Finite Group Databasebeta

← #28335396

Subgroups — #28335396

Order 13696.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#20292760C212yesmaximal, complemented
2#20292544C211yesmaximal, characteristic, complemented
3#20292361C212yesmaximal, complemented
4#20292353C211yesmaximal, characteristic, complemented
5#537×C107C211yesmaximal, characteristic, complemented
6C22×D428C2211yescharacteristic
7C2×#13133803—22no—
8C2×(C2×D107)⋊C4—21no—
9C2×(C2×D107)⋊C4—21no—
10#16812029—22no—
11C22⋊(C107⋊C8)C2212yes—
12C107⋊#95C2211yescharacteristic
13C2×#41×C107—21no—
14C22⋊Q8×C107—22no—
15#121×C107—22no—
16#121×C107—21no—
17#121×C107C2212yes—
18C2×C4⋊C4×C107C2211yescharacteristic
19C23×D107—21no—
20C2×D428—22no—
21C2×D428—42no—
22C22×Dic107—21no—
23#13133803—44no—
24C2×C107⋊C8—22no—
25(C2×D107)⋊C4—42no—
26(C2×D107)⋊C4—42no—
27#41×C107—42no—
28#41×C107—42no—
29C2×Q8×C107—21no—
30C2×D4×C107—21no—
31C2×D4×C107—22no—
32C2×D4×C107—21no—
33C2×D4×C107D412yes—
34C22×C4×C107—22no—
35C22×C4×C107D411yescharacteristic
36C22×C4×C107D411yescharacteristic
37C22×C4×C107C2311yescharacteristic
38C4⋊C4×C107—21no—
39C4⋊C4×C107—22no—
40C4⋊C4×C107D412yes—
41C22⋊C4×C107—22no—
42C22⋊C4×C107—22no—
43C23×C107—22no—
44C23×C107C2×D411yescharacteristic
45C22×D107—22no—
46C22×D107—41no—
47C22×D107—41no—
48C22×D107—41no—
49C22×D107—41no—
50C22×D107—41no—
51C22×D107—41no—
52Q8×C107—42no—
53D4×C107—42no—
54D4×C107—42no—
55D4×C107—42no—
56D4×C107—42no—
57D4×C107—81no—
58C2×C4×C107—21no—
59C2×C4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—22no—
62C2×C4×C107—41no—
63C2×C4×C107—44no—
64C2×C4×C107—41no—
65C2×C4×C107D812yescomplemented
66C2×C4×C107C2×D412yes—
67C2×Dic107—41no—
68C2×Dic107—41no—
69C2×Dic107—41no—
70D428—24no—
71D428—42no—
72C107⋊C8—42no—
73C22×C107—21no—
74C22×C107—21no—
75C22×C107—42no—
76C22×C107—42no—
77C22×C107—42no—
78C22×C107#12011yescharacteristic
79C22×C107C2×D811yescharacteristic
80C22×C107AGL(1,ℤ/8)11yescharacteristic
81C2×D107—42no—
82C2×D107—42no—
83C2×D107—42no—
84C2×D107—81no—
85C2×D107—81no—
86C2×D107—81no—
87C2×D107—81no—
88C4×C107—22no—
89C4×C107—22no—
90C4×C107—42no—
91C4×C107—41no—
92Dic107—81no—
93C2×C107—21no—
94C2×C107—21no—
95C2×C107—82no—
96C2×C107#44711yescharacteristic
97C2×C107#45411yescharacteristic
98C2×C107B(4,𝔽2)11yescharacteristic
99D107—82no—
100#1598—1071nomaximal, Sylow
101C107#159811yescharacteristic, complemented, Sylow
102#537D10711yescharacteristic, complemented
103#447—1072no—
104#394—1071no—
105#327—1072no—
106#324—1071no—
107C2×#41—21no—
108C22×D4—1071no—
109C2×D8—2142no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D10712yes—
114C2×C4⋊C4C2×D10711yescharacteristic
115C2×C22⋊C4—2141no—
116C2×C22⋊C4—2141no—
117#102—2142no—
118C22⋊C8—1072no—
119#95—1071no—
120C24—2141no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—4282no—
128C2×D4—2142no—
129C2×D4C22⋊D10712yes—
130C22×C4—22no—
131C22×C4—2141no—
132C22×C4C22⋊D10711yescharacteristic
133C22×C4C22⋊D10711yescharacteristic
134C22×C4C22×D10711yescharacteristic
135D8—4284no—
136C2×C8—2142no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D10712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—4282no—
143C22⋊C4—4282no—
144C23—22no—
145C23—4281no—
146C23—4281no—
147C23—4281no—
148C23—4281no—
149C23—4281no—
150C23—4281no—
151C23—2142no—
152C23C2×C22⋊D10711yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—4282no—
160D4—2144no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—4281no—
169C2×C4—4281no—
170C2×C4—4281no—
171C2×C4#1313380312yescomplemented
172C2×C4C2×C22⋊D10712yes—
173C8—4282no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—8561no—
180C22—8561no—
181C22—8561no—
182C22—8561no—
183C22—4282no—
184C22—4282no—
185C22—4282no—
186C22#1681207811yescharacteristic
187C22C2×#1313380311yescharacteristic
188C22#1681213511yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—8561no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—8562no—
198C2#2029276111yescharacteristic
199C2#2029280011yescharacteristic
200C2#2029281411yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#537×C107C2
socleC22×C107#120
hypercenter#537D107
hypocenterC107#1598
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20292760C2semidirect
2. #20292544C2semidirect
3. #20292361C2semidirect
4. #20292353C2semidirect
5. #537×C107C2semidirect
65. C2×C4×C107D8semidirect
101. C107#1598semidirect
102. #537D107semidirect
171. C2×C4#13133803semidirect

Maximal subgroups

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

subgroupindex
#202927602
#202925442
#202923612
#202923532
#537×C1072
#1598107