Finite Group Databasebeta

← #28339723

Subgroups — #28339723

Order 13696.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3#20292567C211yesmaximal, characteristic, complemented
4#20292563C211yesmaximal, characteristic, complemented
5#20292558C211yesmaximal, characteristic, complemented
6#20292556C212yesmaximal, complemented
7#20292542C212yesmaximal
8#20292478C212yesmaximal, complemented
9C22⋊C4×Dic107C212yesmaximal, complemented
10#20292474C211yesmaximal, characteristic, complemented
11#397×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15(C22⋊D107)⋊C4—24no—
16#16812067—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2212yes—
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2211yescharacteristic
21C2×C22⋊Dic107C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×C4×Dic107C2212yes—
26C2×C4×Dic107C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C22⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C2×C22⋊D107—22no—
39C2×C22⋊D107—22no—
40C2×C22⋊D107—22no—
41C22×Dic107—22no—
42C22×Dic107—24no—
43C22×Dic107C2312yes—
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C2×C4×D107—24no—
49C2×C4×D107D414yes—
50C4⋊Dic107—22no—
51Dic107⋊C4—22no—
52Dic107⋊C4—24no—
53Dic107⋊C4—24no—
54C22⋊Dic107—24no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—24no—
58C4×Dic107—24no—
59C4×Dic107—22no—
60C4×Dic107—24no—
61C24×C107C2311yescharacteristic
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2312yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C22⋊C4×C107—24no—
68C22⋊C4×C107—22no—
69C23×C107—21no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—21no—
73C23×C107#4112yes—
74C23×C107C2411yescharacteristic
75C22×D107—22no—
76C22×D107—22no—
77C22×D107—21no—
78C22×D107—21no—
79C22×D107C2×D412yes—
80C2×C4×C107—22no—
81C2×C4×C107—21no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D412yes—
90C22⋊D107—28no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—21no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—44no—
101C2×Dic107C2×D412yes—
102C2×Dic107#4114yes—
103C2×Dic107#4112yes—
104C2×Dic107#4114yes—
105C4×D107—28no—
106C22×C107—22no—
107C22×C107—24no—
108C22×C107—41no—
109C22×C107—42no—
110C22×C107—41no—
111C22×C107C22×D411yescharacteristic
112C22×C107C2×#4112yes—
113C22×C107C2×#4111yescharacteristic
114C22×C107C2×#4112yes—
115C22×C107C2×#4111yescharacteristic
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—42no—
120C2×D107—42no—
121C4×C107—22no—
122C4×C107—42no—
123C4×C107—42no—
124C4×C107—41no—
125Dic107—22no—
126Dic107—24no—
127Dic107—41no—
128Dic107—42no—
129Dic107#12314yes—
130C2×C107—42no—
131C2×C107C2×#12311yescharacteristic
132C2×C107#52511yescharacteristic
133C2×C107#52912yes—
134C2×C107#52912yes—
135C2×C107#53211yescharacteristic
136D107—42no—
137#2395—1071nomaximal, Sylow
138C107#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—1071no—
140C2×#117—1071no—
141#400—1071no—
142#399—1071no—
143#398—1071no—
144#397—1072no—
145#397D10711yescharacteristic, complemented
146#394—1072no—
147C4×C4⋊C4—1072no—
148C4×C22⋊C4—1072no—
149C4×C22⋊C4—1071no—
150C22×D4—1071no—
151C23×C4—1071no—
152C23×C4—1071no—
153C4×D4—2144no—
154#117—2144no—
155C2×C4⋊C4—1071no—
156C2×C4⋊C4—1071no—
157C2×C4⋊C4—1072no—
158C2×C4⋊C4—1072no—
159C2×C4⋊C4C2×D10711yescharacteristic
160C2×C22⋊C4—1072no—
161C2×C22⋊C4—1072no—
162C2×C22⋊C4—1071no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4C2×D10711yescharacteristic
165C2×C22⋊C4C2×D10712yes—
166C2×C42—1072no—
167C2×C42—1071no—
168C2×C42—1072no—
169#95—1072no—
170#95—1072no—
171#95—1071no—
172#95—1072no—
173#95C2×D10711yescharacteristic
174#95C2×D10712yes—
175C24—1071no—
176C24C22×D10711yescharacteristic
177C2×D4—2142no—
178C2×D4—2142no—
179C2×D4—2142no—
180C22×C4—1071no—
181C22×C4—1072no—
182C22×C4—1071no—
183C22×C4—1074no—
184C22×C4—1071no—
185C22×C4—1072no—
186C22×C4—2144no—
187C22×C4—2142no—
188C22×C4—2144no—
189C22×C4C22×D10712yes—
190C22×C4C22×D10712yes—
191C22×C4C22×D10711yescharacteristic
192C22×C4C22×D10711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—2144no—
195C4⋊C4—2144no—
196C4⋊C4—2142no—
197C4⋊C4—2142no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—2144no—
201C22⋊C4—2144no—
202C22⋊C4—2142no—
203C22⋊C4—2144no—
204C42—2144no—
205C42—2142no—
206C42—2144no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—1072no—
212C23—2142no—
213C23—2142no—
214C23—2141no—
215C23—2141no—
216C23#1313381712yes—
217C23C23×D10711yescharacteristic
218D4—2148no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—1072no—
229C2×C4—1074no—
230C2×C4—1074no—
231C2×C4—1072no—
232C2×C4—4284no—
233C2×C4—2148no—
234C2×C4—2142no—
235C2×C4—2141no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2141no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4—2141no—
243C2×C4D4×D10712yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—4282no—
250C22—4282no—
251C22—2142no—
252C22—2142no—
253C22—2142no—
254C22C2×D4×D10711yescharacteristic
255C22#41×D10712yes—
256C22#41×D10711yescharacteristic
257C22#41×D10712yes—
258C22C2×#1313381711yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—1074no—
264C4—4281no—
265C4—4282no—
266C4—2142no—
267C4—2144no—
268C2—42no—
269C2—4282no—
270C2#123×D10711yescharacteristic
271C2#2029313711yescharacteristic
272C2#2029315412yes—
273C2#2029316012yes—
274C2#2029317611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#397×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#397D107
hypocenterC107#2395
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#16812067C2semidirect
3. #20292567C2semidirect
4. #20292563C2semidirect
5. #20292558C2semidirect
6. #20292556C2semidirect
8. #20292478C2semidirect
9. C22⋊C4×Dic107C2semidirect
10. #20292474C2semidirect
11. #397×C107C2semidirect
138. C107#2395semidirect
145. #397D107semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D107)⋊C42
C2×#168120672
#202925672
#202925632
#202925582
#202925562
#202925422
#202924782
C22⋊C4×Dic1072
#202924742
#397×C1072
#2395107