Finite Group Databasebeta

← #28335027

Subgroups — #28335027

Order 13696.

340 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#121×D107C211yesmaximal, characteristic
2C22⋊C8×D107C211yesmaximal, characteristic, complemented
3#20292486C211yesmaximal, characteristic, complemented
4#20292362C211yesmaximal, characteristic, complemented
5#20292361C211yesmaximal, characteristic, complemented
6#20292360C211yesmaximal, characteristic, complemented
7#327×C107C211yesmaximal, characteristic, complemented
8C2×D4×D107—21no—
9C2×D4×D107C411yescharacteristic, complemented
10C22×C4×D107C2211yescharacteristic
11C2×C8×D107—21no—
12#16812084C411yescharacteristic, complemented
13#16812083C2211yescharacteristic
14#16812082—21no—
15#16812081—21no—
16#16812080C411yescharacteristic, complemented
17C2×Dic107⋊C4—21no—
18C4⋊C4×D107C411yescharacteristic, complemented
19C2×(C2×D107)⋊C4—21no—
20C22⋊C4×D107—21no—
21C22⋊(C107⋊C8)C2211yescharacteristic, complemented
22(C2×D107)⋊C8—21no—
23C107⋊#95C2211yescharacteristic, complemented
24#121×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic, complemented
26#95×C107C2211yescharacteristic, complemented
27C23×D107—21no—
28C23×D107D411yescharacteristic
29C2×D428—21no—
30C2×D428C2×C411yescharacteristic
31C2×C22⋊D107—21no—
32C2×C22⋊D107—21no—
33C2×C22⋊D107—21no—
34D4×D107—42no—
35D4×D107—42no—
36D4×D107—42no—
37C22×Dic107—21no—
38C22×Dic107D411yescharacteristic
39C2×C4×D107—21no—
40C2×C4×D107—42no—
41C2×C4×D107C2×C411yescharacteristic
42C2×C4×D107C2×C411yescharacteristic
43C2×C4×D107D411yescharacteristic
44C2×C4×D107D411yescharacteristic
45C2×C107⋊C8—21no—
46C8×D107—42no—
47C4⋊Dic107C2×C411yescharacteristic
48Dic107⋊C4—21no—
49Dic107⋊C4—42no—
50C22⋊Dic107—21no—
51(C2×D107)⋊C4—21no—
52(C2×D107)⋊C4—21no—
53(C2×D107)⋊C4—42no—
54C2×D4×C107—21no—
55C2×D4×C107C2×C411yescharacteristic
56C22×C4×C107—21no—
57C22×C4×C107C2311yescharacteristic
58C2×C8×C107—21no—
59C4⋊C4×C107C2×C411yescharacteristic
60C22⋊C4×C107—21no—
61C23×C107—21no—
62C23×C107C2×D411yescharacteristic
63C22×D107—21no—
64C22×D107—41no—
65C22×D107—41no—
66C22×D107—42no—
67C22×D107—42no—
68C22×D107—42no—
69C22×D107C22⋊C411yescharacteristic
70C22×D107C22⋊C411yescharacteristic
71C22×D107D812yes—
72C22×D107SD1612yes—
73D4×C107—41no—
74D4×C107—41no—
75D4×C107—41no—
76C2×C4×C107—21no—
77C2×C4×C107—41no—
78C2×C4×C107—41no—
79C2×C4×C107—41no—
80C2×C4×C107—41no—
81C2×C4×C107C22×C411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C22⋊D107—41no—
84C22⋊D107—41no—
85C22⋊D107—42no—
86C22⋊D107—42no—
87C2×Dic107—22no—
88C2×Dic107—21no—
89C2×Dic107—41no—
90C2×Dic107—41no—
91C2×Dic107—41no—
92C2×Dic107C22⋊C411yescharacteristic
93C2×Dic107C22⋊C411yescharacteristic
94D428—41no—
95D428—42no—
96C4×D107—22no—
97C4×D107—22no—
98C4×D107—41no—
99C4×D107—41no—
100C4×D107—42no—
101C8×C107—41no—
102C107⋊C8—41no—
103C22×C107—21no—
104C22×C107—21no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107C2×C22⋊C411yescharacteristic
109C22×C107C2×D811yescharacteristic
110C22×C107C2×SD1611yescharacteristic
111C2×D107—21no—
112C2×D107—21no—
113C2×D107—21no—
114C2×D107—21no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—41no—
118C2×D107—41no—
119C2×D107—41no—
120C2×D107—84no—
121C2×D107#10212yes—
122C2×D107#10212yes—
123C4×C107—21no—
124C4×C107—21no—
125C4×C107—41no—
126C4×C107—81no—
127Dic107—21no—
128Dic107—41no—
129Dic107—81no—
130Dic107#9912yes—
131C2×C107—21no—
132C2×C107—21no—
133C2×C107—81no—
134C2×C107C2×#9911yescharacteristic
