Finite Group Databasebeta

← #28336419

Subgroups — #28336419

Order 13696.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16812157C211yesmaximal, characteristic
2C2×#16812037C212yesmaximal, complemented
3#20292626C212yesmaximal, complemented
4C2×#16812028C211yesmaximal, characteristic, complemented
5#391×C107C211yesmaximal, characteristic, complemented
6#16812157—24no—
7#16812157—22no—
8#16812157C414yescomplemented
9C2×#13133819—22no—
10C2×#13133818—22no—
11C2×#13133818C2212yes—
12C2×Q8×D107—22no—
13C2×#13133802C2212yes—
14#16812037—24no—
15#16812032—22no—
16#16812029—22no—
17#16812028—22no—
18C22×Q8×C107C2211yescharacteristic
19C2×C4⋊C4×C107C2211yescharacteristic
20C2×C42×C107C2211yescharacteristic
21#95×C107—22no—
22#13133819—28no—
23#13133819—28no—
24#13133818—28no—
25#13133818—28no—
26#13133818—22no—
27#13133818—22no—
28#13133818C2×C414yes—
29#13133818D414yes—
30C2×Dic214—22no—
31C2×Dic214D412yes—
32Q8×D107—28no—
33Q8×D107—28no—
34C2×D428—22no—
35C2×D428D412yes—
36C2×C22⋊D107—22no—
37C2×C4×D107—24no—
38C2×C4×D107—22no—
39#13133802—22no—
40#13133802—22no—
41C2×C107⋊C8—22no—
42C2×Q8×C107—22no—
43C2×Q8×C107—22no—
44C2×Q8×C107C2×C412yes—
45C2×Q8×C107D412yes—
46C22×C4×C107—21no—
47C22×C4×C107—21no—
48C22×C4×C107D412yes—
49C22×C4×C107C2311yescharacteristic
50C4⋊C4×C107—22no—
51C42×C107—22no—
52C23×C107C2×D411yescharacteristic
53C22×D107—22no—
54Q8×C107—24no—
55Q8×C107—24no—
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—41no—
60C2×C4×C107—41no—
61C2×C4×C107—42no—
62C2×C4×C107—41no—
63C2×C4×C107C22⋊C414yes—
64C2×C4×C107C22×C411yescharacteristic
65C2×C4×C107C2×D412yes—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D411yescharacteristic
68C22⋊D107—24no—
69C22⋊D107—24no—
70C2×Dic107—22no—
71D428—28no—
72D428—22no—
73D428C22⋊C414yes—
74C4×D107—216no—
75C4×D107—24no—
76C4×D107—24no—
77Dic214—28no—
78Dic214—22no—
79Dic214C22⋊C414yes—
80C107⋊C8—42no—
81C22×C107—21no—
82C22×C107—21no—
83C22×C107C2×C22⋊C411yescharacteristic
84C22×C107#12011yescharacteristic
85C22×C107#12011yescharacteristic
86C2×D107—24no—
87C2×D107—42no—
88C2×D107—42no—
89C4×C107—24no—
90C4×C107—42no—
91C4×C107—81no—
92C4×C107C2×C22⋊C412yes—
93C4×C107#12012yes—
94Dic107—24no—
95C2×C107—21no—
96C2×C107—21no—
97C2×C107#37911yescharacteristic
98C2×C107#45411yescharacteristic
99C2×C107#45611yescharacteristic
100D107—44no—
101#1793—1071nomaximal, Sylow
102C107#179311yescharacteristic, complemented, Sylow
103C2×2−1+4—1071no—
104C2×#104—1072no—
105#417—1072no—
106C2×#101—1071no—
107#391D10711yescharacteristic, complemented
1082−1+4—1074no—
1092−1+4—2144no—
1102−1+4—2142no—
111C2×#41—1072no—
112C2×#41—2142no—
113C2×#41—2142no—
114C22×Q8—2142no—
115C22×Q8C2×D10711yescharacteristic
116C2×M16—1072no—
117C2×C4⋊C4C2×D10711yescharacteristic
118C2×C42C2×D10711yescharacteristic
119#104—2144no—
120#103—2142no—
121#102—2142no—
122#101—2142no—
123#95—22no—
124#41—1074no—
125#41—1074no—
126#41—2148no—
127#41—2148no—
128#41—2148no—
129#41—2148no—
130#41—2142no—
131#41—2142no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—1072no—
135C2×Q8—2142no—
136C2×Q8—2148no—
137C2×Q8—2148no—
138C2×Q8C4×D10712yes—
139C2×Q8D42812yes—
140C2×D4—1072no—
141C2×D4—2142no—
142C2×D4—2142no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—2144no—
146C22×C4—2142no—
147C22×C4C22⋊D10712yes—
148C22×C4C22×D10711yescharacteristic
149M16—2142no—
150M16—2142no—
151C2×C8—2142no—
152C4⋊C4—22no—
153C42—22no—
154C23—2142no—
155C23C2×C22⋊D10711yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—1074no—
159Q8—2148no—
160Q8—2142no—
161D4—1074no—
162D4—2148no—
163D4—2144no—
164D4—2144no—
165D4—2142no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—21416no—
174C2×C4—2142no—
175C2×C4—2144no—
176C2×C4—2144no—
177C2×C4(C2×D107)⋊C414yes—
178C2×C4C2×C4×D10711yescharacteristic
179C2×C4D4×D10712yes—
180C2×C4D4×D10712yes—
181C2×C4C2×D42811yescharacteristic
182C8—4282no—
183C22—21no—
184C22—21no—
185C22—4282no—
186C22—4282no—
187C22—2144no—
188C22C2×(C2×D107)⋊C411yescharacteristic
189C22#1681207711yescharacteristic
190C22#1681207711yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—2144no—
195C4C22⋊C4×D10712yes—
196C4#1681207912yes—
197C2—21no—
198C2—21no—
199C2—4284no—
200C2#2029248211yescharacteristic
201C2#2029279911yescharacteristic
202C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#391×C107C2
socleC22×C107#120
prime radicalC2×#16812157C2
cosocleC22×C4×C107C23
hypercenter#391D107
hypocenterC107#1793
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#16812037C2semidirect
3. #20292626C2semidirect
4. C2×#16812028C2semidirect
5. #391×C107C2semidirect
8. #16812157C4semidirect
102. C107#1793semidirect
107. #391D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121572
C2×#168120372
#202926262
C2×#168120282
#391×C1072
#1793107