Finite Group Databasebeta

← #28339868

Subgroups — #28339868

Order 13696.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812092C211yesmaximal, characteristic
2C2×#16812092C211yesmaximal, characteristic
3C2×#16812068C211yesmaximal, characteristic, complemented
4#20292560C212yesmaximal, complemented
5#20292549C212yesmaximal, complemented
6#20292522C212yesmaximal, complemented
7#20292517C212yesmaximal, complemented
8#20292506C211yesmaximal, characteristic
9#20292500C211yesmaximal, characteristic, complemented
10#20292490C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13#16812092—24no—
14#16812092—24no—
15#16812068—24no—
16C2×C4⋊Dic107C2212yes—
17C2×Dic107⋊C4C2211yescharacteristic
18C2×Dic107⋊C4C2212yes—
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×C22⋊Dic107C2211yescharacteristic
22C2×C22⋊Dic107C2211yescharacteristic
23C2×C22⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C23×C4×C107C2211yescharacteristic
34C23×C4×C107C2211yescharacteristic
35C23×C4×C107C2211yescharacteristic
36C2×C4⋊C4×C107—24no—
37C2×C4⋊C4×C107C2212yes—
38C2×C4⋊C4×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×C22⋊D107—22no—
41C2×C22⋊D107—22no—
42C2×C22⋊D107—22no—
43C22×Dic107C2312yes—
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2312yes—
48C4⋊Dic107—24no—
49Dic107⋊C4—22no—
50Dic107⋊C4—22no—
51Dic107⋊C4—24no—
52Dic107⋊C4—24no—
53C22⋊Dic107—22no—
54C22⋊Dic107—22no—
55C22⋊Dic107D414yes—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—24no—
60C4×Dic107—24no—
61C24×C107C2311yescharacteristic
62C22×C4×C107—22no—
63C22×C4×C107—22no—
64C22×C4×C107—22no—
65C22×C4×C107—24no—
66C22×C4×C107—22no—
67C22×C4×C107—22no—
68C22×C4×C107—22no—
69C22×C4×C107D414yes—
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2312yes—
74C4⋊C4×C107—24no—
75C4⋊C4×C107—24no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107C2×D412yes—
79C23×C107C2×D412yes—
80C23×C107#4112yes—
81C23×C107C2411yescharacteristic
82C22×D107—21no—
83C22×D107—21no—
84C22×D107—22no—
85C22×D107—21no—
86C22×D107—22no—
87C2×C4×C107—28no—
88C2×C4×C107—28no—
89C2×C4×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—44no—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D412yes—
101C2×C4×C107#4114yes—
102C22⋊D107—44no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—21no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107C2×D414yes—
116C22×C107—22no—
117C22×C107—22no—
118C22×C107—24no—
119C22×C107—24no—
120C22×C107#12114yes—
121C22×C107C22×D411yescharacteristic
122C22×C107C22×D411yescharacteristic
123C22×C107C2×#4111yescharacteristic
124C22×C107C2×#4112yes—
125C22×C1072−1+412yes—
126C2×D107—41no—
127C2×D107—41no—
128C2×D107—41no—
129C2×D107—42no—
130C2×D107—42no—
131C4×C107—22no—
132C4×C107—22no—
133C4×C107—24no—
134C4×C107—42no—
135Dic107—24no—
136Dic107—41no—
137Dic107—41no—
138Dic107—41no—
139Dic107—42no—
140C2×C107—24no—
141C2×C107C2×#12111yescharacteristic
142C2×C107#52511yescharacteristic
143C2×C107#53611yescharacteristic
144C2×C107#54712yes—
145C2×C107#55212yes—
146D107—81no—
147#2413—1071nomaximal, Sylow
148C107#241311yescharacteristic, complemented, Sylow
149C2×#123—1071no—
150C2×#123—1071no—
151C2×#117—1071no—
152C22×C4⋊C4D10711yescharacteristic, complemented
153#398—1072no—
154#396—1072no—
155#389—1072no—
156#388—1072no—
157#386—1071no—
158#385—1071no—
159#381—1071no—
160C22×D4—1071no—
161C23×C4C2×D10711yescharacteristic
162C23×C4C2×D10711yescharacteristic
163C23×C4C2×D10711yescharacteristic
164#123—2144no—
165#123—2144no—
166#117—2144no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—1072no—
169C2×C4⋊C4—1071no—
170C2×C4⋊C4—1072no—
171C2×C4⋊C4—1071no—
172C2×C4⋊C4—1072no—
173C2×C4⋊C4C2×D10712yes—
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⋊C4—1071no—
181C2×C22⋊C4—1072no—
182C2×C42—1072no—
183#95—1072no—
184#95—1072no—
185#95—1072no—
186#95—1072no—
187C24—1071no—
188C24C22×D10711yescharacteristic
189C2×D4—2142no—
190C2×D4—2142no—
191C2×D4—2142no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—1072no—
200C22×C4—1071no—
201C22×C4—1072no—
202C22×C4—1071no—
203C22×C4—1071no—
204C22×C4C22⋊D10714yes—
205C22×C4C22×D10712yes—
206C22×C4C22×D10711yescharacteristic
207C22×C4C22×D10711yescharacteristic
208C22×C4C22×D10712yes—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—2144no—
212C4⋊C4—2144no—
213C4⋊C4—2142no—
214C4⋊C4—2142no—
215C4⋊C4—2144no—
216C22⋊C4—1074no—
217C22⋊C4—2144no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C22⋊C4—2142no—
222C22⋊C4—2144no—
223C42—2144no—
224C23—22no—
225C23—22no—
226C23—2141no—
227C23—2141no—
228C23—2141no—
229C23—2142no—
230C23—2142no—
231C23D4×D10712yes—
232C23C2×C22⋊D10712yes—
233C23#1313381712yes—
234C23C23×D10711yescharacteristic
235D4—4284no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—1074no—
249C2×C4—2142no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4—2141no—
253C2×C4—2141no—
254C2×C4—2141no—
255C2×C4—2142no—
256C2×C4—2142no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4—2142no—
260C2×C4—2142no—
261C2×C4C2×C22⋊D10712yes—
262C2×C4C2×C22⋊D10712yes—
263C2×C4#1313381814yes—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—4281no—
269C22—4281no—
270C22—4282no—
271C22—4282no—
272C22—4281no—
273C22#1681208114yes—
274C22C2×D4×D10711yescharacteristic
275C22C22×C22⋊D10711yescharacteristic
276C22C2×#1313381711yescharacteristic
277C22C2×#1313381812yes—
278C22#1681215712yes—
279C4—24no—
280C4—22no—
281C4—22no—
282C4—42no—
283C4—4281no—
284C4—4281no—
285C4—4282no—
286C4—4281no—
287C4—2144no—
288C2—24no—
289C2—8561no—
290C2C2×#1681208111yescharacteristic
291C2#2029313811yescharacteristic
292C2#2029320111yescharacteristic
293C2#2029327812yes—
294C2#2029331212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C4⋊C4×C107C2
prime radicalC22×C22⋊D107C22
hypercenterC22×C4⋊C4D107
hypocenterC107#2413
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16812068C2semidirect
4. #20292560C2semidirect
5. #20292549C2semidirect
6. #20292522C2semidirect
7. #20292517C2semidirect
9. #20292500C2semidirect
10. #20292490C2semidirect
11. C22×C4⋊C4×C107C2semidirect
148. C107#2413semidirect
152. C22×C4⋊C4D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120922
C2×#168120682
#202925602
#202925492
#202925222
#202925172
#202925062
#202925002
#202924902
C22×C4⋊C4×C1072
#2413107