135C2×C107C2×#10211yescharacteristic
136C2×C107#42111yescharacteristic
137D107—22no—
138D107—22no—
139D107—81no—
140#1519—1071nomaximal, Sylow
141C107#151911yescharacteristic, complemented, Sylow
142C2×#121—1071no—
143C2×C22⋊C8—1071no—
144#380—1071no—
145#327—1071no—
146#327—1071no—
147#327—1071no—
148#327D10711yescharacteristic, complemented
149C22×D4—1071no—
150C22×D4—2141no—
151C23×C4—1071no—
152C22×C8—2141no—
153#121—1071no—
154#121—1071no—
155#121—1071no—
156#121—2141no—
157#121—2141no—
158#121C2×D10711yescharacteristic
159C2×C4⋊C4—1071no—
160C2×C4⋊C4—2141no—
161C2×C22⋊C4—2141no—
162C2×C22⋊C4—2141no—
163C22⋊C8—1071no—
164C22⋊C8—2141no—
165C22⋊C8C2×D10711yescharacteristic, complemented
166#95—1071no—
167#95C2×D10711yescharacteristic, complemented
168C24—1071no—
169C24—2141no—
170C2×D4—21no—
171C2×D4—1071no—
172C2×D4—4282no—
173C2×D4—4282no—
174C2×D4—4282no—
175C2×D4—2141no—
176C2×D4—2141no—
177C2×D4—2141no—
178C2×D4—2141no—
179C2×D4C4×D10711yescharacteristic
180C22×C4—21no—
181C22×C4—1071no—
182C22×C4—1071no—
183C22×C4—1071no—
184C22×C4—1071no—
185C22×C4—1071no—
186C22×C4—4282no—
187C22×C4—2141no—
188C22×C4—2141no—
189C22×C4C22×D10711yescharacteristic
190C2×C8—21no—
191C2×C8—4282no—
192C2×C8—2141no—
193C4⋊C4—1071no—
194C4⋊C4—4282no—
195C4⋊C4—2141no—
196C4⋊C4C4×D10711yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—4282no—
199C22⋊C4—2141no—
200C22⋊C4—2141no—
201C22⋊C4—2141no—
202C23—21no—
203C23—1072no—
204C23—1072no—
205C23—1071no—
206C23—1071no—
207C23—4281no—
208C23—4281no—
209C23—4282no—
210C23—4282no—
211C23—4282no—
212C23—2141no—
213C23D4×D10711yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—4281no—
218D4—4282no—
219D4—4282no—
220D4—4282no—
221D4—4281no—
222D4—4281no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—1071no—
229C2×C4—1071no—
230C2×C4—4281no—
231C2×C4—4281no—
232C2×C4—4282no—
233C2×C4—4281no—
234C2×C4—4281no—
235C2×C4—4281no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2141no—
240C2×C4C2×C4×D10711yescharacteristic
241C2×C4D4×D10711yescharacteristic
242C8—41no—
243C8—4281no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—8564no—
250C22—1072no—
251C22—1072no—
252C22—4282no—
253C22—4282no—
254C22—4281no—
255C22—4281no—
256C22—4281no—
257C22—2141no—
258C22—2141no—
259C22—2141no—
260C22—2141no—
261C22C22⋊C4×D10711yescharacteristic
262C22D8×D10711yescharacteristic
263C22SD16×D10711yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—81no—
268C4—8561no—
269C4—1072no—
270C4—4281no—
271C4—2141no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—8561no—
276C2—2142no—
277C2—2142no—
278C2#99×D10711yescharacteristic
279C2#102×D10711yescharacteristic
280C2#2029264211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107C2×C22⋊C4
prime radical#121×D107C2
cosocleC22×C4×C107C23
hypercenter#327D107
hypocenterC107#1519
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D107C2semidirect
3. #20292486C2semidirect
4. #20292362C2semidirect
5. #20292361C2semidirect
6. #20292360C2semidirect
7. #327×C107C2semidirect
9. C2×D4×D107C4semidirect
12. #16812084C4semidirect
16. #16812080C4semidirect
18. C4⋊C4×D107C4semidirect
21. C22⋊(C107⋊C8)C22semidirect
23. C107⋊#95C22semidirect
25. C22⋊C8×C107C22semidirect
26. #95×C107C22semidirect
141. C107#1519semidirect
148. #327D107semidirect
165. C22⋊C8C2×D107semidirect
167. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
#121×D1072
C22⋊C8×D1072
#202924862
#202923622
#202923612
#202923602
#327×C1072
#1519107