Finite Group Databasebeta

← #28339864

Subgroups — #28339864

Order 13696.

610 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16812090C211yesmaximal, characteristic
2C2×#16812065C211yesmaximal, characteristic, complemented
3C22×Dic107⋊C4C211yesmaximal, characteristic, complemented
4#20292558C212yesmaximal, complemented
5#20292550C212yesmaximal, complemented
6#20292523C212yesmaximal, complemented
7#20292515C212yesmaximal, complemented
8#20292507C211yesmaximal, characteristic
9#20292502C211yesmaximal, characteristic, complemented
10#20292490C211yesmaximal, characteristic, complemented
11C2×#123×C107C211yesmaximal, characteristic
12C23×Dic107C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14#16812090—24no—
15#16812065—24no—
16C2×C4⋊Dic107C2212yes—
17C2×Dic107⋊C4—24no—
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C2×Dic107⋊C4C2212yes—
23C2×C22⋊Dic107C2211yescharacteristic
24C2×C22⋊Dic107C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2212yes—
28C2×C4×Dic107C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C22×D4×C107C2211yescharacteristic
34C23×C4×C107C2211yescharacteristic
35#123×C107—24no—
36C2×C4⋊C4×C107C2212yes—
37C2×C22⋊C4×C107C2212yes—
38C2×C22⋊C4×C107C2211yescharacteristic
39C22×Dic107—22no—
40C22×Dic107—22no—
41C22×Dic107—22no—
42C22×Dic107—24no—
43C22×Dic107D414yes—
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2312yes—
50C4⋊Dic107—24no—
51Dic107⋊C4—24no—
52Dic107⋊C4—24no—
53Dic107⋊C4—24no—
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56C22⋊Dic107—22no—
57C22⋊Dic107—22no—
58C22⋊Dic107—22no—
59C22⋊Dic107—22no—
60C22⋊Dic107—24no—
61C4×Dic107—24no—
62C24×C107C2311yescharacteristic
63C24×C107C2311yescharacteristic
64C2×D4×C107—22no—
65C2×D4×C107—22no—
66C2×D4×C107—22no—
67C22×C4×C107—22no—
68C22×C4×C107—22no—
69C22×C4×C107—22no—
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2312yes—
73C4⋊C4×C107—24no—
74C22⋊C4×C107—24no—
75C22⋊C4×C107D414yes—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—21no—
79C23×C107—21no—
80C23×C107—21no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107C2×D412yes—
84C23×C107C2×D412yes—
85C23×C107#4112yes—
86C23×C107C2411yescharacteristic
87D4×C107—44no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—21no—
96C2×C4×C107—44no—
97C2×C4×C107C2×D414yes—
98C2×Dic107—28no—
99C2×Dic107—28no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107—22no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107C2×D412yes—
114C2×Dic107C2×D412yes—
115C2×Dic107#4114yes—
116C22×C107—22no—
117C22×C107—22no—
118C22×C107—24no—
119C22×C107—24no—
120C22×C107—41no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107#12114yes—
126C22×C107C22×D411yescharacteristic
127C22×C107C22×D411yescharacteristic
128C22×C107C2×#4112yes—
129C22×C107C2×#4111yescharacteristic
130C22×C1072−1+412yes—
131C4×C107—24no—
132C4×C107—42no—
133C4×C107—41no—
134Dic107—22no—
135Dic107—22no—
136Dic107—24no—
137Dic107—41no—
138Dic107—41no—
139Dic107—42no—
140C2×C107—24no—
141C2×C107—81no—
142C2×C107C2×#12111yescharacteristic
143C2×C107#52511yescharacteristic
144C2×C107#53611yescharacteristic
145C2×C107#54712yes—
146C2×C107#55212yes—
