Finite Group Databasebeta

← #28337507

Subgroups — #28337507

Order 13696.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16812140C212yesmaximal, complemented
2#20293208C211yesmaximal, characteristic
3C2×#16812037C212yesmaximal, complemented
4#20292396C211yesmaximal, characteristic, complemented
5#391×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2212yes—
7#16812140—24no—
8#16812140—24no—
9C2×#13133808—22no—
10C2×#13133805—22no—
11C2×#13133802C2212yes—
12#16812089—22no—
13#16812084—22no—
14#16812083—22no—
15#16812037—24no—
16C22×Q8×C107C2211yescharacteristic
17C2×C4⋊C4×C107C2211yescharacteristic
18C2×C42×C107C2211yescharacteristic
19#95×C107—22no—
20#13133818—22no—
21#13133818—22no—
22#13133818—22no—
23#13133818—22no—
24C2×Dic214D412yes—
25C2×D428—22no—
26C2×D428D412yes—
27C2×C22⋊D107—22no—
28C2×C4×D107—22no—
29#13133808—28no—
30#13133805—28no—
31#13133802—22no—
32#13133802D414yescomplemented
33C2×C107⋊C8—22no—
34C4⋊Dic107—22no—
35(C2×D107)⋊C4—24no—
36C2×Q8×C107—21no—
37C2×Q8×C107—22no—
38C2×Q8×C107—22no—
39C2×Q8×C107D412yescomplemented
40C22×C4×C107—21no—
41C22×C4×C107—21no—
42C22×C4×C107D412yes—
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—22no—
45C42×C107—22no—
46C23×C107C2×D411yescharacteristic
47C22×D107—22no—
48Q8×C107—24no—
49Q8×C107—42no—
50C2×C4×C107—22no—
51C2×C4×C107—22no—
52C2×C4×C107—22no—
53C2×C4×C107—41no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107—41no—
57C2×C4×C107—42no—
58C2×C4×C107C2×D412yes—
59C2×C4×C107C2×D412yes—
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107#4111yescharacteristic
62C22⋊D107—42no—
63C22⋊D107—42no—
64C2×Dic107—22no—
65D428—22no—
66D428—22no—
67D428—82no—
68C4×D107—42no—
69C4×D107—42no—
70Dic214—22no—
71Dic214—22no—
72C107⋊C8—24no—
73C22×C107—21no—
74C22×C107—21no—
75C22×C107#12011yescharacteristic
76C22×C107#12111yescharacteristic
77C22×C107#12711yescharacteristic
78C2×D107—42no—
79C2×D107—42no—
80C2×D107—42no—
81C4×C107—42no—
82C4×C107—42no—
83C4×C107—42no—
84C4×C107#12012yes—
85C4×C107#12112yes—
86Dic107—42no—
87C2×C107—21no—
88C2×C107—21no—
89C2×C107#39211yescharacteristic
90C2×C107#45511yescharacteristic
91C2×C107#45611yescharacteristic
92D107—82no—
93#2007—1071nomaximal, Sylow
94C107#200711yescharacteristic, complemented, Sylow
95C2×#137—1072no—
96#537—1071no—
97C2×#104—1072no—
98#391D10711yescharacteristic, complemented
99#343—1071no—
100C2×#41—1072no—
101C22×Q8C2×D10711yescharacteristic
102#137—2144no—
103#137—2144no—
104C2×Q16—2142no—
105C2×SD16—2142no—
106C2×M16—1072no—
107C22⋊Q8—2142no—
108#121—2142no—
109#121—2142no—
110C2×C4⋊C4C2×D10711yescharacteristic
111C2×C42C2×D10711yescharacteristic
112#104—2144no—
113#95—22no—
114#41—2142no—
115#41—2142no—
116#41—2142no—
117#41—2142no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—1072no—
122C2×Q8D42812yescomplemented
123C2×D4—1072no—
124C2×D4—2142no—
125C2×D4—2142no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—2142no—
129C22×C4C22⋊D10712yes—
130C22×C4C22×D10711yescharacteristic
131Q16—2148no—
132SD16—2148no—
133M16—1074no—
134M16—2142no—
135C2×C8—2142no—
136C4⋊C4—22no—
137C4⋊C4—2142no—
138C22⋊C4—2144no—
139C42—22no—
140C23—2142no—
141C23C2×C22⋊D10711yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—2142no—
145Q8—2142no—
146D4—8562no—
147D4—4282no—
148D4—4282no—
149D4—2142no—
150D4—2142no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—4282no—
160C2×C4—4282no—
161C2×C4—2142no—
162C2×C4D4×D10712yes—
163C2×C4D4×D10712yes—
164C2×C4C2×D42811yescharacteristic
165C2×C4#1313381811yescharacteristic
166C8—2144no—
167C22—21no—
168C22—21no—
169C22—4282no—
170C22—4282no—
171C22—4282no—
172C22#1681207711yescharacteristic
173C22#1681208311yescharacteristic
174C22#1681210511yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—4282no—
179C4#1681207912yes—
180C4#1681208212yes—
181C2—21no—
182C2—21no—
183C2—8562no—
184C2#2029253611yescharacteristic
185C2#2029280411yescharacteristic
186C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#391×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#391D107
hypocenterC107#2007
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
3. C2×#16812037C2semidirect
4. #20292396C2semidirect
5. #391×C107C2semidirect
32. #13133802D4semidirect
39. C2×Q8×C107D4semidirect
94. C107#2007semidirect
98. #391D107semidirect
122. C2×Q8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
#202932082
C2×#168120372
#202923962
#391×C1072
#2007107