Finite Group Databasebeta

← #30766200

Subgroups — #30766200

Order 17152.

434 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 95 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, complemented
2#22558533C211yesmaximal, characteristic, complemented
3#22558236C212yesmaximal, complemented
4#22558212C212yesmaximal
5C67×#2398C211yesmaximal, characteristic
6#16863449—21no—
7C2×#14523962—21no—
8C4×#11761612—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×C4×Q8×C67C2211yescharacteristic
13C2×C4×D4×C67C411yescharacteristic
14#400×C67C411yescharacteristic
15#400×C67C2211yescharacteristic
16#397×C67—21no—
17#394×C67C411yescharacteristic
18#394×C67C2211yescharacteristic
19#393×C67—21no—
20C4×C22⋊C4×C67—21no—
21C4×C22⋊C4×C67—21no—
22#376×C67C411yescharacteristic
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—41no—
25#14523962—24no—
26C22⋊(C67⋊C8)—22no—
27#14523954—41no—
28C4×C67⋊C8—41no—
29C4×C67⋊C8—41no—
30C22×Q8×C67D411yescharacteristic
31C22×D4×C67C2×C411yescharacteristic
32C23×C4×C67—21no—
33C4×Q8×C67—41no—
34C4×Q8×C67—41no—
35C4×D4×C67—41no—
36C4×D4×C67—41no—
37C2×C4⋊C4×C67C2×C411yescharacteristic
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C4⋊C4×C67C2311yescharacteristic
41C2×C22⋊C4×C67—21no—
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C42×C67—21no—
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67D411yescharacteristic
49C2×C42×C67D411yescharacteristic
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67C2×C411yescharacteristic
56#95×C67C2×C411yescharacteristic
57#11761612—24no—
58#11761612—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C24×C67—21no—
62C2×Q8×C67—22no—
63C2×Q8×C67—22no—
64C2×Q8×C67—41no—
65C2×D4×C67—41no—
66C2×D4×C67—42no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67—41no—
73C22×C4×C67—41no—
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22×C411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80C22×C4×C67C2×D411yescharacteristic
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—41no—
84C4⋊C4×C67—41no—
85C22⋊C4×C67—42no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C22⋊C4×C67—42no—
89C22⋊C4×C67—42no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—21no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67—21no—
96C42×C67—42no—
97C23×C67—22no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67—41no—
104C23×C67C2×C22⋊C411yescharacteristic
105Q8×C67—44no—
106D4×C67—42no—
107D4×C67—42no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—22no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—82no—
132C2×C4×C67—82no—
133C2×C4×C67#9911yescharacteristic
134C2×C4×C67#9911yescharacteristic
135C2×C4×C67#10412yes—
136C2×C4×C67#13512yes—
137C67⋊C8—44no—
138C67⋊C8—82no—
139C22×C67—21no—
140C22×C67—21no—
141C22×C67—42no—
142C22×C67—42no—
143C22×C67—42no—
144C22×C67—81no—
145C22×C67—81no—
146C22×C67—81no—
147C22×C67—81no—
148C22×C67C2×#9911yescharacteristic
149C22×C67#41611yescharacteristic
150C22×C67C2×#10411yescharacteristic
151C4×C67—21no—
152C4×C67—22no—
153C4×C67—22no—
154C4×C67—41no—
155C4×C67—41no—
156C4×C67—41no—
157C4×C67—41no—
158C4×C67—81no—
159C4×C67—81no—
160C4×C67—81no—
161C4×C67#35512yes—
162#11734—671nomaximal, Sylow
163C2×C67—21no—
164C2×C67—21no—
165C2×C67—82no—
166C2×C67#150111yescharacteristic
167C2×C67#212011yescharacteristic
168C2×C67C2×#35511yescharacteristic
169#2398D6711yescharacteristic
170C4×#101—671no—
171#1461—671no—
172#1353—672no—
173#1345—672no—
174C67#1173411yescharacteristic, complemented, Sylow
175C2×C4×Q8C2×D6711yescharacteristic
176C2×C4×D4Dic6711yescharacteristic
177#431—1341no—
178C2×#101—1341no—
179C4×M16—1341no—
180#400Dic6711yescharacteristic
181#400C2×D6711yescharacteristic
182#397—21no—
183#394Dic6711yescharacteristic
184#394C2×D6711yescharacteristic
185#393—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188#376Dic6711yescharacteristic
189#343—672no—
190#324—672no—
191#324—1341no—
192C22×Q8C22⋊D6711yescharacteristic
193C22×D4C2×Dic6711yescharacteristic
194C23×C4—21no—
195C2×M16—1342no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4C2×Dic6711yescharacteristic
201C2×C4⋊C4C2×Dic6711yescharacteristic
202C2×C4⋊C4C22⋊D6711yescharacteristic
203C2×C4⋊C4C22×D6711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C2×Dic6711yescharacteristic
211C2×C42C22⋊D6711yescharacteristic
212C2×C42C22⋊D6711yescharacteristic
213C4⋊C8—2681no—
214#101—1344no—
215C22⋊C8—1342no—
216#97—2681no—
217C4×C8—2681no—
218C4×C8—2681no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C2×Dic6711yescharacteristic
225#95C2×Dic6711yescharacteristic
226C24—21no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—41no—
231C2×D4—42no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4C22⋊Dic6711yescharacteristic
240C22×C4C22⋊Dic6711yescharacteristic
241C22×C4C22⋊Dic6711yescharacteristic
242C22×C4C22⋊Dic6711yescharacteristic
243C22×C4C22×Dic6711yescharacteristic
244C22×C4C2×C22⋊D6711yescharacteristic
245C22×C4C2×C22⋊D6711yescharacteristic
246M16—2682no—
247M16—1344no—
248C2×C8—2682no—
249C2×C8—2682no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—42no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C2×C22⋊Dic6711yescharacteristic
274Q8—44no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#1452395811yescharacteristic
303C2×C4#1452395811yescharacteristic
304C2×C4#1452397012yes—
305C2×C4#1452406312yes—
306C8—2684no—
307C8—5362no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22C2×#1452395811yescharacteristic
318C22#1686339211yescharacteristic
319C22C2×#1452397011yescharacteristic
320C4—21no—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4#1686319312yes—
331C2—21no—
332C2—21no—
333C2—82no—
334C2#2255870011yescharacteristic
335C2#2256185011yescharacteristic
336C2C2×#1686319311yescharacteristic
subgroupG/H
centerC2×C4#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2398C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2398D67
hypocenterC67#11734
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. C4×#14523962C2semidirect
2. #22558533C2semidirect
3. #22558236C2semidirect
174. C67#11734semidirect

Maximal subgroups

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

subgroupindex
C4×#145239622
#225585332
#225582362
#225582122
C67×#23982
#1173467