Finite Group Databasebeta

← #28347902

Subgroups — #28347902

Order 13696.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293232C211yesmaximal, characteristic
2#20292934C212yesmaximal, complemented
3#20292924C212yesmaximal, complemented
4#20292875C212yesmaximal, complemented
5#20292858C212yesmaximal, complemented
6#20292829C212yesmaximal, complemented
7#20292827C212yesmaximal, complemented
8#20292654C211yesmaximal, characteristic, complemented
9#541×C107C211yesmaximal, characteristic, complemented
10C2×#13133805C2212yes—
11C2×#13133805C2212yes—
12C2×#13133802C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812106C2211yescharacteristic
15#16812101C2211yescharacteristic
16#16812094—22no—
17#16812083C2212yes—
18#16812082—22no—
19C4×D428C2212yes—
20#16812042C2212yes—
21#16812040C2212yes—
22C4⋊(C107⋊C8)C2212yes—
23C4⋊(C107⋊C8)C2212yes—
24#16812033C2212yes—
25#16812033C2212yes—
26#16812029C2212yes—
27#16812029C2212yes—
28C4⋊Q8×C107C2211yescharacteristic
29#126×C107—22no—
30#125×C107—22no—
31#125×C107C2211yescharacteristic
32C22⋊Q8×C107C2212yes—
33C4×Q8×C107C2212yes—
34#117×C107C2211yescharacteristic
35C2×D428—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133805—42no—
40#13133805—42no—
41#13133802—41no—
42C2×C107⋊C8—41no—
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—22no—
47(C2×D107)⋊C4—22no—
48(C2×D107)⋊C4—22no—
49C2×Q8×C107C2312yes—
50C22×C4×C107C2311yescharacteristic
51C4⋊C4×C107—22no—
52C4⋊C4×C107—22no—
53C4⋊C4×C107—22no—
54C4⋊C4×C107—22no—
55C4⋊C4×C107D412yes—
56C4⋊C4×C107C2312yes—
57C22⋊C4×C107—22no—
58C22⋊C4×C107D412yes—
59C42×C107—22no—
60C42×C107C2311yescharacteristic
61C42×C107C2311yescharacteristic
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64Q8×C107—22no—
65Q8×C107—22no—
66C2×C4×C107—21no—
67C2×C4×C107—21no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D412yes—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2411yescharacteristic
74C22⋊D107—82no—
75C2×Dic107—22no—
76D428—22no—
77D428—22no—
78D428—82no—
79C4×D107—82no—
80C107⋊C8—22no—
81C107⋊C8—22no—
82C22×C107—21no—
83C22×C107—22no—
84C22×C107C22×D411yescharacteristic
85C2×D107—42no—
86C2×D107—42no—
87C2×D107—42no—
88C4×C107—21no—
89C4×C107—22no—
90C4×C107—22no—
91C4×C107—42no—
92C4×C107—42no—
93C4×C107H(5,𝔽2)11yescharacteristic
94C4×C1072−1+411yescharacteristic
95Dic107—42no—
96C2×C107—41no—
97C2×C107#53711yescharacteristic
98C2×C107#57712yes—
99D107—82no—
100#3260—1071nomaximal, Sylow
101C107#326011yescharacteristic, complemented, Sylow
102#541D10711yescharacteristic, complemented
103#540—1071no—
104#479—1072no—
105#478—1072no—
106#468—1072no—
107#465—1072no—
108#461—1072no—
109#460—1072no—
110#424—1071no—
111C2×SD16—1072no—
112C2×SD16—1072no—
113C2×M16—1071no—
114C22×C8—1071no—
115C4⋊Q8C2×D10711yescharacteristic
116#127—1071no—
117#126—22no—
118#125—22no—
119#125—1071no—
120#125C2×D10711yescharacteristic
121#123—2142no—
122C22⋊Q8C2×D10712yes—
123#121—1072no—
124#121—2142no—
125C4×Q8C2×D10712yes—
126C4×D4—1072no—
127#117C2×D10711yescharacteristic
128#107—1072no—
129#106—1072no—
130C4⋊C8—1072no—
131C4⋊C8—1072no—
132#103—1072no—
133#103—1072no—
134#102—1072no—
135#102—1072no—
136C2×Q8C22×D10712yes—
137C2×D4—1072no—
138C2×D4—2142no—
139C2×D4—2142no—
140C22×C4—2142no—
141C22×C4C22×D10711yescharacteristic
142SD16—4282no—
143SD16—4282no—
144M16—4281no—
145C2×C8—1072no—
146C2×C8—1072no—
147C2×C8—4281no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1072no—
153C4⋊C4—2142no—
154C4⋊C4C22⋊D10712yes—
155C4⋊C4C22×D10712yes—
156C22⋊C4—22no—
157C22⋊C4—2142no—
158C22⋊C4—2142no—
159C22⋊C4C22⋊D10712yes—
160C42—22no—
161C42C22×D10711yescharacteristic
162C42C22×D10711yescharacteristic
163C23—2142no—
164C23C2×C22⋊D10711yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—8562no—
168D4—8562no—
169D4—2142no—
170D4—2142no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—8562no—
176C2×C4—2142no—
177C2×C4C2×C22⋊D10712yes—
178C2×C4C2×C22⋊D10712yes—
179C2×C4C2×C22⋊D10711yescharacteristic
180C2×C4C23×D10711yescharacteristic
181C8—2142no—
182C8—2142no—
183C22—22no—
184C22—21no—
185C22—4282no—
186C22—4282no—
187C22—4282no—
188C22C22×C22⋊D10711yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—4282no—
195C4#1681215511yescharacteristic
196C4#1681215611yescharacteristic
197C2—41no—
198C2—8562no—
199C2#2029320511yescharacteristic
200C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#541×C107C2
socleC22×C107C22×D4
prime radical#20293232C2
hypercenter#541D107
hypocenterC107#3260
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20292934C2semidirect
3. #20292924C2semidirect
4. #20292875C2semidirect
5. #20292858C2semidirect
6. #20292829C2semidirect
7. #20292827C2semidirect
8. #20292654C2semidirect
9. #541×C107C2semidirect
101. C107#3260semidirect
102. #541D107semidirect

Maximal subgroups

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

subgroupindex
#202932322
#202929342
#202929242
#202928752
#202928582
#202928292
#202928272
#202926542
#541×C1072
#3260107