147#2413—1071nomaximal, Sylow
148C107#241311yescharacteristic, complemented, Sylow
149C2×#123—1071no—
150C2×#123D10711yescharacteristic
151C2×#117—1071no—
152C22×C4⋊C4—1071no—
153#398—1072no—
154#396—1072no—
155#389—1072no—
156#388—1072no—
157#386—1071no—
158#385—1071no—
159#381—1071no—
160C22×D4C2×D10711yescharacteristic
161C23×C4—1071no—
162C23×C4—1071no—
163C23×C4C2×D10711yescharacteristic
164#123—24no—
165#123—2144no—
166#117—2144no—
167C2×C4⋊C4—1072no—
168C2×C4⋊C4—1072no—
169C2×C4⋊C4—1072no—
170C2×C4⋊C4—1071no—
171C2×C4⋊C4—1071no—
172C2×C4⋊C4—1072no—
173C2×C4⋊C4—2144no—
174C2×C4⋊C4C2×D10712yes—
175C2×C22⋊C4—1071no—
176C2×C22⋊C4—1071no—
177C2×C22⋊C4—1071no—
178C2×C22⋊C4—1071no—
179C2×C22⋊C4—1072no—
180C2×C22⋊C4C2×D10712yes—
181C2×C22⋊C4C2×D10711yescharacteristic
182C2×C42—1072no—
183#95—1072no—
184#95—1072no—
185#95—1072no—
186#95—1072no—
187C24C22×D10711yescharacteristic
188C24C22×D10711yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—1074no—
196C22×C4—1072no—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—1072no—
201C22×C4—1071no—
202C22×C4—2144no—
203C22×C4—2142no—
204C22×C4—2142no—
205C22×C4—2142no—
206C22×C4C22×D10712yes—
207C22×C4C22×D10711yescharacteristic
208C22×C4C22×D10712yes—
209C4⋊C4—24no—
210C4⋊C4—2144no—
211C4⋊C4—2144no—
212C4⋊C4—2142no—
213C4⋊C4—2142no—
214C4⋊C4—2144no—
215C4⋊C4—2144no—
216C22⋊C4—24no—
217C22⋊C4—2142no—
218C22⋊C4—2142no—
219C22⋊C4—2144no—
220C22⋊C4—2142no—
221C22⋊C4—2142no—
222C22⋊C4C22⋊D10714yes—
223C42—2144no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—22no—
231C23D4×D10712yes—
232C23C2×C22⋊D10712yes—
233C23#1313381712yes—
234C23C23×D10711yescharacteristic
235D4—44no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—1072no—
246C2×C4—1072no—
247C2×C4—1074no—
248C2×C4—2141no—
249C2×C4—2141no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2148no—
256C2×C4—2148no—
257C2×C4—2142no—
258C2×C4—2141no—
259C2×C4—2141no—
260C2×C4—2142no—
261C2×C4—2141no—
262C2×C4—2141no—
263C2×C4C2×C22⋊D10714yes—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—41no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—41no—
273C22#1681208014yes—
274C22C2×D4×D10711yescharacteristic
275C22C22×C22⋊D10711yescharacteristic
276C22#41×D10712yes—
277C22C2×#1313381711yescharacteristic
278C22#1681215612yes—
279C4—24no—
280C4—42no—
281C4—41no—
282C4—4281no—
283C4—4281no—
284C4—4282no—
285C4—2144no—
286C4—2142no—
287C4—2142no—
288C2—24no—
289C2—81no—
290C2C2×#1681208011yescharacteristic
291C2#2029313511yescharacteristic
292C2#2029319911yescharacteristic
293C2#2029327612yes—
294C2#2029331412yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
prime radicalC22×D4×C107C22
hypercenterC2×#123D107
hypocenterC107#2413
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16812065C2semidirect
3. C22×Dic107⋊C4C2semidirect
4. #20292558C2semidirect
5. #20292550C2semidirect
6. #20292523C2semidirect
7. #20292515C2semidirect
9. #20292502C2semidirect
10. #20292490C2semidirect
148. C107#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#168120902
C2×#168120652
C22×Dic107⋊C42
#202925582
#202925502
#202925232
#202925152
#202925072
#202925022
#202924902
C2×#123×C1072
#2413107