Finite Group Databasebeta

← #28349037

Subgroups — #28349037

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#20293318C212yesmaximal
2#20293029C211yesmaximal, characteristic, complemented
3#20292967C212yesmaximal, complemented
4#20292961C212yesmaximal, complemented
5#20292934C212yesmaximal, complemented
6#20292882C212yesmaximal, complemented
7#20292706C211yesmaximal, characteristic
8#20292668C212yesmaximal, complemented
9C4×Q16×C107C211yesmaximal, characteristic, complemented
10C2×#13133801C2212yes—
11#16812103—22no—
12#16812101C2212yes—
13#16812099—22no—
14#16812094—22no—
15#16812085C2212yes—
16#16812085C2212yes—
17C4×D428C2211yescharacteristic
18#16812067C2212yes—
19#16812066—22no—
20C4⋊C4×D107C2212yes—
21#16812043C2211yescharacteristic
22#16812043C2211yescharacteristic
23#16812042C2212yes—
24#16812041C2212yes—
25#16812040C2212yes—
26C4⋊(C107⋊C8)C2211yescharacteristic
27#16812033C2212yes—
28#16812033C2212yes—
29(C2×D107)⋊C8C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C4×Q8×C107C2212yes—
32#107×C107C2211yescharacteristic
33#103×C107C2212yes—
34C4×C8×C107C2211yescharacteristic
35C2×D428D411yescharacteristic
36C2×C4×D107—22no—
37C2×C4×D107C2312yes—
38#13133801—24no—
39C2×C107⋊C8C2312yes—
40C4⋊Dic107D411yescharacteristic
41C4⋊Dic107C2312yes—
42C4⋊Dic107C2312yes—
43Dic107⋊C4—22no—
44Dic107⋊C4—22no—
45Dic107⋊C4—22no—
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—
53C2×Q8×C107C2312yes—
54Q16×C107—22no—
55C2×C8×C107C2311yescharacteristic
56C2×C8×C107C2311yescharacteristic
57C4⋊C4×C107—22no—
58C4⋊C4×C107C2312yes—
59C42×C107—22no—
60C42×C107C2311yescharacteristic
61C22×D107C2×D412yes—
62Q8×C107—22no—
63Q8×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2411yescharacteristic
69C2×Dic107—22no—
70C2×Dic107—22no—
71C2×Dic107C2×D412yes—
72D428—41no—
73C4×D107—22no—
74C4×D107—22no—
75C4×D107—44no—
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#499—1071no—
100#485—1072no—
101#484—1072no—
102#479—1072no—
103#470—1072no—
104C4×Q16D10711yescharacteristic, complemented
105#436—1071no—
106#428—1072no—
107C2×Q16C2×D10711yescharacteristic
108C2×M16—1072no—
109#126—2142no—
110#125—1072no—
111#125—2142no—
112#123—2142no—
113C22⋊Q8—1072no—
114C22⋊Q8—1072no—
115C4×Q8C2×D10712yes—
116C4×D4—1071no—
117#117—1072no—
118#117—2142no—
119C2×C4⋊C4—1072no—
120#107—1071no—
121#107—1071no—
122#107—1072no—
123#107C2×D10711yescharacteristic
124#106—1072no—
125#106—1072no—
126C4⋊C8—1071no—
127#103—1072no—
128#103—1072no—
129#103C2×D10712yes—
130C22⋊C8—1072no—
131C4×C8C2×D10711yescharacteristic
132C2×Q8C22×D10712yes—
133C2×D4—1071no—
134C22×C4—1072no—
135C22×C4—2142no—
136Q16—22no—
137M16—2144no—
138C2×C8—1072no—
139C2×C8C22×D10711yescharacteristic
140C2×C8C22×D10711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—1071no—
143C4⋊C4—1072no—
144C4⋊C4—1072no—
145C4⋊C4—2142no—
146C4⋊C4—2142no—
147C4⋊C4—2142no—
148C4⋊C4—2142no—
149C4⋊C4—2142no—
150C4⋊C4C22×D10712yes—
151C22⋊C4—1072no—
152C22⋊C4—2142no—
153C22⋊C4—2142no—
154C22⋊C4—2142no—
155C42—22no—
156C42—2142no—
157C42C22×D10711yescharacteristic
158C23—1072no—
159Q8—22no—
160Q8—22no—
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#1313381812yes—
176C22—2142no—
177C22—2142no—
178C22—2142no—
179C22C2×D4×D10711yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—4282no—
185C4—4282no—
186C4—2142no—
187C4C2×#1313381811yescharacteristic
188C4#1681215611yescharacteristic
189C2—4282no—
190C2#2029328211yescharacteristic
191C2#2029342411yescharacteristic
192C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×Q16×C107C2
socleC22×C107C22×D4
prime radicalC2×D428D4
hypercenterC4×Q16D107
hypocenterC107#3352
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #20293029C2semidirect
3. #20292967C2semidirect
4. #20292961C2semidirect
5. #20292934C2semidirect
6. #20292882C2semidirect
8. #20292668C2semidirect
9. C4×Q16×C107C2semidirect
97. C107#3352semidirect
104. C4×Q16D107semidirect

Maximal subgroups

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

subgroupindex
#202933182
#202930292
#202929672
#202929612
#202929342
#202928822
#202927062
#202926682
C4×Q16×C1072
#3352107