Finite Group Databasebeta

← #28347922

Subgroups — #28347922

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#20292923C212yesmaximal, complemented
3#20292908C212yesmaximal, complemented
4#20292873C212yesmaximal, complemented
5#20292868C211yesmaximal, characteristic, complemented
6#20292863C211yesmaximal, characteristic
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#16812043C2211yescharacteristic
24#16812043C2212yes—
25#16812043C2211yescharacteristic
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×C107D411yescharacteristic
61C4⋊C4×C107D411yescharacteristic
62C22⋊C4×C107D412yes—
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—21no—
83C8×C107—21no—
84C8×C107—22no—
85C22×C107—21no—
86C22×C107—21no—
87C22×C107—21no—
88C22×C107C22×D411yescharacteristic
89C2×D107—42no—
90C2×D107—42no—
91C2×D107—42no—
92C4×C107—21no—
93C4×C107—22no—
94C4×C107—22no—
95C4×C107H(5,𝔽2)11yescharacteristic
96C4×C1072−1+411yescharacteristic
97Dic107—42no—
98Dic107—42no—
99Dic107—42no—
100C2×C107—41no—
101C2×C107#53711yescharacteristic
102C2×C107#57611yescharacteristic
103C2×C107#57811yescharacteristic
104D107—82no—
105#3262—1071nomaximal, Sylow
106C107#326211yescharacteristic, complemented, Sylow
107#542—1072no—
108#477—1072no—
109#474—1072no—
110#468—1072no—
111#467—1071no—
112#466—1071no—
113#464—1072no—
114#463—1072no—
115#424D10711yescharacteristic, complemented
116C2×Q16—1071no—
117C2×SD16—1072no—
118C2×D8—1071no—
119C2×M16C2×D10711yescharacteristic
120C22×C8C2×D10711yescharacteristic
121C4⋊Q8—1072no—
122#126—2142no—
123#124—1072no—
124#124—2142no—
125#123—2142no—
126C22⋊Q8—1072no—
127C22⋊Q8—2142no—
128#121—1072no—
129C4×Q8—1072no—
130C4×D4—1072no—
131#117C2×D10711yescharacteristic
132#107—1071no—
133#107—1071no—
134#107—1072no—
135C4⋊C8C2×D10712yes—
136C4⋊C8C2×D10712yes—
137#103—1072no—
138#103—1072no—
139#102—1072no—
140#102—1072no—
141C2×Q8—1072no—
142C2×Q8—2142no—
143C2×D4—1072no—
144C2×D4—2142no—
145C22×C4—2142no—
146C22×C4C22×D10711yescharacteristic
147Q16—4281no—
148SD16—4282no—
149D8—4281no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D10711yescharacteristic
153C2×C8C22×D10712yes—
154C2×C8C22×D10711yescharacteristic
155C4⋊C4—1072no—
156C4⋊C4—1072no—
157C4⋊C4—2142no—
158C4⋊C4—2142no—
159C4⋊C4—2142no—
160C4⋊C4D42811yescharacteristic
161C4⋊C4D42811yescharacteristic
162C22⋊C4—2142no—
163C22⋊C4—2142no—
164C22⋊C4—2142no—
165C22⋊C4—2142no—
166C22⋊C4—2142no—
167C22⋊C4D42812yes—
168C42—2142no—
169C42C22×D10712yes—
170C23—2142no—
171C23C2×D42811yescharacteristic
172Q8—8562no—
173Q8—2142no—
174Q8—2142no—
175D4—8562no—
176D4—2142no—
177D4—2142no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—8562no—
181C2×C4—2142no—
182C2×C4—2142no—
183C2×C4—2142no—
184C2×C4C2×D42812yes—
185C2×C4C2×D42812yes—
186C2×C4C2×D42811yescharacteristic
187C2×C4C23×D10711yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—4282no—
195C22—4282no—
196C22—4282no—
197C22C22×D42811yescharacteristic
198C4—22no—
199C4—22no—
200C4—21no—
201C4—4282no—
202C4—4282no—
203C4—4282no—
204C4#1681215411yescharacteristic
205C4#1681215711yescharacteristic
206C2—41no—
207C2—8562no—
208C2#2029320811yescharacteristic
209C2#2029343211yescharacteristic
210C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
prime radical#20292863C2
hypercenter#424D107
hypocenterC107#3262
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. #20292923C2semidirect
3. #20292908C2semidirect
4. #20292873C2semidirect
5. #20292868C2semidirect
7. #20292850C2semidirect
8. #20292843C2semidirect
9. #424×C107C2semidirect
106. C107#3262semidirect
115. #424D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202929232
#202929082
#202928732
#202928682
#202928632
#202928502
#202928432
#424×C1072
#3262107