Dic97C2211yescharacteristic, complemented
8C2×C22⋊Dic97—22no—
9C2×C4×Dic97C2211yescharacteristic, complemented
10#15690737—24no—
11#15690734—22no—
12#15690731—22no—
13#15690727—22no—
14H(5,𝔽2)×C97—24no—
15H(5,𝔽2)×C97—22no—
16H(5,𝔽2)×C97C414yescomplemented
17C2×#41×C97—22no—
18C2×#41×C97C2212yes—
19C22×D4×C97—22no—
20C22×D4×C97—22no—
21C22×D4×C97C2211yescharacteristic
22C22×Dic97—21no—
23C22×Dic97—21no—
24#13095495—22no—
25#13095495—22no—
26C2×#11715895—22no—
27C4⋊Dic97—22no—
28C22⋊Dic97—44no—
29C4×Dic97—22no—
30C24×C97—22no—
31C24×C97D412yes—
32#41×C97—28no—
33#41×C97—28no—
34#41×C97—22no—
35#41×C97—22no—
36#41×C97C2×C414yes—
37#41×C97D414yes—
38C2×Q8×C97D412yes—
39C2×D4×C97—24no—
40C2×D4×C97—28no—
41C2×D4×C97—28no—
42C2×D4×C97—28no—
43C2×D4×C97—22no—
44C2×D4×C97—28no—
45C2×D4×C97—22no—
46C2×D4×C97—22no—
47C2×D4×C97C2×C412yes—
48C2×D4×C97D412yes—
49C2×D4×C97D412yes—
50C22×C4×C97—22no—
51C22×C4×C97—22no—
52C22×C4×C97C2311yescharacteristic
53C23×C97—28no—
54C23×C97—22no—
55C23×C97—22no—
56C23×C97—22no—
57C23×C97—42no—
58C23×C97—42no—
59C23×C97—82no—
60C23×C97—82no—
61C23×C97C22⋊C414yes—
62C23×C97C2×D411yescharacteristic
63Q8×C97—22no—
64Q8×C97C22⋊C414yescomplemented
65D4×C97—216no—
66D4×C97—24no—
67D4×C97—24no—
68D4×C97—24no—
69D4×C97—24no—
70D4×C97—22no—
71D4×C97C22⋊C414yescomplemented
72C2×C4×C97—28no—
73C2×C4×C97—22no—
74C2×C4×C97—24no—
75C2×C4×C97—24no—
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C2×D412yes—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D411yescharacteristic
80C2×Dic97—22no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97—42no—
84C2×Dic97—41no—
85#11715895—42no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—24no—
89C22×C97—24no—
90C22×C97—44no—
91C22×C97—42no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97—82no—
96C22×C97—88no—
97C22×C97C2×C22⋊C411yescharacteristic
98C22×C97#12011yescharacteristic
99C22×C97#12011yescharacteristic
100C4×C97—24no—
101C4×C97C2×C22⋊C412yes—
102C4×C97#12012yes—
103Dic97—42no—
104Dic97—81no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97—44no—
108C2×C97—44no—
109C2×C97#37911yescharacteristic
110C2×C97#45311yescharacteristic
111C2×C97#45511yescharacteristic
112#1792—971nomaximal, Sylow
113C97#179211yescharacteristic, complemented, Sylow
114C2×H(5,𝔽2)D9711yescharacteristic
115C2×#104—972no—
116#417—972no—
117C2×#100—971no—
118#390—971no—
119H(5,𝔽2)—24no—
120H(5,𝔽2)—22no—
121H(5,𝔽2)Dic9714yescomplemented
122C2×#41—22no—
123C2×#41C2×D9712yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D9711yescharacteristic
127C2×M16—972no—
128C2×C4⋊C4—971no—
129C2×C22⋊C4—1942no—
130C2×C42—971no—
131#104—1944no—
132#103—1942no—
133#102—1942no—
134#100—1942no—
135C24—22no—
136C24C22⋊D9712yes—
137#41—28no—
138#41—28no—
139#41—22no—
140#41—22no—
141#41C2×Dic9714yes—
142#41C22⋊D9714yes—
143C2×Q8C22⋊D9712yes—
144C2×D4—24no—
145C2×D4—28no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—22no—
149C2×D4—28no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4C2×Dic9712yes—
153C2×D4C22⋊D9712yes—
154C2×D4C22⋊D9712yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—1941no—
158C22×C4—1941no—
159C22×C4C22×D9711yescharacteristic
160M16—1942no—
161M16—1942no—
162C2×C8—1942no—
163C4⋊C4—1942no—
164C22⋊C4—3884no—
165C42—1942no—
166C23—28no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23C22⋊Dic9714yes—
175C23C2×C22⋊D9711yescharacteristic
176Q8—22no—
177Q8C22⋊Dic9714yescomplemented
178D4—216no—
179D4—24no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C22⋊Dic9714yescomplemented
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—3882no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—3881no—
193C2×C4—1942no—
194C2×C4C22×Dic9711yescharacteristic
195C2×C4C2×C22⋊D9712yes—
196C2×C4C2×C22⋊D9712yes—
197C2×C4C2×C22⋊D9711yescharacteristic
198C8—3882no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—88no—
210C22C2×C22⋊Dic9711yescharacteristic
211C22#1569077911yescharacteristic
212C22#1569077911yescharacteristic
213C4—24no—
214C4—7761no—
215C4—3882no—
216C4C2×C22⋊Dic9712yes—
217C4#1569077912yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2#2026059111yescharacteristic
223C2#2026090311yescharacteristic
224C2#2026091311yescharacteristic
subgroupG/H
centerC22#15690779
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×H(5,𝔽2)×C97C2
socleC22×C97#120
cosocleC22×C4×C97C23
hypercenterC2×H(5,𝔽2)D97
hypocenterC97#1792
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15690737C2semidirect
2. #20260736C2semidirect
3. C2×#15690727C2semidirect
4. #20260638C2semidirect
6. C2×#13095495C22semidirect
7. C2×C4⋊Dic97C22semidirect
9. C2×C4×Dic97C22semidirect
16. H(5,𝔽2)×C97C4semidirect
64. Q8×C97C22⋊C4semidirect
71. D4×C97C22⋊C4semidirect
113. C97#1792semidirect
121. H(5,𝔽2)Dic97semidirect
177. Q8C22⋊Dic97semidirect
184. D4C22⋊Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907372
#202607362
C2×#156907272
#202606382
C2×H(5,𝔽2)×C972
#179297