Finite Group Databasebeta

← #27142783

Subgroups — #27142783

Order 12416.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15690840C212yesmaximal, complemented
2C2×#15690737C212yesmaximal, complemented
3#20260640C211yesmaximal, characteristic, complemented
4#20260504C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7#15690840—24no—
8#15690840—24no—
9C2×#13095501—22no—
10C2×#13095499—22no—
11C2×#13095495C2212yes—
12C2×C4×Dic97C2211yescharacteristic
13#15690737—24no—
14#15690717—22no—
15C2×#41×C97C2212yes—
16C22⋊Q8×C97—22no—
17#121×C97—22no—
18#121×C97—22no—
19C2×C4⋊C4×C97C2211yescharacteristic
20C2×Dic194—22no—
21C2×Dic194—22no—
22C2×Dic194—21no—
23C2×Dic194D412yescomplemented
24C22×Dic97—21no—
25C22×Dic97—21no—
26#13095501—28no—
27#13095499—28no—
28#13095495—22no—
29#13095495D414yescomplemented
30C2×#11715895—22no—
31C4×Dic97—22no—
32#41×C97—22no—
33#41×C97—22no—
34#41×C97—22no—
35#41×C97—22no—
36C2×Q8×C97D412yes—
37C2×D4×C97—22no—
38C2×D4×C97—22no—
39C2×D4×C97D412yes—
40C22×C4×C97—22no—
41C22×C4×C97D412yes—
42C22×C4×C97C2311yescharacteristic
43C4⋊C4×C97—22no—
44C4⋊C4×C97—22no—
45C22⋊C4×C97—24no—
46C23×C97—22no—
47C23×C97C2×D411yescharacteristic
48Q8×C97—22no—
49Q8×C97—22no—
50D4×C97—22no—
51D4×C97—22no—
52D4×C97—42no—
53D4×C97—42no—
54D4×C97—82no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—42no—
58C2×C4×C97—42no—
59C2×C4×C97—42no—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D412yes—
62C2×C4×C97C2×D412yes—
63C2×C4×C97#4111yescharacteristic
64C2×Dic97—22no—
65C2×Dic97—22no—
66C2×Dic97—41no—
67C2×Dic97—41no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70Dic194—24no—
71Dic194—42no—
72#11715895—24no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97—42no—
78C22×C97#12011yescharacteristic
79C22×C97#12111yescharacteristic
80C22×C97#12711yescharacteristic
81C4×C97—42no—
82C4×C97—42no—
83C4×C97#12012yes—
84C4×C97#12112yes—
85Dic97—42no—
86Dic97—42no—
87C2×C97—21no—
88C2×C97—21no—
89C2×C97—82no—
90C2×C97#39211yescharacteristic
91C2×C97#45511yescharacteristic
92C2×C97#45611yescharacteristic
93#2007—971nomaximal, Sylow
94C97#200711yescharacteristic, complemented, Sylow
95C2×#137—972no—
96#537D9711yescharacteristic
97C2×#104—972no—
98#391—971no—
99#343—971no—
100C2×#41C2×D9712yes—
101C22×Q8—971no—
102#137—1944no—
103#137—1944no—
104C2×Q16—1942no—
105C2×SD16—1942no—
106C2×M16—972no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D9711yescharacteristic
111C2×C42—971no—
112#104—1944no—
113#95—1942no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—972no—
119C2×Q8—1942no—
120C2×Q8—1942no—
121C2×Q8—1941no—
122C2×Q8C22⋊D9712yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D9712yes—
126C22×C4—22no—
127C22×C4—1941no—
128C22×C4—1941no—
129C22×C4C22⋊D9712yes—
130C22×C4C22×D9711yescharacteristic
131Q16—1948no—
132SD16—1948no—
133M16—974no—
134M16—1942no—
135C2×C8—1942no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—1942no—
140C23—22no—
141C23C2×C22⋊D9711yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—3882no—
145Q8—1944no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—3881no—
157C2×C4—3881no—
158C2×C4—3881no—
159C2×C4—3881no—
160C2×C4—1942no—
161C2×C4—1942no—
162C2×C4D4×D9711yescharacteristic
163C2×C4D4×D9712yes—
164C2×C4C2×C22⋊D9712yes—
165C2×C4#1309551011yescharacteristic
166C8—1944no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1569077911yescharacteristic
173C22#1569078111yescharacteristic
174C22#1569080611yescharacteristic
175C4—42no—
176C4—42no—
177C4—3882no—
178C4—3882no—
179C4#1569077812yes—
180C4#1569078212yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2026064311yescharacteristic
185C2#2026091311yescharacteristic
186C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#2007
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#15690840C2semidirect
2. C2×#15690737C2semidirect
3. #20260640C2semidirect
4. #20260504C2semidirect
23. C2×Dic194D4semidirect
29. #13095495D4semidirect
94. C97#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
C2×#156907372
#202606402
#202605042
#537×C972
#200797