Finite Group Databasebeta

← #30774158

Subgroups — #30774158

Order 17152.

472 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16863195C214yesmaximal
2#22560205C211yesmaximal, characteristic
3C4×#14523962C211yesmaximal, characteristic
4C67×#2546C211yesmaximal, characteristic
5#16863420—21no—
6C2×#14523962C2212yes—
7C2×#14523962C2212yes—
8C4×#11761612—21no—
9#16863195—28no—
10#16863165—21no—
11#16863165—21no—
12C2×C4⋊Q8×C67C411yescharacteristic
13C2×C4⋊Q8×C67C2212yes—
14C2×C4×Q8×C67C2211yescharacteristic
15#398×C67—21no—
16#398×C67—21no—
17#391×C67C412yes—
18#389×C67—21no—
19#389×C67—21no—
20C4×C4⋊C4×C67C411yescharacteristic
21C2×#11761612—22no—
22C2×#11761612—22no—
23C4⋊(C67⋊C8)—41no—
24C4⋊(C67⋊C8)—41no—
25#14523962—24no—
26#14523962—22no—
27#14523962D414yes—
28#14523954—41no—
29C4×C67⋊C8—41no—
30C22×Q8×C67C2×C412yes—
31C22×Q8×C67C2311yescharacteristic
32C4⋊Q8×C67—24no—
33C4⋊Q8×C67—24no—
34C4⋊Q8×C67—41no—
35C4⋊Q8×C67—41no—
36C4×Q8×C67—41no—
37C4×Q8×C67—41no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—22no—
42C2×C4⋊C4×C67—22no—
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C4⋊C4×C67D411yescharacteristic
46C2×C42×C67C2×C412yes—
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67D411yescharacteristic
49C2×C42×C67D411yescharacteristic
50#95×C67—21no—
51#95×C67—22no—
52#95×C67—22no—
53#11761612—24no—
54#11761612—42no—
55#11761612—42no—
56#11761612—42no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C2×Q8×C67—24no—
60C2×Q8×C67—41no—
61C2×Q8×C67—41no—
62C2×Q8×C67—42no—
63C2×Q8×C67—42no—
64C2×Q8×C67C2×D412yes—
65C2×Q8×C67#4112yes—
66C22×C4×C67—22no—
67C22×C4×C67—22no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—24no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—44no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—21no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—24no—
92C23×C67C2×C22⋊C411yescharacteristic
93Q8×C67—42no—
94Q8×C67—42no—
95Q8×C67—84no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—22no—
103C2×C4×C67—24no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67#9911yescharacteristic
116C2×C4×C67#9911yescharacteristic
117C2×C4×C67C4×D412yes—
118C2×C4×C67#12112yes—
119C2×C4×C67#12411yescharacteristic
120C2×C4×C67#12711yescharacteristic
121C67⋊C8—44no—
122C67⋊C8—82no—
123C22×C67—21no—
124C22×C67—21no—
125C22×C67#39011yescharacteristic
126C22×C67C2×#9911yescharacteristic
127C22×C67#41011yescharacteristic
128C4×C67—21no—
129C4×C67—22no—
130C4×C67—21no—
131C4×C67—22no—
132C4×C67—21no—
133C4×C67—44no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—41no—
137C4×C67—82no—
138C4×C67#35612yescomplemented
139#13227—671nomaximal, Sylow
140C2×C67—21no—
141C2×C67—21no—
142C2×C67#196411yescharacteristic
143C2×C67C2×#35611yescharacteristic
144C2×C67#213511yescharacteristic
145#2546D6711yescharacteristic
146C2×#356—674no—
147#1801—671no—
148C4×#101—671no—
149C67#1322711yescharacteristic, complemented, Sylow
150C2×C4⋊Q8Dic6711yescharacteristic
151C2×C4⋊Q8C2×D6712yes—
152C2×C4×Q8C2×D6711yescharacteristic
153#423—1341no—
154C2×#101—672no—
155C2×#101—672no—
156C4×M16—1341no—
157#398—21no—
158#398—21no—
159#391Dic6712yes—
160#389—21no—
161#389—21no—
162C4×C4⋊C4Dic6711yescharacteristic
163#356—1348no—
164#343—1341no—
165#343—1341no—
166C22×Q8C2×Dic6712yes—
167C22×Q8C22×D6711yescharacteristic
168C2×M16—1342no—
169C2×M16—1342no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×Dic6711yescharacteristic
182C2×C4⋊C4C22⋊D6711yescharacteristic
183C2×C4⋊C4C22⋊D6711yescharacteristic
184C2×C42C2×Dic6712yes—
185C2×C42C2×Dic6711yescharacteristic
186C2×C42C22⋊D6711yescharacteristic
187C2×C42C22⋊D6711yescharacteristic
188C4⋊C8—2681no—
189C4⋊C8—2681no—
190#101—674no—
191#101—1344no—
192#101—1342no—
193#97—2681no—
194C4×C8—2681no—
195#95—21no—
196#95—22no—
197#95—22no—
198C2×Q8—24no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8D4×D6712yes—
204C2×Q8#1176162712yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4C22⋊Dic6711yescharacteristic
208C22×C4C22⋊Dic6711yescharacteristic
209C22×C4C22⋊Dic6711yescharacteristic
210C22×C4C22⋊Dic6711yescharacteristic
211C22×C4C22×Dic6711yescharacteristic
212C22×C4C2×C22⋊D6711yescharacteristic
213C22×C4C2×C22⋊D6711yescharacteristic
214M16—2682no—
215M16—2682no—
216M16—2682no—
217M16—1344no—
218C2×C8—2682no—
219C2×C8—2682no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—24no—
237C23C2×C22⋊Dic6711yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4#1452395811yescharacteristic
261C2×C4#1452395811yescharacteristic
262C2×C4D4×Dic6712yes—
263C2×C4#1452401412yes—
264C2×C4#1452403011yescharacteristic
265C2×C4#1452403911yescharacteristic
266C8—2684no—
267C8—5362no—
268C22—21no—
269C22—21no—
270C22#1686329911yescharacteristic
271C22C2×#1452395811yescharacteristic
272C22#1686337011yescharacteristic
273C4—22no—
274C4—21no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4#1686319512yescomplemented
284C2—21no—
285C2—21no—
286C2#2256104611yescharacteristic
287C2C2×#1686319511yescharacteristic
288C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC67×#2546C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC22×Q8×C67C23
hypercenter#2546D67
hypocenterC67#13227
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
138. C4×C67#356semidirect
149. C67#13227semidirect
283. C4#16863195semidirect

Maximal subgroups

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

subgroupindex
C2×#168631952
#225602052
C4×#145239622
C67×#25462
#1322767