Finite Group Databasebeta

← #28344490

Subgroups — #28344490

Order 13696.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812089C211yesmaximal, characteristic, complemented
2C2×#16812082C211yesmaximal, characteristic
3#20292562C211yesmaximal, characteristic, complemented
4#20292560C212yesmaximal, complemented
5#20292544C212yesmaximal, complemented
6#20292543C212yesmaximal, complemented
7#20292542C212yesmaximal, complemented
8#20292540C211yesmaximal, characteristic, complemented
9#20292539C211yesmaximal, characteristic, complemented
10#20292489C211yesmaximal, characteristic, complemented
11#399×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×C22⋊D107C2211yescharacteristic
15C22×C22⋊D107C2211yescharacteristic
16C22×C4×D107C2211yescharacteristic
17#16812089—24no—
18#16812082—24no—
19C2×C4⋊Dic107C2212yes—
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C23×D107C2311yescharacteristic
39C2×Dic214—22no—
40C2×Dic214—42no—
41C2×D428—22no—
42C2×D428—42no—
43C2×C22⋊D107—22no—
44C2×C22⋊D107—22no—
45C2×C22⋊D107—42no—
46C2×C22⋊D107—42no—
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2311yescharacteristic
51C2×C4×D107—42no—
52C2×C4×D107D414yes—
53C4⋊Dic107—24no—
54Dic107⋊C4—22no—
55Dic107⋊C4—42no—
56C22⋊Dic107—42no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—42no—
60(C2×D107)⋊C4—42no—
61(C2×D107)⋊C4—42no—
62C24×C107C2311yescharacteristic
63C22×C4×C107C2312yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107—22no—
68C4⋊C4×C107—42no—
69C22⋊C4×C107—22no—
70C22⋊C4×C107—42no—
71C23×C107—21no—
72C23×C107—22no—
73C23×C107—22no—
74C23×C107—21no—
75C23×C107—21no—
76C23×C107C2411yescharacteristic
77C22×D107—22no—
78C22×D107—22no—
79C22×D107—21no—
80C22×D107—21no—
81C22×D107—21no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—21no—
85C22×D107—21no—
86C22×D107C2×D412yes—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107C2×D412yes—
97C22⋊D107—44no—
98C22⋊D107—44no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107C2×D412yes—
109D428—44no—
110C4×D107—44no—
111Dic214—44no—
112C22×C107—41no—
113C22×C107—41no—
114C22×C107—41no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C22×C107H(5,𝔽2)12yes—
119C22×C107H(5,𝔽2)12yes—
120C22×C1072−1+411yescharacteristic
121C22×C1072−1+411yescharacteristic
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—41no—
128C2×D107—41no—
129C2×D107—41no—
130C2×D107—42no—
131C2×D107—42no—
132C4×C107—22no—
133C4×C107—41no—
134C4×C107—42no—
135C4×C107—42no—
136Dic107—22no—
137Dic107—42no—
138Dic107—41no—
139Dic107—42no—
140C2×C107—81no—
141C2×C107#53411yescharacteristic
142C2×C107#53511yescharacteristic
143C2×C107#53611yescharacteristic
144C2×C107#56212yes—
145C2×C107#56212yes—
146D107—42no—
147D107—81no—
148#2816—1071nomaximal, Sylow
149C107#281611yescharacteristic, complemented, Sylow
150C2×C22⋊Q8—1071no—
151C2×#121—1071no—
152#399—1071no—
153#399D10711yescharacteristic, complemented
154#398—1072no—
155#394—1072no—
156#394—1072no—
157#394—1072no—
158#393—1071no—
159#393—1071no—
160#381—1071no—
161C22×Q8—1071no—
162C22×D4—1071no—
163C22×D4—1071no—
164C22×D4—1071no—
165C23×C4—1071no—
166C22⋊Q8—2144no—
167#121—2144no—
168C2×C4⋊C4—1072no—
169C2×C4⋊C4—1071no—
170C2×C4⋊C4—1072no—
171C2×C4⋊C4C2×D10711yescharacteristic
172C2×C4⋊C4C2×D10712yes—
173C2×C22⋊C4—1072no—
174C2×C22⋊C4—1072no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4—1072no—
177C2×C22⋊C4—1071no—
178C2×C22⋊C4—1072no—
179C2×C22⋊C4C2×D10711yescharacteristic
180C2×C22⋊C4C2×D10712yes—
181#95—1071no—
182#95—1072no—
183#95—1072no—
184#95—1072no—
185#95C2×D10711yescharacteristic
186C24—1071no—
187C24—1071no—
188C24C22×D10711yescharacteristic
189C2×Q8—4282no—
190C2×Q8—2142no—
191C2×D4—4282no—
192C2×D4—4282no—
193C2×D4—4282no—
194C2×D4—2142no—
195C2×D4—2142no—
196C2×D4—2142no—
197C22×C4—1071no—
198C22×C4—1074no—
199C22×C4—1072no—
200C22×C4—1072no—
201C22×C4—1071no—
202C22×C4—4282no—
203C22×C4C22×D10712yes—
204C22×C4C22×D10711yescharacteristic
205C22×C4C22×D10712yes—
206C22×C4C22×D10711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—4282no—
210C4⋊C4—2142no—
211C4⋊C4—2144no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—4282no—
215C22⋊C4—4282no—
216C22⋊C4—4282no—
217C22⋊C4—4282no—
218C22⋊C4—2144no—
219C22⋊C4—2142no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—1072no—
226C23—2141no—
227C23—2142no—
228C23—2142no—
229C23—2141no—
230C23—2141no—
231C23—2142no—
232C23—2142no—
233C23—2141no—
234C23—2141no—
235C23C23×D10711yescharacteristic
236Q8—4284no—
237D4—4284no—
238D4—4284no—
239D4—4284no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—1072no—
250C2×C4—4284no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2142no—
256C2×C4—2142no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4—2142no—
260C2×C4D4×D10712yes—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—42no—
266C22—4281no—
267C22—4281no—
268C22—4281no—
269C22—4282no—
270C22—4282no—
271C22—4282no—
272C22—4282no—
273C22—2142no—
274C22—2142no—
275C22—2142no—
276C22C2×D4×D10711yescharacteristic
277C22#1681215412yes—
278C22#1681215512yes—
279C22#1681215611yescharacteristic
280C22#1681215711yescharacteristic
281C4—22no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—4282no—
286C4—4281no—
287C4—4282no—
288C4—2142no—
289C2—81no—
290C2—8561no—
291C2—4282no—
292C2#2029318811yescharacteristic
293C2#2029319011yescharacteristic
294C2#2029320011yescharacteristic
295C2#2029337312yes—
296C2#2029337412yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
prime radicalC2×#16812082C2
hypercenter#399D107
hypocenterC107#2816
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16812089C2semidirect
3. #20292562C2semidirect
4. #20292560C2semidirect
5. #20292544C2semidirect
6. #20292543C2semidirect
7. #20292542C2semidirect
8. #20292540C2semidirect
9. #20292539C2semidirect
10. #20292489C2semidirect
11. #399×C107C2semidirect
149. C107#2816semidirect
153. #399D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
C2×#168120822
#202925622
#202925602
#202925442
#202925432
#202925422
#202925402
#202925392
#202924892
#399×C1072
#2816107