Finite Group Databasebeta

← #27142584

Subgroups — #27142584

Order 12416.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15690799C211yesmaximal, characteristic
2#20260758C211yesmaximal, characteristic, complemented
3C2×#15690729C211yesmaximal, characteristic, complemented
4C2×#15690726C211yesmaximal, characteristic
5C2×#15690726C211yesmaximal, characteristic
6M16×Dic97C211yesmaximal, characteristic, complemented
7C2×#101×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C2×#13095495C2211yescharacteristic
11#15690799—21no—
12#15690799—21no—
13#15690799—22no—
14C2×#13095488C414yes—
15C2×C4×Dic97C2211yescharacteristic
16#15690739—22no—
17#15690729—22no—
18#15690726—22no—
19#15690726—22no—
20#15690720—21no—
21C8×Dic97—21no—
22C22×Q8×C97C2211yescharacteristic
23C2×M16×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25#101×C97—22no—
26C23×D97C2×C412yes—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—21no—
33C22×Dic97C2×C412yes—
34#13095495—21no—
35#13095495—21no—
36#13095495—21no—
37#13095495D412yes—
38C2×#11715895—21no—
39C2×#11715895—21no—
40#13095488—24no—
41#13095488—24no—
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97D412yes—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97—21no—
51C22×C4×C97C2×C412yes—
52C22×C4×C97C2311yescharacteristic
53M16×C97—21no—
54M16×C97—21no—
55M16×C97—21no—
56M16×C97D412yes—
57C2×C8×C97—21no—
58C2×C8×C97—21no—
59C23×C97C22×C411yescharacteristic
60C22×D97—22no—
61C22×D97—22no—
62C22×D97—22no—
63C22×D97—44no—
64Q8×C97—41no—
65Q8×C97—41no—
66Q8×C97—41no—
67Q8×C97—41no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97#4111yescharacteristic
76C2×C4×C97#4111yescharacteristic
77C2×Dic97—42no—
78C2×Dic97C22⋊C412yes—
79C2×Dic97C22⋊C412yes—
80D388—41no—
81D388—41no—
82D388—41no—
83D388—41no—
84C8×C97—22no—
85C8×C97—41no—
86#11715895—22no—
87#11715895—41no—
88C22×C97—22no—
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97C4×D411yescharacteristic
91C22×C97C4×D411yescharacteristic
92C2×D97—42no—
93C2×D97—44no—
94C2×D97—82no—
95C2×D97—82no—
96C4×C97—42no—
97C4×C97#12111yescharacteristic
98C4×C97#12111yescharacteristic
99C4×C97#12411yescharacteristic
100C4×C97#12711yescharacteristic
101Dic97—22no—
102Dic97—22no—
103C2×C97—21no—
104C2×C97—21no—
105C2×C97#39011yescharacteristic
106C2×C97#41312yes—
107D97—82no—
108#1967—971nomaximal, Sylow
109C97#196711yescharacteristic, complemented, Sylow
110C2×#124—971no—
111#423—971no—
112C2×#101—971no—
113C2×#101D9711yescharacteristic, complemented
114C2×#100—971no—
115C2×#100—971no—
116C4×M16—971no—
117C22×Q8C2×D9711yescharacteristic
118C22×D4—971no—
119C2×M16—971no—
120C2×M16—971no—
121C2×M16C2×D9711yescharacteristic
122C2×M16C2×D9711yescharacteristic
123#124—1941no—
124#124—1941no—
125#124—1942no—
126C2×C22⋊C4—974no—
127C2×C42—971no—
128C4⋊C8—1942no—
129#101—22no—
130#101—1942no—
131#100—1942no—
132#100—1942no—
133#97—1941no—
134C4×C8—1941no—
135C24—972no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×D4—1941no—
143C2×D4—1941no—
144C2×D4—1941no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4—1941no—
148C22×C4—972no—
149C22×C4C4×D9712yes—
150C22×C4C22×D9711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—972no—
155M16—1941no—
156M16—1941no—
157M16—1941no—
158M16C22⋊D9712yes—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1941no—
162C2×C8—1941no—
163C22⋊C4—1944no—
164C22⋊C4—1944no—
165C42—971no—
166C42—971no—
167C42—972no—
168C23—3884no—
169C23—1942no—
170C23—1942no—
171C23—1942no—
172C23C2×C4×D9711yescharacteristic
173Q8—41no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—3881no—
178D4—3881no—
179D4—3881no—
180D4—3881no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—972no—
185C2×C4—972no—
186C2×C4—3882no—
187C2×C4D4×D9711yescharacteristic
188C2×C4D4×D9711yescharacteristic
189C2×C4D4×D9711yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C2×C4#1309551011yescharacteristic
192C2×C4#1309551111yescharacteristic
193C8—22no—
194C8—41no—
195C8—3881no—
196C8—1942no—
197C22—22no—
198C22—7762no—
199C22—7762no—
200C22—3882no—
201C22—3884no—
202C22C22⋊C4×D9711yescharacteristic
203C22#1569077011yescharacteristic
204C22C4×C22⋊D9711yescharacteristic
205C4—42no—
206C4—1942no—
207C4—1942no—
208C4#1569078211yescharacteristic
209C4#1569078311yescharacteristic
210C4#1569079611yescharacteristic
211C4#1569080611yescharacteristic
212C2—21no—
213C2—21no—
214C2—7762no—
215C2#2026063511yescharacteristic
216C2#2026071812yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#101×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenterC2×#101D97
hypocenterC97#1967
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #20260758C2semidirect
3. C2×#15690729C2semidirect
6. M16×Dic97C2semidirect
7. C2×#101×C97C2semidirect
109. C97#1967semidirect
113. C2×#101D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
#202607582
C2×#156907292
C2×#156907262
M16×Dic972
C2×#101×C972
#196797