Finite Group Databasebeta

← #28346113

Subgroups — #28346113

Order 13696.

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

normal subgroups only

This group has 460 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
2C2×#16812116C211yesmaximal, characteristic
3#20293198C211yesmaximal, characteristic
4#20292883C214yesmaximal, complemented
5#20292866C214yesmaximal, complemented
6#20292663C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292635C211yesmaximal, characteristic, complemented
9#536×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C2×#13133808—24no—
12C2×#13133808C2212yes—
13C2×#13133808C2212yes—
14#16812116—24no—
15C2×#13133802C2212yes—
16C2×#13133802C2211yescharacteristic
17C22×C107⋊C8C2212yes—
18C22×C107⋊C8C2211yescharacteristic
19#16812107—21no—
20#16812096—21no—
21#16812090—22no—
22#16812088C2214yes—
23#16812068C2211yescharacteristic
24#16812042C2212yes—
25#16812040C2212yes—
26#16812033C2214yes—
27#16812032C2214yes—
28C2×#41×C107C2211yescharacteristic
29C22×Q8×C107C2211yescharacteristic
30C4⋊Q8×C107—21no—
31#124×C107—21no—
32#123×C107—22no—
33C22⋊Q8×C107C2214yes—
34#117×C107C2211yescharacteristic
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214—21no—
38C2×Dic214C2312yes—
39C22×Dic107—21no—
40#13133808—24no—
41#13133808—24no—
42#13133802—22no—
43#13133802—24no—
44C2×C107⋊C8—24no—
45C2×C107⋊C8D414yes—
46C2×C107⋊C8C2312yes—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2311yescharacteristic
49C4⋊Dic107C2312yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—21no—
52C22⋊Dic107—24no—
53C4×Dic107—21no—
54#41×C107—22no—
55#41×C107—22no—
56C2×Q8×C107—21no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107D411yescharacteristic
60C2×Q8×C107C2312yes—
61C2×D4×C107D412yes—
62C2×D4×C107D411yescharacteristic
63C22×C4×C107—21no—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—24no—
67C4⋊C4×C107C2312yes—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—24no—
70C42×C107—21no—
71C23×C107C2×D412yes—
72C23×C107C2×D411yescharacteristic
73Q8×C107—22no—
74Q8×C107—22no—
75Q8×C107—22no—
76Q8×C107—42no—
77D4×C107—24no—
78D4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4112yes—
88C2×C4×C107C2411yescharacteristic
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—42no—
92Dic214—22no—
93Dic214—22no—
94Dic214—42no—
95C107⋊C8—22no—
96C107⋊C8C2×D412yes—
97C107⋊C8C2×D412yes—
98C22×C107—22no—
99C22×C107—22no—
100C22×C107—24no—
101C22×C107#12112yes—
102C22×C107C22×D411yescharacteristic
103C4×C107—22no—
104C4×C107—42no—
105C4×C107—42no—
106C4×C107#12112yes—
107C4×C107C22×D411yescharacteristic
108C4×C107C2×#4111yescharacteristic
109Dic107—42no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107—42no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57812yes—
115#3059—1071nomaximal, Sylow
116C107#305911yescharacteristic, complemented, Sylow
117C22×Q16—1071no—
118C2×#131—1071no—
119#536—1071no—
120#536D10711yescharacteristic
121#470—1074no—
122#467—1074no—
123#427—1071no—
124#419—1071no—
125#419—1071no—
126C2×#41C2×D10711yescharacteristic
127C22×Q8—1071no—
128C22×Q8C2×D10711yescharacteristic
129C2×Q16—1072no—
130C2×Q16—1072no—
131C2×Q16—2144no—
132#131—2144no—
133C2×M16—1071no—
134C2×M16—1072no—
135C22×C8—1071no—
136C22×C8—1072no—
137C4⋊Q8—21no—
138C4⋊Q8—2141no—
139#124—21no—
140#124—2141no—
141#123—22no—
142#123—2142no—
143C22⋊Q8—1074no—
144C22⋊Q8C2×D10714yes—
145#117—1071no—
146#117C2×D10711yescharacteristic
147#107—1072no—
148#106—1072no—
149#103—1074no—
150#103—1074no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1072no—
157C2×Q8—2142no—
158C2×Q8—2142no—
159C2×Q8—2141no—
160C2×Q8C22⋊D10711yescharacteristic
161C2×Q8C22×D10712yes—
162C2×D4C22⋊D10712yes—
163C2×D4C22⋊D10711yescharacteristic
164C22×C4—21no—
165C22×C4—2141no—
166C22×C4C22×D10712yes—
167C22×C4C22×D10711yescharacteristic
168Q16—2144no—
169Q16—2144no—
170M16—2144no—
171M16—2142no—
172C2×C8—1074no—
173C2×C8—1071no—
174C2×C8—1071no—
175C2×C8—1072no—
176C2×C8—2144no—
177C4⋊C4—24no—
178C4⋊C4—1072no—
179C4⋊C4—2144no—
180C4⋊C4C22×D10712yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—2141no—
184C22⋊C4—2144no—
185C42—21no—
186C42—2141no—
187C23C2×C22⋊D10712yes—
188C23C2×C22⋊D10711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—4282no—
194Q8—2142no—
195Q8—2142no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—4282no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4D4×D10712yes—
207C2×C4C2×C22⋊D10712yes—
208C2×C4C2×C22⋊D10711yescharacteristic
209C2×C4#1313381712yes—
210C2×C4C23×D10711yescharacteristic
211C8—1072no—
212C8—1072no—
213C8—2142no—
214C22—22no—
215C22—24no—
216C22—22no—
217C22#1681208112yes—
218C22C22×C22⋊D10711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—4282no—
223C4—4282no—
224C4#1681208112yes—
225C4C2×D4×D10711yescharacteristic
226C4C2×#1313381711yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1681208111yescharacteristic
230C2#2029344212yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×C107C22
hypercenter#536D107
hypocenterC107#3059
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#13133808C2semidirect
4. #20292883C2semidirect
5. #20292866C2semidirect
6. #20292663C2semidirect
7. #20292636C2semidirect
8. #20292635C2semidirect
116. C107#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
C2×#168121162
#202931982
#202928832
#202928662
#202926632
#202926362
#202926352
#536×C1072
#3059107