Finite Group Databasebeta

← #28346990

Subgroups — #28346990

Order 13696.

340 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 137 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#20293257C212yesmaximal
2#20293037C212yesmaximal, complemented
3#20293035C212yesmaximal, complemented
4#20293027C211yesmaximal, characteristic, complemented
5#20293025C211yesmaximal, characteristic, complemented
6#20293024C211yesmaximal, characteristic, complemented
7#20292668C212yesmaximal
8#20292662C211yesmaximal, characteristic
9#106×D107C211yesmaximal, characteristic
10#20292581C211yesmaximal, characteristic
11#499×C107C211yesmaximal, characteristic, complemented
12C2×#13133801C2211yescharacteristic
13C2×C8×D107C2211yescharacteristic
14#16812109C2212yes—
15#16812108—22no—
16#16812108C2212yes—
17#16812100C2212yes—
18#16812099—22no—
19#16812089—22no—
20#16812087—22no—
21#16812067C2212yes—
22#16812066C2211yescharacteristic
23C4⋊C4×D107C2212yes—
24#16812042C2212yes—
25#16812042C2212yes—
26#16812041C2211yescharacteristic
27#16812041C2211yescharacteristic
28#16812041C2212yes—
29#16812040C2212yes—
30#16812040C2212yes—
31#16812020C2211yescharacteristic
32#16812019C2211yescharacteristic
33C8×Dic107C2211yescharacteristic
34#125×C107C2212yes—
35#107×C107C2212yes—
36#106×C107C2211yescharacteristic
37#106×C107C2211yescharacteristic
38C4×C8×C107C2211yescharacteristic
39C2×Dic214—22no—
40C2×Dic214—22no—
41C2×C4×D107—22no—
42C2×C4×D107C2311yescharacteristic
43#13133801Q814yescomplemented
44C2×C107⋊C8C2311yescharacteristic
45C2×C107⋊C8C2311yescharacteristic
46C8×D107Q814yescomplemented
47C4⋊Dic107—22no—
48C4⋊Dic107—22no—
49C4⋊Dic107C2312yes—
50C4⋊Dic107C2312yes—
51Dic107⋊C4—22no—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4D412yes—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4D412yes—
57C4×Dic107—22no—
58C4×Dic107C2311yescharacteristic
59C2×C8×C107C2311yescharacteristic
60C2×C8×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107—22no—
63C4⋊C4×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C42×C107C2311yescharacteristic
66C22×D107C2×D411yescharacteristic
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2411yescharacteristic
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107C2×D412yes—
74C2×Dic107C2×D411yescharacteristic
75C4×D107—82no—
76C4×D107C2×Q812yes—
77C4×D107C2×Q812yes—
78Dic214—82no—
79Dic214—82no—
80C8×C107C2×Q812yes—
81C8×C107C2×Q812yes—
82C107⋊C8C2×Q812yes—
83C107⋊C8C2×Q812yes—
84C22×C107C22×D411yescharacteristic
85C2×D107—22no—
86C2×D107C4⋊Q812yes—
87C4×C107—22no—
88C4×C107—42no—
89C4×C107—42no—
90C4×C107C22×Q811yescharacteristic
91C4×C107C22×Q811yescharacteristic
92Dic107—22no—
93Dic107—42no—
94Dic107—42no—
95Dic107C4⋊Q812yes—
96C2×C107C2×C4⋊Q811yescharacteristic
97C2×C107#57712yes—
98D107—22no—
99#3163—1071nomaximal, Sylow
100C107#316311yescharacteristic, complemented, Sylow
101#544—1072no—
102#501—1072no—
103#501—1072no—
104#499—1071no—
105#499D10711yescharacteristic, complemented
106#498—1071no—
107#498—1071no—
108#428—1072no—
109#427—1071no—
110C2×#106—1071no—
111#405—1071no—
112C2×M16—1071no—
113C22×C8—1071no—
114C4⋊Q8—1072no—
115C4⋊Q8—1072no—
116C4⋊Q8—2142no—
117#125—1072no—
118#125—2142no—
119#125C2×D10712yes—
120C22⋊Q8—2142no—
121C22⋊Q8—2142no—
122#117—1071no—
123#117—1072no—
124C2×C4⋊C4—1072no—
125#107—1072no—
126#107—1072no—
127#107C2×D10712yes—
128#106—1071no—
129#106—1071no—
130#106—1072no—
131#106—1072no—
132#106—1072no—
133#106C2×D10711yescharacteristic
134#106C2×D10711yescharacteristic
135#97—1071no—
136#97—1071no—
137C4×C8—1071no—
138C4×C8C2×D10711yescharacteristic
139C2×Q8—2142no—
140C2×Q8—2142no—
141C22×C4—1071no—
142C22×C4—2142no—
143M16—1074no—
144C2×C8—1071no—
145C2×C8—1071no—
146C2×C8—1074no—
147C2×C8C22×D10711yescharacteristic
148C2×C8C22×D10711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—1072no—
152C4⋊C4—1072no—
153C4⋊C4—1072no—
154C4⋊C4—2142no—
155C4⋊C4—2142no—
156C4⋊C4—2142no—
157C4⋊C4—2142no—
158C4⋊C4—2142no—
159C4⋊C4C22×D10712yes—
160C4⋊C4C22×D10712yes—
161C22⋊C4—1072no—
162C22⋊C4—2142no—
163C42—1071no—
164C42—2142no—
165C42C22×D10711yescharacteristic
166C23—1071no—
167Q8—8562no—
168Q8—8562no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—8562no—
172C2×C4—1071no—
173C2×C4—1072no—
174C2×C4—1072no—
175C2×C4—1072no—
176C2×C4—2142no—
177C2×C4—2142no—
178C2×C4D4×D10712yes—
179C2×C4C23×D10711yescharacteristic
180C8—1072no—
181C8—1072no—
182C8Q8×D10712yes—
183C8Q8×D10712yes—
184C22—1072no—
185C22—2142no—
186C22C2×D4×D10711yescharacteristic
187C4—22no—
188C4—42no—
189C4—42no—
190C4—1072no—
191C4—4282no—
192C4—4282no—
193C4—2142no—
194C4C2×Q8×D10711yescharacteristic
195C4C2×Q8×D10711yescharacteristic
196C2—2142no—
197C2C4⋊Q8×D10711yescharacteristic
198C2#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#499×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#499D107
hypocenterC107#3163
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. #20293037C2semidirect
3. #20293035C2semidirect
4. #20293027C2semidirect
5. #20293025C2semidirect
6. #20293024C2semidirect
11. #499×C107C2semidirect
43. #13133801Q8semidirect
46. C8×D107Q8semidirect
100. C107#3163semidirect
105. #499D107semidirect

Maximal subgroups

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

subgroupindex
#202932572
#202930372
#202930352
#202930272
#202930252
#202930242
#202926682
#202926622
#106×D1072
#202925812
#499×C1072
#3163107