Finite Group Databasebeta

← #30749462

Subgroups — #30749462

Order 17152.

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

normal subgroups only

This group has 480 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×#14523958C211yesmaximal, characteristic
2#22558547C211yesmaximal, characteristic
3#22558095C212yesmaximal, complemented
4#22558061C212yesmaximal, complemented
5C67×#2353C211yesmaximal, characteristic
6C2×#14523958—21no—
7#16863360C2212yes—
8C4×C22⋊Dic67C2212yes—
9#16863163—21no—
10#16863122—21no—
11#16863120—21no—
12C2×C4×D4×C67C2211yescharacteristic
13#388×C67—21no—
14#387×C67—21no—
15#386×C67—21no—
16#383×C67C411yescharacteristic, complemented
17#381×C67C2211yescharacteristic
18#380×C67C411yescharacteristic, complemented
19C4×C22⋊C4×C67—21no—
20C4×C22⋊C4×C67C2211yescharacteristic
21#376×C67C411yescharacteristic, complemented
22C2×#95×C67C411yescharacteristic, complemented
23C2×#11761612—22no—
24C2×C22⋊Dic67—22no—
25C2×C4×Dic67—22no—
26#14523958—24no—
27C22⋊(C67⋊C8)—22no—
28C22⋊(C67⋊C8)—22no—
29C67⋊#95—22no—
30C22×D4×C67D411yescharacteristic
31C23×C4×C67—21no—
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67C2311yescharacteristic
34C4×D4×C67—22no—
35C4×D4×C67—22no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C22⋊C4×C67—21no—
39C2×C22⋊C4×C67—21no—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67—21no—
45C2×C42×C67C2×C411yescharacteristic
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
55C22×Dic67—42no—
56#11761612—82no—
57C2×C67⋊C8—42no—
58C22⋊Dic67—24no—
59C22⋊Dic67—42no—
60C4×Dic67—82no—
61C24×C67C22⋊C411yescharacteristic
62C24×C67C2×D411yescharacteristic
63C2×D4×C67—22no—
64C2×D4×C67—22no—
65C2×D4×C67—41no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—42no—
73C22×C4×C67—42no—
74C22×C4×C67—81no—
75C22×C4×C67—81no—
76C22×C4×C67—81no—
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C2×D411yescharacteristic
82C22×C4×C67#4112yes—
83C22×C4×C67#4112yes—
84C4⋊C4×C67—22no—
85C4⋊C4×C67—81no—
86C22⋊C4×C67—22no—
87C22⋊C4×C67—22no—
88C22⋊C4×C67—42no—
89C22⋊C4×C67—42no—
90C22⋊C4×C67—81no—
91C42×C67—21no—
92C42×C67—21no—
93C42×C67—42no—
94C42×C67—81no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67—41no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67#10412yes—
104C23×C67C2×C22⋊C411yescharacteristic
105C23×C67#12312yes—
106D4×C67—44no—
107C2×C4×C67—22no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
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—81no—
131C2×C4×C67—81no—
132C2×C4×C67—84no—
133C2×C4×C67#11711yescharacteristic
134C2×C4×C67#11711yescharacteristic
135C2×C4×C67#12312yes—
136C2×Dic67—44no—
137C2×Dic67—82no—
138C2×Dic67—82no—
139C67⋊C8—82no—
140C22×C67—21no—
141C22×C67—21no—
142C22×C67—22no—
143C22×C67—22no—
144C22×C67—22no—
145C22×C67—22no—
146C22×C67—42no—
147C22×C67—42no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67—81no—
151C22×C67—81no—
152C22×C67#38111yescharacteristic
153C22×C67C2×#10411yescharacteristic
154C22×C67#42111yescharacteristic
155C4×C67—22no—
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—42no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—81no—
163C4×C67—81no—
164C4×C67—81no—
165Dic67—84no—
166#8777—671nomaximal, Sylow
167C2×C67—21no—
168C2×C67—21no—
169C2×C67—42no—
170C2×C67—42no—
171C2×C67#182611yescharacteristic
