Finite Group Databasebeta

← #28347932

Subgroups — #28347932

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#20293248C212yesmaximal, complemented
2#20292918C212yesmaximal, complemented
3#20292913C212yesmaximal, complemented
4#20292885C211yesmaximal, characteristic, complemented
5#20292880C211yesmaximal, characteristic
6#20292856C212yesmaximal, complemented
7#20292850C212yesmaximal, complemented
8#20292843C212yesmaximal, complemented
9#424×C107C211yesmaximal, characteristic, complemented
10C2×Dic428C2211yescharacteristic
11C2×#13133807C2212yes—
12C2×D856C2211yescharacteristic
13#16812109C2212yes—
14#16812104—22no—
15#16812097C2212yes—
16#16812095—22no—
17#16812091—22no—
18#16812088C2212yes—
19#16812087—22no—
20#16812083C2212yes—
21C4×Dic214C2212yes—
22C4×D428C2212yes—
23#16812041C2211yescharacteristic
24#16812041C2211yescharacteristic
25#16812041C2212yes—
26#16812034C2212yes—
27#16812034C2212yes—
28#16812031C2212yes—
29#16812031C2212yes—
30C2×M16×C107C2211yescharacteristic
31C22×C8×C107C2211yescharacteristic
32#117×C107C2211yescharacteristic
33C4⋊C8×C107C2212yes—
34C4⋊C8×C107C2212yes—
35C2×Dic214—22no—
36C2×Dic214C2312yes—
37C2×D428C2312yes—
38C2×C22⋊D107—22no—
39C2×C4×D107—22no—
40Dic428—41no—
41#13133807—42no—
42D856—41no—
43C4⋊Dic107—22no—
44C4⋊Dic107C2312yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—22no—
47Dic107⋊C4—22no—
48C22⋊Dic107—22no—
49(C2×D107)⋊C4—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53C4×Dic107—22no—
54C22×C4×C107C2311yescharacteristic
55M16×C107—41no—
56C2×C8×C107—41no—
57C2×C8×C107C2311yescharacteristic
58C2×C8×C107C2312yes—
59C2×C8×C107C2311yescharacteristic
60C4⋊C4×C107D412yes—
61C22⋊C4×C107D411yescharacteristic
62C22⋊C4×C107D411yescharacteristic
63C42×C107C2312yes—
64C23×C107C2×D411yescharacteristic
65C22×D107—22no—
66C2×C4×C107—21no—
67C2×C4×C107—21no—
68C2×C4×C107C2×D412yes—
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2411yescharacteristic
72C22⋊D107—82no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76D428—22no—
77D428—22no—
78C4×D107—82no—
79Dic214—22no—
80Dic214—22no—
81Dic214—82no—
82C8×C107—22no—
83C8×C107—21no—
84C8×C107—21no—
85C22×C107—21no—
86C22×C107—22no—
87C22×C107C22×D411yescharacteristic
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—42no—
91C4×C107—21no—
92C4×C107—22no—
93C4×C107—22no—
94C4×C107H(5,𝔽2)11yescharacteristic
95C4×C1072−1+411yescharacteristic
96Dic107—42no—
97Dic107—42no—
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#470—1071no—
109#469—1071no—
110#465—1072no—
111#464—1072no—
112#463—1072no—
113#424D10711yescharacteristic, complemented
114C2×Q16—1071no—
115C2×SD16—1072no—
116C2×D8—1071no—
117C2×M16C2×D10711yescharacteristic
118C22×C8C2×D10711yescharacteristic
119C4⋊Q8—1072no—
120#126—2142no—
121#124—1072no—
122#124—2142no—
123#123—2142no—
124C22⋊Q8—1072no—
125C22⋊Q8—2142no—
126#121—1072no—
127C4×Q8—1072no—
128C4×D4—1072no—
129#117C2×D10711yescharacteristic
130#106—1072no—
131#106—1071no—
132#106—1071no—
133C4⋊C8C2×D10712yes—
134C4⋊C8C2×D10712yes—
135#103—1072no—
136#103—1072no—
137#102—1072no—
138#102—1072no—
139C2×Q8—1072no—
140C2×Q8—2142no—
141C2×D4—1072no—
142C2×D4—2142no—
143C22×C4—2142no—
144C22×C4C22×D10711yescharacteristic
145Q16—4281no—
146SD16—4282no—
147D8—4281no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D10711yescharacteristic
151C2×C8C22×D10712yes—
152C2×C8C22×D10711yescharacteristic
153C4⋊C4—1072no—
154C4⋊C4—1072no—
155C4⋊C4—2142no—
156C4⋊C4—2142no—
157C4⋊C4—2142no—
158C4⋊C4D42812yes—
159C22⋊C4—2142no—
160C22⋊C4—2142no—
161C22⋊C4—2142no—
162C22⋊C4—2142no—
163C22⋊C4—2142no—
164C22⋊C4D42811yescharacteristic
165C22⋊C4D42811yescharacteristic
166C42—2142no—
167C42C22×D10712yes—
168C23—2142no—
169C23C2×D42811yescharacteristic
170Q8—8562no—
171Q8—2142no—
172Q8—2142no—
173D4—8562no—
174D4—2142no—
175D4—2142no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—8562no—
179C2×C4—2142no—
180C2×C4—2142no—
181C2×C4—2142no—
182C2×C4C2×D42812yes—
183C2×C4C2×D42812yes—
184C2×C4C2×D42811yescharacteristic
185C2×C4C23×D10711yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—4282no—
192C22—4282no—
193C22—4282no—
194C22C22×D42811yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—4282no—
199C4—4282no—
200C4—4282no—
201C4#1681215411yescharacteristic
202C4#1681215711yescharacteristic
203C2—41no—
204C2—8562no—
205C2#2029320811yescharacteristic
206C2#2029343912yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
prime radical#20292880C2
hypercenter#424D107
hypocenterC107#3263
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #20293248C2semidirect
2. #20292918C2semidirect
3. #20292913C2semidirect
4. #20292885C2semidirect
6. #20292856C2semidirect
7. #20292850C2semidirect
8. #20292843C2semidirect
9. #424×C107C2semidirect
104. C107#3263semidirect
113. #424D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202929182
#202929132
#202928852
#202928802
#202928562
#202928502
#202928432
#424×C1072
#3263107