Finite Group Databasebeta

← #28347927

Subgroups — #28347927

Order 13696.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#20293250C212yesmaximal, complemented
2#20292916C212yesmaximal, complemented
3#20292910C212yesmaximal, complemented
4#20292880C211yesmaximal, characteristic
5#20292858C212yesmaximal, complemented
6#20292851C212yesmaximal, complemented
7#20292845C212yesmaximal, complemented
8#20292655C211yesmaximal, characteristic, complemented
9#470×C107C211yesmaximal, characteristic, complemented
10C2×#13133805C2212yes—
11C2×D856C2211yescharacteristic
12C22×C107⋊C8C2211yescharacteristic
13#16812108C2212yes—
14#16812104—22no—
15#16812098C2212yes—
16#16812095—22no—
17#16812091—22no—
18#16812089—22no—
19#16812083C2212yes—
20Q8×Dic107C2212yes—
21#16812071C2212yes—
22#16812068C2211yescharacteristic
23#16812041C2211yescharacteristic
24#16812040C2212yes—
25Dic107⋊C8C2212yes—
26Dic107⋊C8C2212yes—
27#16812033C2212yes—
28#16812031C2212yes—
29#16812029C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C2×M16×C107C2211yescharacteristic
32C22⋊Q8×C107C2212yes—
33#106×C107C2211yescharacteristic
34#103×C107C2212yes—
35C2×Dic214—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133805—42no—
40D856—41no—
41C2×C107⋊C8—41no—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107—22no—
44C4⋊Dic107D412yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—22no—
47C22⋊Dic107D411yescharacteristic
48C22⋊Dic107D411yescharacteristic
49(C2×D107)⋊C4—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53C4×Dic107—22no—
54C4×Dic107C2312yes—
55C2×Q8×C107C2312yes—
56C22×C4×C107C2311yescharacteristic
57Q16×C107—41no—
58M16×C107—41no—
59C2×C8×C107C2311yescharacteristic
60C2×C8×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107C2312yes—
63C22⋊C4×C107—22no—
64C23×C107C2×D411yescharacteristic
65C22×D107—22no—
66Q8×C107—22no—
67Q8×C107—22no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2411yescharacteristic
74C22⋊D107—82no—
75C2×Dic107—22no—
76C2×Dic107C2×D412yes—
77C2×Dic107C2×D412yes—
78D428—22no—
79D428—22no—
80C4×D107—82no—
81Dic214—82no—
82C8×C107—21no—
83C8×C107—21no—
84C107⋊C8—22no—
85C22×C107—21no—
86C22×C107—22no—
87C22×C107C22×D411yescharacteristic
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—42no—
91C4×C107—21no—
92C4×C107—42no—
93C4×C107—42no—
94C4×C107H(5,𝔽2)11yescharacteristic
95C4×C1072−1+411yescharacteristic
96Dic107—22no—
97Dic107—22no—
98Dic107—42no—
99C2×C107—41no—
100C2×C107#53711yescharacteristic
101C2×C107#57712yes—
102D107—82no—
103#3263—1071nomaximal, Sylow
104C107#326311yescharacteristic, complemented, Sylow
105#542—1072no—
106#476—1072no—
107#475—1072no—
108#470D10711yescharacteristic, complemented
109#469—1071no—
110#465—1072no—
111#464—1072no—
112#463—1072no—
113#424—1071no—
114C2×Q16C2×D10711yescharacteristic
115C2×SD16—1072no—
116C2×D8—1071no—
117C2×M16C2×D10711yescharacteristic
118C22×C8—1071no—
119C4⋊Q8—1072no—
120#126—2142no—
121#124—1072no—
122#124—2142no—
123#123—2142no—
124C22⋊Q8—2142no—
125C22⋊Q8C2×D10712yes—
126#121—1072no—
127C4×Q8—1072no—
128C4×D4—1072no—
129#117—1071no—
130#106—1071no—
131#106—1072no—
132#106C2×D10711yescharacteristic
133C4⋊C8—1072no—
134C4⋊C8—1072no—
135#103—1072no—
136#103C2×D10712yes—
137#102—1072no—
138#102—1072no—
139C2×Q8—2142no—
140C2×Q8C22×D10712yes—
141C2×D4—1072no—
142C2×D4—2142no—
143C22×C4—2142no—
144C22×C4C22×D10711yescharacteristic
145Q16—41no—
146SD16—4282no—
147D8—4281no—
148M16—41no—
149C2×C8—1072no—
150C2×C8—4281no—
151C2×C8C22×D10711yescharacteristic
152C2×C8C22×D10711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—1072no—
155C4⋊C4—1072no—
156C4⋊C4—2142no—
157C4⋊C4—2142no—
158C4⋊C4C22×D10712yes—
159C22⋊C4—22no—
160C22⋊C4—1071no—
161C22⋊C4—1071no—
162C22⋊C4—2142no—
163C22⋊C4—2142no—
164C22⋊C4—2142no—
165C22⋊C4—2142no—
166C42—1072no—
167C42—2142no—
168C23—2142no—
169C23D4×D10711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—8562no—
173D4—8562no—
174D4—2142no—
175D4—2142no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—8562no—
181C2×C4—1072no—
182C2×C4—1072no—
183C2×C4—2142no—
184C2×C4D4×D10711yescharacteristic
185C2×C4C23×D10711yescharacteristic
186C8—21no—
187C8—21no—
188C8—2142no—
189C22—22no—
190C22—21no—
191C22—4282no—
192C22—4282no—
193C22—4282no—
194C22C2×D4×D10711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—4282no—
199C4—2142no—
200C4—2142no—
201C4#1681215411yescharacteristic
202C4#1681215611yescharacteristic
203C2—41no—
204C2—8562no—
205C2#2029320411yescharacteristic
206C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
prime radical#20292880C2
hypercenter#470D107
hypocenterC107#3263
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #20293250C2semidirect
2. #20292916C2semidirect
3. #20292910C2semidirect
5. #20292858C2semidirect
6. #20292851C2semidirect
7. #20292845C2semidirect
8. #20292655C2semidirect
9. #470×C107C2semidirect
104. C107#3263semidirect
108. #470D107semidirect

Maximal subgroups

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

subgroupindex
#202932502
#202929162
#202929102
#202928802
#202928582
#202928512
#202928452
#202926552
#470×C1072
#3263107