Finite Group Databasebeta

← #28339725

Subgroups — #28339725

Order 13696.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16812071C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3#20292566C211yesmaximal, characteristic, complemented
4#20292564C211yesmaximal, characteristic, complemented
5#20292554C212yesmaximal, complemented
6#20292553C211yesmaximal, characteristic, complemented
7#20292544C212yesmaximal
8C4⋊C4×Dic107C212yesmaximal, complemented
9#20292474C211yesmaximal, characteristic, complemented
10#20292474C212yesmaximal, complemented
11#398×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15#16812071—24no—
16#16812067—24no—
17C2×C4⋊Dic107C2212yes—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×C4×Dic107C2212yes—
26C2×C4×Dic107C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2212yes—
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2211yescharacteristic
35#95×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C23×D107C2311yescharacteristic
39C2×D428—22no—
40C2×D428—22no—
41C2×D428—22no—
42C22×Dic107C2311yescharacteristic
43C22×Dic107C2312yes—
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C2×C4×D107—24no—
47C2×C4×D107—22no—
48C2×C4×D107—24no—
49C2×C4×D107D414yes—
50C4⋊Dic107—22no—
51C4⋊Dic107—24no—
52(C2×D107)⋊C4—24no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—24no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—24no—
57(C2×D107)⋊C4—24no—
58C4×Dic107—24no—
59C4×Dic107—22no—
60C4×Dic107—24no—
61C22×C4×C107C2312yes—
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C4⋊C4×C107—22no—
68C4⋊C4×C107—24no—
69C23×C107C2411yescharacteristic
70C22×D107—21no—
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—22no—
74C22×D107—22no—
75C22×D107—21no—
76C22×D107—21no—
77C22×D107—21no—
78C22×D107C2×D412yes—
79C22×D107#4112yes—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—22no—
86C2×C4×C107—21no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107C2×D412yes—
91C2×C4×C107#4112yes—
92C2×C4×C107#4114yes—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107C2×D412yes—
102C2×Dic107#4114yes—
103D428—28no—
104C4×D107—28no—
105C4×D107—44no—
106C22×C107C22×D411yescharacteristic
107C22×C107C2×#4112yes—
108C22×C107C2×#4111yescharacteristic
109C22×C107C2×#4111yescharacteristic
110C22×C107C2×#4112yes—
111C2×D107—22no—
112C2×D107—22no—
113C2×D107—22no—
114C2×D107—24no—
115C2×D107—22no—
116C2×D107—41no—
117C2×D107—42no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—41no—
121C4×C107—22no—
122C4×C107—22no—
123C4×C107—24no—
124C4×C107—41no—
125C4×C107—42no—
126Dic107—42no—
127Dic107—41no—
128Dic107—42no—
129Dic107#12314yes—
130C2×C107C2×#12311yescharacteristic
131C2×C107#52511yescharacteristic
132C2×C107#52912yes—
133C2×C107#52912yes—
134C2×C107#53211yescharacteristic
135D107—42no—
136D107—42no—
137#2395—1071nomaximal, Sylow
138C107#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—1071no—
140C2×#117—1071no—
141#400—1071no—
142#399—1071no—
143#398D10711yescharacteristic, complemented
144#397—1071no—
145#397—1072no—
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—1072no—
157C2×C4⋊C4C2×D10711yescharacteristic
158C2×C4⋊C4C2×D10711yescharacteristic
159C2×C4⋊C4C2×D10712yes—
160C2×C22⋊C4—1072no—
161C2×C22⋊C4—1071no—
162C2×C22⋊C4—1071no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4—1072no—
166C2×C42—1072no—
167C2×C42—1071no—
168C2×C42—1072no—
169#95—1071no—
170#95—1072no—
171#95—1072no—
172#95—1072no—
173#95C2×D10712yes—
174#95C2×D10711yescharacteristic
175C24—1071no—
176C24—1071no—
177C2×D4—2142no—
178C2×D4—2142no—
179C2×D4—2142no—
180C22×C4—1071no—
181C22×C4—1074no—
182C22×C4—1072no—
183C22×C4—1071no—
184C22×C4—1072no—
185C22×C4—2142no—
186C22×C4—2144no—
187C22×C4—2144no—
188C22×C4C22×D10711yescharacteristic
189C22×C4C22×D10712yes—
190C22×C4C22×D10711yescharacteristic
191C22×C4C22×D10711yescharacteristic
192C22×C4C22×D10712yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—2142no—
197C4⋊C4—2144no—
198C22⋊C4—2144no—
199C22⋊C4—2142no—
200C22⋊C4—2144no—
201C22⋊C4—2142no—
202C22⋊C4—2144no—
203C22⋊C4—2144no—
204C42—2144no—
205C42—2142no—
206C42—2144no—
207C23—1072no—
208C23—1072no—
209C23—2141no—
210C23—2142no—
211C23—2142no—
212C23—2141no—
213C23—2141no—
214C23—2141no—
215C23—2142no—
216C23—2142no—
217C23C23×D10711yescharacteristic
218D4—2148no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1072no—
230C2×C4—1074no—
231C2×C4—4284no—
232C2×C4—2148no—
233C2×C4—2141no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10712yes—
242C2×C4#1313381712yes—
243C2×C4#1313381914yes—
244C22—4281no—
245C22—4282no—
246C22—4282no—
247C22—4282no—
248C22—4281no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22—2144no—
253C22—2142no—
254C22C2×D4×D10711yescharacteristic
255C22#41×D10712yes—
256C22#41×D10711yescharacteristic
257C22C2×#1313381711yescharacteristic
258C22C2×#1313381912yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—1074no—
265C4—4282no—
266C4—4281no—
267C4—4282no—
268C2—4282no—
269C2—4282no—
270C2#123×D10711yescharacteristic
271C2#2029313711yescharacteristic
272C2#2029316212yes—
273C2#2029316312yes—
274C2#2029317811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC22×D428C22
hypercenter#398D107
hypocenterC107#2395
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#16812067C2semidirect
3. #20292566C2semidirect
4. #20292564C2semidirect
5. #20292554C2semidirect
6. #20292553C2semidirect
8. C4⋊C4×Dic107C2semidirect
9. #20292474C2semidirect
10. #20292474C2semidirect
11. #398×C107C2semidirect
138. C107#2395semidirect
143. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120712
C2×#168120672
#202925662
#202925642
#202925542
#202925532
#202925442
C4⋊C4×Dic1072
#202924742
#398×C1072
#2395107