Finite Group Databasebeta

← #28340102

Subgroups — #28340102

Order 13696.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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×Q8×D107C211yesmaximal, characteristic
2C2×#16812107C211yesmaximal, characteristic, complemented
3C2×#16812085C211yesmaximal, characteristic
4#20292547C214yesmaximal, complemented
5#20292534C211yesmaximal, characteristic, complemented
6#20292533C211yesmaximal, characteristic, complemented
7#20292498C214yesmaximal
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic
9C2×C4⋊Q8×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2212yes—
11C22×Dic214C2211yescharacteristic
12C2×Q8×D107—24no—
13C2×Q8×D107—24no—
14C2×Q8×D107—24no—
15C22×C4×D107C2212yes—
16C22×C4×D107C2211yescharacteristic
17#16812107—24no—
18#16812085—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2214yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C107⋊#95C2212yes—
26C107⋊#95C2214yes—
27C107⋊#95C2214yes—
28C22×Q8×C107C2211yescharacteristic
29C22×Q8×C107C2211yescharacteristic
30C4⋊Q8×C107—24no—
31C2×C4⋊C4×C107C2214yes—
32C2×C42×C107C2211yescharacteristic
33C23×D107C2311yescharacteristic
34C2×Dic214—22no—
35C2×Dic214—24no—
36C2×Dic214—22no—
37C2×Dic214—24no—
38C2×Dic214—24no—
39C2×Dic214D414yes—
40Q8×D107—232no—
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2312yes—
45C2×C4×D107—24no—
46C2×C4×D107—22no—
47C2×C4×D107—24no—
48C2×C4×D107—22no—
49C2×C4×D107—24no—
50C2×C4×D107D414yes—
51C4⋊Dic107—22no—
52Dic107⋊C4—24no—
53Dic107⋊C4—28no—
54(C2×D107)⋊C4—24no—
55(C2×D107)⋊C4Q818yescomplemented
56C4×Dic107—22no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107—22no—
60C2×Q8×C107—22no—
61C2×Q8×C107—22no—
62C2×Q8×C107D414yes—
63C22×C4×C107C2312yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107—28no—
68C42×C107—22no—
69C23×C107C2411yescharacteristic
70C22×D107—21no—
71C22×D107—21no—
72C22×D107—22no—
73C22×D107C2×D412yes—
74C22×D107C2×Q814yes—
75Q8×C107—28no—
76Q8×C107—44no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D414yes—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×Q814yescomplemented
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—24no—
97C2×Dic107C2×D414yes—
98C2×Dic107C2×D412yes—
99C2×Dic107C2×Q814yescomplemented
100C4×D107—216no—
101C4×D107—28no—
102Dic214—216no—
103Dic214—28no—
104C22×C107C22×D411yescharacteristic
105C22×C107C22×D411yescharacteristic
106C22×C107C22×D411yescharacteristic
107C22×C107C22×Q812yes—
108C22×C1072−1+412yes—
109C2×D107—22no—
110C2×D107—22no—
111C2×D107—24no—
112C2×D107—24no—
113C2×D107C4⋊Q814yescomplemented
114C4×C107—24no—
115C4×C107—24no—
116C4×C107—42no—
117C4×C107#12014yes—
118Dic107—22no—
119Dic107—24no—
120Dic107—24no—
121Dic107—42no—
122C2×C107C2×#12011yescharacteristic
123C2×C107C2×C4⋊Q811yescharacteristic
124C2×C107#53611yescharacteristic
125C2×C107D4×Q812yes—
126C2×C107D4×Q812yes—
127D107—24no—
128#2434—1071nomaximal, Sylow
129C107#243411yescharacteristic, complemented, Sylow
