Finite Group Databasebeta

← #28345993

Subgroups — #28345993

Order 13696.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13133808C211yesmaximal, characteristic, complemented
2#20293198C211yesmaximal, characteristic
3#20293105C211yesmaximal, characteristic, complemented
4#20292847C214yesmaximal, complemented
5#20292835C214yesmaximal, complemented
6#20292655C211yesmaximal, characteristic, complemented
7#20292637C211yesmaximal, characteristic, complemented
8#20292623C211yesmaximal, characteristic
9C2×#137×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C2×#13133808—24no—
12C2×#13133808C2214yes—
13C22×C107⋊C8C2211yescharacteristic
14#16812107C2212yes—
15#16812096C2212yes—
16#16812090—22no—
17#16812088—22no—
18Q8×Dic107—22no—
19Q8×Dic107C2214yes—
20#16812068—22no—
21#16812068C2211yescharacteristic
22#16812068C2211yescharacteristic
23Dic107⋊C8C2214yes—
24#16812034C2214yes—
25#16812033C2212yes—
26#16812030C2212yes—
27C2×#41×C107C2211yescharacteristic
28C22×Q8×C107C2211yescharacteristic
29#137×C107—24no—
30C2×Q16×C107C2212yes—
31C2×SD16×C107C2212yes—
32C2×M16×C107C2211yescharacteristic
33C2×Dic214—22no—
34C2×Dic214—22no—
35C2×Dic214—21no—
36C2×Dic214C2312yes—
37C22×Dic107—21no—
38#13133808—28no—
39C2×C107⋊C8—22no—
40C2×C107⋊C8C2312yes—
41C4⋊Dic107—24no—
42C4⋊Dic107D412yes—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—24no—
45C22⋊Dic107—21no—
46C22⋊Dic107—24no—
47C22⋊Dic107D412yes—
48C4×Dic107—21no—
49C4×Dic107—24no—
50C4×Dic107C2312yes—
51#41×C107—22no—
52#41×C107—22no—
53C2×Q8×C107—21no—
54C2×Q8×C107—22no—
55C2×Q8×C107D414yes—
56C2×Q8×C107C2311yescharacteristic
57C2×Q8×C107C2312yes—
58C2×D4×C107—21no—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107—21no—
61C22×C4×C107—21no—
62C22×C4×C107C2311yescharacteristic
63Q16×C107—24no—
64SD16×C107—24no—
65M16×C107—22no—
66C2×C8×C107C2312yes—
67C23×C107—21no—
68C23×C107C2×D411yescharacteristic
69Q8×C107—24no—
70Q8×C107—22no—
71Q8×C107—21no—
72Q8×C107—21no—
73Q8×C107C2×D414yes—
74D4×C107—21no—
75D4×C107—21no—
76D4×C107—42no—
77C2×C4×C107—24no—
78C2×C4×C107—22no—
79C2×C4×C107—21no—
80C2×C4×C107—42no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D412yes—
83C2×C4×C107#4112yes—
84C2×C4×C107C2411yescharacteristic
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—42no—
88C2×Dic107C2×D414yes—
89Dic214—22no—
90Dic214—22no—
91Dic214—42no—
92C8×C107—22no—
93C107⋊C8—22no—
94C22×C107—22no—
95C22×C107—41no—
96C22×C107—42no—
97C22×C107#12112yes—
98C22×C107C22×D411yescharacteristic
99C4×C107—24no—
100C4×C107—41no—
101C4×C107#12112yes—
102C4×C107C22×D411yescharacteristic
103C4×C107C2×#4111yescharacteristic
104Dic107—24no—
105Dic107—42no—
106Dic107—42no—
107C2×C107—22no—
108C2×C107—81no—
109C2×C107C2×#12111yescharacteristic
110C2×C107#57812yes—
111#3044—1071nomaximal, Sylow
112C107#304411yescharacteristic, complemented, Sylow
113C2×#137D10711yescharacteristic
114C22×Q16—1071no—
115#536—1071no—
116#519—1071no—
117#464—1074no—
118#462—1074no—
119#424—1071no—
120#419—1071no—
121#416—1071no—
122C2×#41C2×D10711yescharacteristic
123C22×Q8—1071no—
124C22×Q8C2×D10711yescharacteristic
125#137—24no—
126C2×Q16—1074no—
127C2×Q16—2144no—
128C2×Q16C2×D10712yes—
129C2×SD16C2×D10712yes—
130C2×M16C2×D10711yescharacteristic
131C22×C8—1071no—
132C4⋊Q8—1072no—
133#124—1072no—
134#123—2142no—
135C22⋊Q8—2142no—
136C4×Q8—1074no—
137C4×Q8—2142no—
138#117—1071no—
139#117—1071no—
140#117—2142no—
141C4⋊C8—1074no—
142#103—1072no—
143#103—1074no—
144#102—1072no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—1072no—
150C2×Q8—2142no—
151C2×Q8—2142no—
152C2×Q8—2141no—
153C2×Q8C22⋊D10714yes—
154C2×Q8C22×D10711yescharacteristic
155C2×Q8C22×D10712yes—
156C2×D4—21no—
157C2×D4C22×D10711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—24no—
163Q16—2148no—
164SD16—24no—
165M16—22no—
166C2×C8—1072no—
167C2×C8—2142no—
168C2×C8C22×D10712yes—
169C4⋊C4—1072no—
170C4⋊C4—1072no—
171C4⋊C4—2144no—
172C4⋊C4—2144no—
173C22⋊C4—1072no—
174C22⋊C4—2144no—
175C22⋊C4—2141no—
176C42—1072no—
177C42—2144no—
178C42—2141no—
179C23—21no—
180C23D4×D10711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—4282no—
186Q8—2142no—
187Q8—2142no—
188Q8C2×C22⋊D10714yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—1074no—
197C2×C4—4282no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4D4×D10711yescharacteristic
201C2×C4C2×C22⋊D10712yes—
202C2×C4#1313381712yes—
203C2×C4C23×D10711yescharacteristic
204C8—22no—
205C8—2142no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1681208112yes—
210C22C2×D4×D10711yescharacteristic
211C4—24no—
212C4—41no—
213C4—4282no—
214C4—4282no—
215C4—2144no—
216C4#1681208112yes—
217C4C22×C22⋊D10711yescharacteristic
218C4C2×#1313381711yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1681208111yescharacteristic
222C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×C107C22
hypercenterC2×#137D107
hypocenterC107#3044
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13133808C2semidirect
3. #20293105C2semidirect
4. #20292847C2semidirect
5. #20292835C2semidirect
6. #20292655C2semidirect
7. #20292637C2semidirect
112. C107#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
#202931982
#202931052
#202928472
#202928352
#202926552
#202926372
#202926232
C2×#137×C1072
#3044107