Finite Group Databasebeta

← #30749447

Subgroups — #30749447

Order 17152.

384 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#14523962C211yesmaximal, characteristic
2#22558553C211yesmaximal, characteristic
3#22558102C212yesmaximal
4#22558073C212yesmaximal
5C67×#2356C211yesmaximal, characteristic
6#16863453C2212yes—
7C2×#14523962—21no—
8C4×#11761612C2212yes—
9#16863165—21no—
10#16863122—21no—
11#16863122—21no—
12C2×C4×Q8×C67C2211yescharacteristic
13#391×C67—21no—
14#387×C67—21no—
15#387×C67—21no—
16#383×C67C411yescharacteristic
17#383×C67C2211yescharacteristic
18#382×C67C411yescharacteristic
19C4×C4⋊C4×C67—21no—
20C4×C4⋊C4×C67C411yescharacteristic
21#376×C67C411yescharacteristic
22#376×C67C2211yescharacteristic
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—22no—
25#14523962—24no—
26C22⋊(C67⋊C8)—22no—
27#14523954—22no—
28#14523954—22no—
29C4×C67⋊C8—22no—
30C22×Q8×C67D411yescharacteristic
31C4×Q8×C67—22no—
32C4×Q8×C67—22no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67D411yescharacteristic
36C2×C4⋊C4×C67D411yescharacteristic
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C42×C67—21no—
39C2×C42×C67—21no—
40C2×C42×C67—21no—
41C2×C42×C67C2×C411yescharacteristic
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67C2311yescharacteristic
44#95×C67—21no—
45#95×C67—21no—
46#95×C67—21no—
47#95×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67C2×C411yescharacteristic
52#95×C67C2×C411yescharacteristic
53#95×C67C2×C411yescharacteristic
54#95×C67C2×C411yescharacteristic
55#11761612—24no—
56#11761612—42no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C2×Q8×C67—22no—
60C2×Q8×C67—22no—
61C2×Q8×C67—41no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22×C411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C4⋊C4×C67—22no—
74C4⋊C4×C67—22no—
75C4⋊C4×C67—22no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—42no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—42no—
83C42×C67—42no—
84C42×C67—42no—
85C42×C67#4112yes—
86C42×C67#4112yes—
87C23×C67C2×C22⋊C411yescharacteristic
88Q8×C67—44no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—22no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67#10412yes—
115C2×C4×C67#11711yescharacteristic
116C2×C4×C67#11711yescharacteristic
117C2×C4×C67#12312yes—
118C2×C4×C67#12312yes—
119C67⋊C8—44no—
120C67⋊C8—82no—
121C22×C67—21no—
122C22×C67—21no—
123C22×C67#38111yescharacteristic
124C22×C67C2×#10411yescharacteristic
125C22×C67#42111yescharacteristic
126C4×C67—22no—
127C4×C67—22no—
128C4×C67—22no—
129C4×C67—21no—
130C4×C67—21no—
131C4×C67—42no—
132C4×C67—41no—
133C4×C67—41no—
134C4×C67—41no—
135C4×C67—81no—
136C4×C67—81no—
137C4×C67—81no—
138#8774—671nomaximal, Sylow
139C2×C67—21no—
140C2×C67—21no—
141C2×C67#182611yescharacteristic
142C2×C67#182711yescharacteristic
143C2×C67#183011yescharacteristic
144#2356D6711yescharacteristic
145C4×#101—671no—
146#1467—671no—
147#1300—672no—
148#1288—672no—
149C67#877411yescharacteristic, complemented, Sylow
150C2×C4×Q8C2×D6711yescharacteristic
151#432—672no—
152C2×#101—1341no—
153C4×M16—672no—
154#391—21no—
155#387—21no—
156#387—21no—
157#383Dic6711yescharacteristic
158#383C2×D6711yescharacteristic
159#382Dic6711yescharacteristic
160C4×C4⋊C4—21no—
161C4×C4⋊C4Dic6711yescharacteristic
162#376Dic6711yescharacteristic
163#376C2×D6711yescharacteristic
164#343—1341no—
165#324—1341no—
166#324—1341no—
167C22×Q8C22⋊D6711yescharacteristic
168C2×M16—1342no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C22⋊D6711yescharacteristic
174C2×C4⋊C4C22⋊D6711yescharacteristic
175C2×C4⋊C4C22⋊D6711yescharacteristic
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×Dic6711yescharacteristic
180C2×C42C2×Dic6711yescharacteristic
181C2×C42C22×D6711yescharacteristic
182C4⋊C8—1342no—
183#101—1344no—
184C22⋊C8—1342no—
185#97—1342no—
186#97—1342no—
187C4×C8—1342no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95C2×Dic6711yescharacteristic
196#95C2×Dic6711yescharacteristic
197#95C2×Dic6711yescharacteristic
198#95C2×Dic6711yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4C22⋊Dic6711yescharacteristic
207C22×C4C22⋊Dic6711yescharacteristic
208C22×C4C22⋊Dic6711yescharacteristic
209C22×C4C22⋊Dic6711yescharacteristic
210C22×C4C22×Dic6711yescharacteristic
211C22×C4C2×C22⋊D6711yescharacteristic
212C22×C4C2×C22⋊D6711yescharacteristic
213M16—2682no—
214M16—1344no—
215C2×C8—2682no—
216C2×C8—2682no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42#1176162812yes—
230C42#1176162812yes—
231C23C2×C22⋊Dic6711yescharacteristic
232Q8—44no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4#1452397012yes—
259C2×C4#1452400211yescharacteristic
260C2×C4#1452400211yescharacteristic
261C2×C4#1452402612yes—
262C2×C4#1452402612yes—
263C8—2684no—
264C8—5362no—
265C22—21no—
266C22—21no—
267C22#1686325911yescharacteristic
268C22C2×#1452397011yescharacteristic
269C22#1686341311yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C2—21no—
283C2—21no—
284C2#2256030811yescharacteristic
285C2#2256031311yescharacteristic
286C2#2256032311yescharacteristic
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2356C2
socleC22×C67#381
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2356D67
hypocenterC67#8774
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
149. C67#8774semidirect

Maximal subgroups

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

subgroupindex
C4×#145239622
#225585532
#225581022
#225580732
C67×#23562
#877467