130C23×Q8—1071no—
131C2×C4⋊Q8—1071no—
132C2×C4⋊Q8D10711yescharacteristic, complemented
133C2×C22⋊Q8—1071no—
134#395—1074no—
135#391—1071no—
136#391—1071no—
137#385—1074no—
138C4×C22⋊C4—1071no—
139C22×Q8—1071no—
140C22×Q8—1072no—
141C22×Q8—2144no—
142C22×Q8—2144no—
143C22×Q8—2144no—
144C22×Q8C2×D10711yescharacteristic
145C22×Q8C2×D10711yescharacteristic
146C23×C4—1071no—
147C23×C4—1072no—
148C4⋊Q8—24no—
149C4⋊Q8—2144no—
150C22⋊Q8—2144no—
151C2×C4⋊C4—1072no—
152C2×C4⋊C4—1071no—
153C2×C4⋊C4—1074no—
154C2×C4⋊C4C2×D10714yes—
155C2×C22⋊C4—1072no—
156C2×C22⋊C4—1072no—
157C2×C42—1071no—
158C2×C42C2×D10711yescharacteristic
159#95—1072no—
160#95—1074no—
161#95—1074no—
162C24—1071no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—1074no—
169C2×Q8—2144no—
170C2×Q8—2144no—
171C2×Q8—21432no—
172C2×Q8—2142no—
173C2×Q8—2144no—
174C2×Q8—2142no—
175C2×Q8C22⋊D10714yes—
176C22×C4—1072no—
177C22×C4—1071no—
178C22×C4—1072no—
179C22×C4—1074no—
180C22×C4—1072no—
181C22×C4—2144no—
182C22×C4—2144no—
183C22×C4—2142no—
184C22×C4—2144no—
185C22×C4—2142no—
186C22×C4C22×D10712yes—
187C22×C4C22×D10712yes—
188C22×C4C22×D10712yes—
189C22×C4C22×D10711yescharacteristic
190C4⋊C4—28no—
191C4⋊C4—2144no—
192C4⋊C4—2148no—
193C4⋊C4—2142no—
194C22⋊C4—1078no—
195C22⋊C4—2144no—
196C42—22no—
197C42—2142no—
198C23—1072no—
199C23—1074no—
200C23—2141no—
201C23—2141no—
202C23—2142no—
203C23C23×D10711yescharacteristic
204Q8—28no—
205Q8—44no—
206Q8—21416no—
207Q8—2148no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1072no—
216C2×C4—1074no—
217C2×C4—1074no—
218C2×C4—21416no—
219C2×C4—2142no—
220C2×C4—2144no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4—2148no—
228C2×C4D4×D10712yes—
229C2×C4D4×D10712yes—
230C2×C4C2×C22⋊D10714yes—
231C2×C4C2×C22⋊D10712yes—
232C2×C4Q8×D10714yescomplemented
233C22—1074no—
234C22—2142no—
235C22—2144no—
236C22—2144no—
237C22—2142no—
238C22C2×D4×D10711yescharacteristic
239C22C2×D4×D10711yescharacteristic
240C22C22×C22⋊D10711yescharacteristic
241C22C2×Q8×D10712yes—
242C22#1681215712yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—4282no—
247C4—2144no—
248C4—2142no—
249C4—2144no—
250C4#1681207714yes—
251C2—2144no—
252C2C2×#1681207711yescharacteristic
253C2C4⋊Q8×D10711yescharacteristic
254C2#2029320211yescharacteristic
255C2Q8×C22⋊D10712yes—
256C2#2029329212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C4⋊Q8×C107C2
prime radicalC23×D107C23
hypercenterC2×C4⋊Q8D107
hypocenterC107#2434
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812107C2semidirect
4. #20292547C2semidirect
5. #20292534C2semidirect
6. #20292533C2semidirect
9. C2×C4⋊Q8×C107C2semidirect
55. (C2×D107)⋊C4Q8semidirect
88. C2×C4×C107C2×Q8semidirect
99. C2×Dic107C2×Q8semidirect
113. C2×D107C4⋊Q8semidirect
129. C107#2434semidirect
132. C2×C4⋊Q8D107semidirect
232. C2×C4Q8×D107semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168121072
C2×#168120852
#202925472
#202925342
#202925332
#202924982
C4×(C2×D107)⋊C42
C2×C4⋊Q8×C1072
#2434107