Finite Group Databasebeta

← #28349034

Subgroups — #28349034

Order 13696.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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#20293315C212yesmaximal
2#20292970C212yesmaximal, complemented
3#20292964C212yesmaximal, complemented
4#20292931C212yesmaximal, complemented
5#20292886C212yesmaximal, complemented
6C4×Dic428C211yesmaximal, characteristic, complemented
7#20292706C211yesmaximal, characteristic
8#20292666C212yesmaximal, complemented
9#499×C107C211yesmaximal, characteristic, complemented
10C2×Dic428C2211yescharacteristic
11C2×#13133801C2212yes—
12#16812104—22no—
13#16812100—22no—
14#16812094—22no—
15#16812089C2212yes—
16#16812089C2212yes—
17C4×Dic214C2212yes—
18C4×D428C2211yescharacteristic
19#16812067—22no—
20#16812067C2212yes—
21C4⋊C4×D107C2212yes—
22#16812043C2211yescharacteristic
23#16812042C2212yes—
24#16812040C2212yes—
25C4⋊(C107⋊C8)C2211yescharacteristic
26#16812034C2212yes—
27#16812032C2212yes—
28#16812032C2212yes—
29(C2×D107)⋊C8C2212yes—
30#125×C107C2212yes—
31#107×C107C2211yescharacteristic
32#107×C107C2211yescharacteristic
33#106×C107C2212yes—
34C4×C8×C107C2211yescharacteristic
35C2×Dic214C2312yes—
36C2×D428D411yescharacteristic
37C2×C4×D107—22no—
38C2×C4×D107C2312yes—
39Dic428—22no—
40#13133801—24no—
41C2×C107⋊C8C2312yes—
42C4⋊Dic107—22no—
43C4⋊Dic107—22no—
44C4⋊Dic107D411yescharacteristic
45C4⋊Dic107C2312yes—
46Dic107⋊C4—22no—
47Dic107⋊C4—22no—
48(C2×D107)⋊C4—22no—
49(C2×D107)⋊C4—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4D412yes—
52C4×Dic107—22no—
53C4×Dic107—22no—
54C2×C8×C107C2311yescharacteristic
55C2×C8×C107C2311yescharacteristic
56C4⋊C4×C107—22no—
57C4⋊C4×C107—22no—
58C4⋊C4×C107C2312yes—
59C4⋊C4×C107C2312yes—
60C42×C107C2311yescharacteristic
61C22×D107C2×D412yes—
62C2×C4×C107—22no—
63C2×C4×C107—22no—
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2411yescharacteristic
67C2×Dic107—22no—
68C2×Dic107—22no—
69C2×Dic107C2×D412yes—
70D428—41no—
71C4×D107—22no—
72C4×D107—22no—
73C4×D107—44no—
74Dic214—22no—
75Dic214—22no—
76C8×C107—21no—
77C8×C107#4112yes—
78C107⋊C8—22no—
79C22×C107C22×D411yescharacteristic
80C2×D107—22no—
81C2×D107—22no—
82C2×D107—22no—
83C4×C107—21no—
84C4×C107—21no—
85C4×C107—42no—
86C4×C107—42no—
87C4×C107C2×#4111yescharacteristic
88C4×C1072−1+411yescharacteristic
89Dic107—22no—
90Dic107—42no—
91Dic107—42no—
92C2×C107#54711yescharacteristic
93C2×C107#57511yescharacteristic
94C2×C107#57811yescharacteristic
95D107—42no—
96#3352—1071nomaximal, Sylow
97C107#335211yescharacteristic, complemented, Sylow
98#552—1072no—
99#499D10711yescharacteristic, complemented
100#485—1072no—
101#484—1072no—
102#479—1072no—
103#470—1072no—
104C4×Q16—1071no—
105#436—1071no—
106#428—1072no—
107C2×Q16—1071no—
108C2×M16—1072no—
109#126—2142no—
110#125—2142no—
111#125C2×D10712yes—
112#123—2142no—
113C22⋊Q8—1072no—
114C22⋊Q8—1072no—
115C4×Q8—1072no—
116C4×D4—1071no—
117#117—1072no—
118#117—2142no—
119C2×C4⋊C4—1072no—
120#107—1071no—
121#107—1072no—
122#107C2×D10711yescharacteristic
123#107C2×D10711yescharacteristic
124#106—1072no—
125#106C2×D10712yes—
126C4⋊C8—1071no—
127#103—1072no—
128#103—1072no—
129#103—1072no—
130C22⋊C8—1072no—
131C4×C8C2×D10711yescharacteristic
132C2×Q8—1072no—
133C2×D4—1071no—
134C22×C4—1072no—
135C22×C4—2142no—
136Q16—2142no—
137M16—2144no—
138C2×C8—1072no—
139C2×C8C22×D10711yescharacteristic
140C2×C8C22×D10711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—1071no—
144C4⋊C4—1072no—
145C4⋊C4—2142no—
146C4⋊C4—2142no—
147C4⋊C4—2142no—
148C4⋊C4—2142no—
149C4⋊C4C22×D10712yes—
150C4⋊C4C22×D10712yes—
151C22⋊C4—1072no—
152C22⋊C4—2142no—
153C22⋊C4—2142no—
154C22⋊C4—2142no—
155C42—2142no—
156C42—2142no—
157C42C22×D10711yescharacteristic
158C23—1072no—
159Q8—2142no—
160Q8—2142no—
161D4—4281no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—1072no—
165C2×C4—4284no—
166C2×C4—2142no—
167C2×C4—2142no—
168C2×C4—2142no—
169C2×C4—2142no—
170C2×C4D4×D10711yescharacteristic
171C2×C4D4×D10711yescharacteristic
172C2×C4C23×D10711yescharacteristic
173C8—21no—
174C8—2142no—
175C8#1313381912yes—
176C22—2142no—
177C22—2142no—
178C22—2142no—
179C22C2×D4×D10711yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—4282no—
185C4—4282no—
186C4—2142no—
187C4C2×#1313381911yescharacteristic
188C4#1681215711yescharacteristic
189C2—4282no—
190C2#2029328011yescharacteristic
191C2#2029342411yescharacteristic
192C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#499×C107C2
socleC22×C107C22×D4
prime radicalC2×D428D4
hypercenter#499D107
hypocenterC107#3352
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
2. #20292970C2semidirect
3. #20292964C2semidirect
4. #20292931C2semidirect
5. #20292886C2semidirect
6. C4×Dic428C2semidirect
8. #20292666C2semidirect
9. #499×C107C2semidirect
97. C107#3352semidirect
99. #499D107semidirect

Maximal subgroups

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

subgroupindex
#202933152
#202929702
#202929642
#202929312
#202928862
C4×Dic4282
#202927062
#202926662
#499×C1072
#3352107