Finite Group Databasebeta

← #28337508

Subgroups — #28337508

Order 13696.

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×#16812139C212yesmaximal, complemented
2C2×#16812036C212yesmaximal, complemented
3#20292532C211yesmaximal, characteristic, complemented
4#20292396C211yesmaximal, characteristic, complemented
5#537×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7#16812139—24no—
8#16812139—24no—
9C2×#13133808—22no—
10C2×#13133806—22no—
11C2×#13133802C2212yes—
12C2×C4×Dic107C2211yescharacteristic
13#16812036—24no—
14C107⋊#95—22no—
15C2×#41×C107C2212yes—
16C22⋊Q8×C107—22no—
17#121×C107—22no—
18#121×C107—22no—
19C2×C4⋊C4×C107C2211yescharacteristic
20C2×Dic214—22no—
21C2×Dic214—22no—
22C2×Dic214—21no—
23C2×Dic214D412yescomplemented
24C22×Dic107—21no—
25C22×Dic107—21no—
26#13133808—28no—
27#13133806—28no—
28#13133802—22no—
29#13133802D414yescomplemented
30C2×C107⋊C8—22no—
31C4×Dic107—22no—
32#41×C107—22no—
33#41×C107—22no—
34#41×C107—22no—
35#41×C107—22no—
36C2×Q8×C107D412yes—
37C2×D4×C107—22no—
38C2×D4×C107—22no—
39C2×D4×C107D412yes—
40C22×C4×C107—22no—
41C22×C4×C107D412yes—
42C22×C4×C107C2311yescharacteristic
43C4⋊C4×C107—22no—
44C4⋊C4×C107—22no—
45C22⋊C4×C107—24no—
46C23×C107—22no—
47C23×C107C2×D411yescharacteristic
48Q8×C107—22no—
49Q8×C107—22no—
50D4×C107—22no—
51D4×C107—22no—
52D4×C107—42no—
53D4×C107—42no—
54D4×C107—82no—
55C2×C4×C107—22no—
56C2×C4×C107—22no—
57C2×C4×C107—42no—
58C2×C4×C107—42no—
59C2×C4×C107—42no—
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107C2×D412yes—
62C2×C4×C107C2×D412yes—
63C2×C4×C107#4111yescharacteristic
64C2×Dic107—22no—
65C2×Dic107—22no—
66C2×Dic107—41no—
67C2×Dic107—41no—
68C2×Dic107—41no—
69C2×Dic107—41no—
70Dic214—24no—
71Dic214—42no—
72C107⋊C8—24no—
73C22×C107—21no—
74C22×C107—21no—
75C22×C107—42no—
76C22×C107—42no—
77C22×C107—42no—
78C22×C107#12011yescharacteristic
79C22×C107#12111yescharacteristic
80C22×C107#12711yescharacteristic
81C4×C107—42no—
82C4×C107—42no—
83C4×C107#12012yes—
84C4×C107#12112yes—
85Dic107—42no—
86Dic107—42no—
87C2×C107—21no—
88C2×C107—21no—
89C2×C107—82no—
90C2×C107#39211yescharacteristic
91C2×C107#45511yescharacteristic
92C2×C107#45611yescharacteristic
93#2007—1071nomaximal, Sylow
94C107#200711yescharacteristic, complemented, Sylow
95C2×#137—1072no—
96#537D10711yescharacteristic
97C2×#104—1072no—
98#391—1071no—
99#343—1071no—
100C2×#41C2×D10712yes—
101C22×Q8—1071no—
102#137—2144no—
103#137—2144no—
104C2×Q16—2142no—
105C2×SD16—2142no—
106C2×M16—1072no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D10711yescharacteristic
111C2×C42—1071no—
112#104—2144no—
113#95—2142no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—1072no—
119C2×Q8—2142no—
120C2×Q8—2142no—
121C2×Q8—2141no—
122C2×Q8C22⋊D10712yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D10712yes—
126C22×C4—22no—
127C22×C4—2141no—
128C22×C4—2141no—
129C22×C4C22⋊D10712yes—
130C22×C4C22×D10711yescharacteristic
131Q16—2148no—
132SD16—2148no—
133M16—1074no—
134M16—2142no—
135C2×C8—2142no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—2142no—
140C23—22no—
141C23C2×C22⋊D10711yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—4282no—
145Q8—2144no—
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—4281no—
157C2×C4—4281no—
158C2×C4—4281no—
159C2×C4—4281no—
160C2×C4—2142no—
161C2×C4—2142no—
162C2×C4D4×D10711yescharacteristic
163C2×C4D4×D10712yes—
164C2×C4C2×C22⋊D10712yes—
165C2×C4#1313381711yescharacteristic
166C8—2144no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1681207811yescharacteristic
173C22#1681208011yescharacteristic
174C22#1681210511yescharacteristic
175C4—42no—
176C4—42no—
177C4—4282no—
178C4—4282no—
179C4#1681207712yes—
180C4#1681208112yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2029253511yescharacteristic
185C2#2029280511yescharacteristic
186C2#2029281011yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting, prime radical#537×C107C2
socleC22×C107#120
hypercenter#537D107
hypocenterC107#2007
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C2×#16812036C2semidirect
3. #20292532C2semidirect
4. #20292396C2semidirect
23. C2×Dic214D4semidirect
29. #13133802D4semidirect
94. C107#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C2×#168120362
#202925322
#202923962
#537×C1072
#2007107