172C2×C67#182711yescharacteristic
173C2×C67#182811yescharacteristic
174#2353D6711yescharacteristic
175C4×#99—671no—
176#1465—671no—
177#1297—672no—
178#1284—672no—
179C67#877711yescharacteristic, complemented, Sylow
180C2×C4×D4C2×D6711yescharacteristic
181C2×#99—1341no—
182#407—672no—
183#388—21no—
184#387—21no—
185#386—21no—
186#383Dic6711yescharacteristic, complemented
187#381C2×D6711yescharacteristic
188#380Dic6711yescharacteristic, complemented
189C4×C22⋊C4—21no—
190C4×C22⋊C4—672no—
191C4×C22⋊C4C2×D6711yescharacteristic
192#376Dic6711yescharacteristic, complemented
193C2×#95Dic6711yescharacteristic, complemented
194#342—1341no—
195#324—1341no—
196#323—1341no—
197C22×D4C22⋊D6711yescharacteristic
198C23×C4—21no—
199C23×C4C2×Dic6711yescharacteristic
200C23×C4C22×D6711yescharacteristic
201C2×M16—1342no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D6711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—1342no—
210C2×C22⋊C4C22⋊D6711yescharacteristic
211C2×C22⋊C4C22⋊D6711yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—1342no—
215C2×C42C2×Dic6711yescharacteristic
216#99—1344no—
217C22⋊C8—1342no—
218C22⋊C8—1342no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—1342no—
225#95C2×Dic6711yescharacteristic
226#95C2×Dic6711yescharacteristic
227#95C2×Dic6711yescharacteristic
228#95C2×Dic6711yescharacteristic
229C24C22⋊Dic6711yescharacteristic
230C24C2×C22⋊D6711yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—2682no—
246C22×C4C22⋊Dic6711yescharacteristic
247C22×C4C22⋊Dic6711yescharacteristic
248C22×C4C22⋊Dic6711yescharacteristic
249C22×C4C22×Dic6711yescharacteristic
250C22×C4C2×C22⋊D6711yescharacteristic
251C22×C4#1176162812yes—
252C22×C4#1176162812yes—
253M16—5362no—
254C2×C8—2682no—
255C4⋊C4—22no—
256C4⋊C4—81no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—2682no—
263C22⋊C4—1344no—
264C42—21no—
265C42—21no—
266C42—42no—
267C42—81no—
268C42—5362no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#1452397012yes—
278C23C2×C22⋊Dic6711yescharacteristic
279C23#1452402612yes—
280D4—44no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—84no—
307C2×C4—2684no—
308C2×C4—5362no—
309C2×C4—5362no—
310C2×C4#1452400211yescharacteristic
311C2×C4#1452400211yescharacteristic
312C2×C4#1452402612yes—
313C8—5362no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22#1686325911yescharacteristic
327C22C2×#1452397011yescharacteristic
328C22#1686341311yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—5364no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—42no—
344C2#2256030811yescharacteristic
345C2#2256031311yescharacteristic
346C2#2256031711yescharacteristic
subgroupG/H
centerC22#16863259
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2353C2
socleC22×C67#381
prime radicalC22×D4×C67D4
cosocleC23×C4×C67C23
hypercenter#2353D67
hypocenterC67#8777
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #22558095C2semidirect
4. #22558061C2semidirect
16. #383×C67C4semidirect
18. #380×C67C4semidirect
21. #376×C67C4semidirect
22. C2×#95×C67C4semidirect
179. C67#8777semidirect
186. #383Dic67semidirect
188. #380Dic67semidirect
192. #376Dic67semidirect
193. C2×#95Dic67semidirect

Maximal subgroups

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

subgroupindex
C4×#145239582
#225585472
#225580952
#225580612
C67×#23532
#877767