Finite Group Databasebeta

← #28347908

Subgroups — #28347908

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#20293239C211yesmaximal, characteristic, complemented
2#20293232C211yesmaximal, characteristic
3#20292935C212yesmaximal, complemented
4#20292928C212yesmaximal, complemented
5#20292873C212yesmaximal, complemented
6#20292856C212yesmaximal, complemented
7#20292833C212yesmaximal, complemented
8#20292828C212yesmaximal, complemented
9#424×C107C211yesmaximal, characteristic, complemented
10C2×#13133807C2212yes—
11C2×#13133807C2212yes—
12#16812109C2211yescharacteristic
13#16812106C2211yescharacteristic
14#16812102—22no—
15#16812101C2211yescharacteristic
16#16812101C2211yescharacteristic
17#16812099—22no—
18#16812094—22no—
19#16812088C2212yes—
20#16812083C2212yes—
21#16812082—22no—
22C4×Dic214C2212yes—
23C4×D428C2212yes—
24#16812043C2212yes—
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×Dic214C2312yes—
36C2×D428—22no—
37C2×D428C2312yes—
38C2×C22⋊D107—22no—
39C2×C4×D107—22no—
40#13133807—42no—
41#13133807—42no—
42C4⋊Dic107—22no—
43C4⋊Dic107C2312yes—
44C4⋊Dic107C2312yes—
45Dic107⋊C4—22no—
46Dic107⋊C4—22no—
47Dic107⋊C4—22no—
48Dic107⋊C4—22no—
49C22⋊Dic107—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4—22no—
52C4×Dic107—22no—
53C22×C4×C107C2311yescharacteristic
54M16×C107—41no—
55C2×C8×C107—41no—
56C2×C8×C107C2312yes—
57C2×C8×C107C2312yes—
58C4⋊C4×C107D412yes—
59C22⋊C4×C107D412yes—
60C42×C107C2311yescharacteristic
61C42×C107C2311yescharacteristic
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64C2×C4×C107—21no—
65C2×C4×C107—21no—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D412yes—
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2411yescharacteristic
70C22⋊D107—82no—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74D428—22no—
75D428—22no—
76D428—82no—
77C4×D107—82no—
78Dic214—22no—
79Dic214—22no—
80C8×C107—22no—
81C8×C107—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×C107H(5,𝔽2)11yescharacteristic
92C4×C1072−1+411yescharacteristic
93Dic107—42no—
94Dic107—42no—
95Dic107—42no—
96C2×C107—41no—
97C2×C107#53711yescharacteristic
98C2×C107#57712yes—
99D107—82no—
100#3260—1071nomaximal, Sylow
101C107#326011yescharacteristic, complemented, Sylow
102#541—1071no—
103#540—1071no—
104#479—1072no—
105#478—1072no—
106#468—1072no—
107#465—1072no—
108#461—1072no—
109#460—1072no—
110#424D10711yescharacteristic, complemented
111C2×SD16—1072no—
112C2×SD16—1072no—
113C2×M16C2×D10711yescharacteristic
114C22×C8C2×D10711yescharacteristic
115C4⋊Q8—1071no—
116#127—1071no—
117#126—2142no—
118#125—1071no—
119#125—1071no—
120#125—2142no—
121#123—2142no—
122C22⋊Q8—1072no—
123#121—1072no—
124#121—2142no—
125C4×Q8—1072no—
126C4×D4—1072no—
127#117C2×D10711yescharacteristic
128#107—1072no—
129#106—1072no—
130C4⋊C8C2×D10712yes—
131C4⋊C8C2×D10712yes—
132#103—1072no—
133#103—1072no—
134#102—1072no—
135#102—1072no—
136C2×Q8—1072no—
137C2×D4—1072no—
138C2×D4—2142no—
139C2×D4—2142no—
140C22×C4—2142no—
141C22×C4C22×D10711yescharacteristic
142SD16—4282no—
143SD16—4282no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D10712yes—
147C2×C8C22×D10712yes—
148C4⋊C4—1072no—
149C4⋊C4—1072no—
150C4⋊C4—2142no—
151C4⋊C4—2142no—
152C4⋊C4—2142no—
153C4⋊C4—2142no—
154C4⋊C4—2142no—
155C4⋊C4D42812yes—
156C22⋊C4—2142no—
157C22⋊C4—2142no—
158C22⋊C4—2142no—
159C22⋊C4D42812yes—
160C42—2142no—
161C42C22×D10711yescharacteristic
162C42C22×D10711yescharacteristic
163C23—2142no—
164C23C2×D42811yescharacteristic
165Q8—2142no—
166Q8—2142no—
167D4—8562no—
168D4—8562no—
169D4—2142no—
170D4—2142no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—8562no—
174C2×C4—2142no—
175C2×C4—2142no—
176C2×C4—2142no—
177C2×C4C2×D42812yes—
178C2×C4C2×D42812yes—
179C2×C4C2×D42811yescharacteristic
180C2×C4C23×D10711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—4282no—
186C22—4282no—
187C22—4282no—
188C22C22×D42811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—4282no—
193C4—4282no—
194C4—4282no—
195C4#1681215411yescharacteristic
196C4#1681215711yescharacteristic
197C2—41no—
198C2—8562no—
199C2#2029320811yescharacteristic
200C2#2029343912yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
prime radical#20293232C2
hypercenter#424D107
hypocenterC107#3260
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. #20293239C2semidirect
3. #20292935C2semidirect
4. #20292928C2semidirect
5. #20292873C2semidirect
6. #20292856C2semidirect
7. #20292833C2semidirect
8. #20292828C2semidirect
9. #424×C107C2semidirect
101. C107#3260semidirect
110. #424D107semidirect

Maximal subgroups

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

subgroupindex
#202932392
#202932322
#202929352
#202929282
#202928732
#202928562
#202928332
#202928282
#424×C1072
#3260107