Finite Group Databasebeta

← #30774518

Subgroups — #30774518

Order 17152.

420 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#22565414C211yesmaximal, characteristic
2#22560205C211yesmaximal, characteristic
3#22558335C214yesmaximal
4C4×#101×C67C211yesmaximal, characteristic
5C2×#14524043C411yescharacteristic
6#16863420—21no—
7C2×#14523962C2212yes—
8#16863316—21no—
9#16863316—21no—
10#16863303C2212yes—
11#16863294—21no—
12#16863294—21no—
13#16863265C412yes—
14C4×C4⋊Dic67C411yescharacteristic
15#16863165—21no—
16C2×C4×Q8×C67C2211yescharacteristic
17C2×#101×C67C2212yes—
18C4×M16×C67—21no—
19#343×C67—21no—
20C22×Dic134—21no—
21C2×#11761612—22no—
22#14524043—41no—
23#14524043—41no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—21no—
26C2×C4⋊Dic67C2×C412yes—
27C2×Dic67⋊C4—22no—
28C2×Dic67⋊C4—22no—
29C2×C4×Dic67C2×C412yes—
30C4⋊(C67⋊C8)—41no—
31C4⋊(C67⋊C8)—41no—
32#14523962—24no—
33C67⋊#95—21no—
34C67⋊#95—22no—
35C67⋊#95—22no—
36C22×Q8×C67C2311yescharacteristic
37C2×M16×C67—22no—
38C4×Q8×C67—41no—
39C4×Q8×C67—41no—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46#101×C67—22no—
47#101×C67Q814yescomplemented
48#97×C67—41no—
49C4×C8×C67—41no—
50C2×Dic134—42no—
51C2×Dic134—82no—
52C22×Dic67—22no—
53C22×Dic67—22no—
54#11761612—42no—
55#11761612—42no—
56C2×C67⋊C8—42no—
57C4⋊Dic67—24no—
58C4⋊Dic67—41no—
59C4⋊Dic67—41no—
60C4⋊Dic67—42no—
61Dic67⋊C4—82no—
62Dic67⋊C4—82no—
63C4×Dic67—42no—
64C2×Q8×C67—41no—
65C2×Q8×C67—41no—
66C2×Q8×C67C2×Q812yes—
67C2×Q8×C67#4112yes—
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
75M16×C67—24no—
76M16×C67—42no—
77C2×C8×C67—42no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67—21no—
87C23×C67C2×C22⋊C411yescharacteristic
88Q8×C67—42no—
89Q8×C67—42no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—22no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67#9911yescharacteristic
104C2×C4×C67#9911yescharacteristic
105C2×C4×C67C4×Q812yes—
106C2×C4×C67C22⋊Q812yes—
107C2×C4×C67#12411yescharacteristic
108C2×C4×C67C4⋊Q811yescharacteristic
109C2×Dic67—24no—
110C2×Dic67—42no—
111C2×Dic67—42no—
112C2×Dic67—42no—
113C2×Dic67—42no—
114C2×Dic67—42no—
115Dic134—84no—
116C8×C67—44no—
117C67⋊C8—82no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67#39111yescharacteristic
121C22×C67C2×#9911yescharacteristic
122C22×C67#41011yescharacteristic
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—22no—
126C4×C67—22no—
127C4×C67—21no—
128C4×C67—41no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67#35612yes—
132Dic67—44no—
133Dic67—82no—
134#13302—671nomaximal, Sylow
135C2×C67—21no—
136C2×C67—21no—
137C2×C67#199611yescharacteristic
138C2×C67C2×#35611yescharacteristic
139C2×C67#213411yescharacteristic
140#2567—671no—
141#1801—671no—
142C4×#101D6711yescharacteristic
143#1394—674no—
144C67#1330211yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—671no—
146C2×C4×Q8C2×D6711yescharacteristic
147#423—1341no—
148C2×#101—672no—
149C2×#101C2×D6712yes—
150C4×M16—21no—
151#395—1341no—
152#395—1341no—
153#391—672no—
154#389—1341no—
155#389—1341no—
156#384—672no—
157C4×C4⋊C4—671no—
158#343—21no—
159#343—1341no—
160C22×Q8—1341no—
161C22×Q8C22×D6711yescharacteristic
162C2×M16—22no—
163C2×M16—1342no—
164C4⋊Q8—2681no—
165C4⋊Q8—2681no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—672no—
169C2×C4⋊C4—1341no—
170C2×C4⋊C4—1342no—
171C2×C4⋊C4—1342no—
172C2×C4⋊C4—1341no—
173C2×C4⋊C4C4×D6711yescharacteristic
174C2×C4⋊C4D26811yescharacteristic
175C2×C4⋊C4C22⋊D6711yescharacteristic
176C2×C42—672no—
177C2×C42C4×D6711yescharacteristic
178C2×C42D26811yescharacteristic
179C2×C42C22⋊D6711yescharacteristic
180C4⋊C8—2681no—
181C4⋊C8—2681no—
182#101—22no—
183#101—1344no—
184#101Dic13414yescomplemented
185#97—41no—
186C4×C8—41no—
187#95—1341no—
188#95—1342no—
189#95—1342no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—2682no—
193C2×Q8—5362no—
194C2×Q8C2×Dic13412yes—
195C2×Q8#1176162812yes—
196C22×C4—1342no—
197C22×C4—1342no—
198C22×C4(C2×D67)⋊C411yescharacteristic
199C22×C4(C2×D67)⋊C411yescharacteristic
200C22×C4(C2×D67)⋊C411yescharacteristic
201C22×C4(C2×D67)⋊C411yescharacteristic
202C22×C4C2×C4×D6711yescharacteristic
203C22×C4C2×C22⋊D6711yescharacteristic
204C22×C4C2×D26811yescharacteristic
205M16—24no—
206M16—42no—
207M16—2682no—
208M16—2682no—
209C2×C8—42no—
210C2×C8—2682no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—2682no—
215C4⋊C4—2681no—
216C4⋊C4—2681no—
217C4⋊C4—5362no—
218C4⋊C4—5362no—
219C4⋊C4—1344no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—2682no—
227C23C2×(C2×D67)⋊C411yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—5364no—
231C2×C4—21no—
232C2×C4—21no—
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—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—2682no—
245C2×C4—2682no—
246C2×C4—2682no—
247C2×C4—2682no—
248C2×C4—2682no—
249C2×C4—1344no—
250C2×C4#1452395711yescharacteristic
251C2×C4#1452395711yescharacteristic
252C2×C4C4×Dic13412yes—
253C2×C4#1452402212yes—
254C2×C4#1452403111yescharacteristic
255C2×C4#1452404311yescharacteristic
256C8—44no—
257C8—5362no—
258C22—21no—
259C22—21no—
260C22#1686330211yescharacteristic
261C22C2×#1452395711yescharacteristic
262C22#1686336911yescharacteristic
263C4—21no—
264C4—22no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—2684no—
272C4—5362no—
273C4#1686319412yes—
274C2—21no—
275C2—21no—
276C2#2256119411yescharacteristic
277C2C2×#1686319411yescharacteristic
278C2#2256190011yescharacteristic
subgroupG/H
centerC22C2×#14523957
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC4×#101×C67C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC22×Q8×C67C23
hypercenterC4×#101D67
hypocenterC67#13302
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
47. #101×C67Q8semidirect
144. C67#13302semidirect
184. #101Dic134semidirect

Maximal subgroups

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

subgroupindex
#225654142
#225602052
#225583352
C4×#101×C672
#1330267