Finite Group Databasebeta

← #30766548

Subgroups — #30766548

Order 17152.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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#22560286C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3#22558233C212yesmaximal
4#22558230C212yesmaximal
5C67×#2396C211yesmaximal, characteristic
6#16863453—21no—
7#16863259—21no—
8#16863163—21no—
9#16863163C2212yes—
10#16863122—21no—
11#16863120C2212yes—
12C2×C4×Q8×C67C411yescharacteristic, complemented
13C2×#117×C67C2211yescharacteristic
14#400×C67C411yescharacteristic, complemented
15#399×C67C2211yescharacteristic
16#398×C67—21no—
17#396×C67—21no—
18#395×C67C411yescharacteristic, complemented
19#393×C67C2211yescharacteristic
20C4×C4⋊C4×C67—21no—
21C4×C4⋊C4×C67C411yescharacteristic, complemented
22C4×C22⋊C4×C67—21no—
23C2×C22⋊Dic67—22no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26#14523954—41no—
27C67⋊#95—41no—
28C67⋊#95—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4×Q8×C67—41no—
32C4×Q8×C67—41no—
33#117×C67—41no—
34#117×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67C2311yescharacteristic
46C2×C42×C67—21no—
47C2×C42×C67—21no—
48C2×C42×C67C2×C411yescharacteristic
49C2×C42×C67D411yescharacteristic
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67C2×C411yescharacteristic
55#95×C67C2×C411yescharacteristic
56C22×Dic67—42no—
57C2×C67⋊C8—42no—
58C22⋊Dic67—42no—
59C22⋊Dic67—42no—
60C24×C67C2×D411yescharacteristic
61C2×Q8×C67—42no—
62C2×Q8×C67—41no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—41no—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—42no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—42no—
93C42×C67—41no—
94C42×C67—41no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67C2×C22⋊C411yescharacteristic
100C23×C67#13712yes—
101Q8×C67—42no—
102Q8×C67—42no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—22no—
106C2×C4×C67—22no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—41no—
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#9911yescharacteristic
131C2×C4×C67#9911yescharacteristic
132C2×Dic67—82no—
133C2×Dic67—82no—
134C2×Dic67—82no—
135C67⋊C8—82no—
136C22×C67—21no—
137C22×C67—21no—
138C22×C67—22no—
139C22×C67—42no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67—81no—
143C22×C67C2×#9911yescharacteristic
144C22×C67#41911yescharacteristic
145C22×C67#42111yescharacteristic
146C4×C67—21no—
147C4×C67—21no—
148C4×C67—42no—
149C4×C67—41no—
150C4×C67—42no—
151C4×C67—41no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—41no—
155C4×C67—81no—
156C4×C67—81no—
157Dic67—162no—
158#11810—671nomaximal, Sylow
159C2×C67—21no—
160C2×C67—21no—
161C2×C67—42no—
162C2×C67#151111yescharacteristic
163C2×C67#212011yescharacteristic
164C2×C67#212611yescharacteristic
165#2396D6711yescharacteristic
166#1820—671no—
167#1464—671no—
168#1352—672no—
169#1351—672no—
170C67#1181011yescharacteristic, complemented, Sylow
171C2×C4×Q8Dic6711yescharacteristic, complemented
172C2×#117C2×D6711yescharacteristic
173#432—1341no—
174#400Dic6711yescharacteristic, complemented
175#399C2×D6711yescharacteristic
176#398—21no—
177#396—21no—
178#395Dic6711yescharacteristic, complemented
179#393C2×D6711yescharacteristic
180#381—1341no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4Dic6711yescharacteristic, complemented
183C4×C22⋊C4—21no—
184#342—672no—
185#342—1341no—
186#324—1341no—
187#323—672no—
188C22×Q8C2×Dic6711yescharacteristic
189C23×C4C22⋊D6711yescharacteristic
190C4×Q8—41no—
191C4×Q8—41no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic6711yescharacteristic
199C2×C4⋊C4C2×Dic6711yescharacteristic
200C2×C4⋊C4C22⋊D6711yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—1342no—
204C2×C22⋊C4C22⋊D6711yescharacteristic
205C2×C22⋊C4C22×D6711yescharacteristic
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C2×Dic6711yescharacteristic
209C2×C42C22⋊D6711yescharacteristic
210C4⋊C8—2681no—
211C22⋊C8—1342no—
212#97—2681no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—2681no—
218#95—2681no—
219#95C2×Dic6711yescharacteristic
220#95C2×Dic6711yescharacteristic
221C24C2×C22⋊D6711yescharacteristic
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—2682no—
233C22×C4C22⋊Dic6711yescharacteristic
234C22×C4C22⋊Dic6711yescharacteristic
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22×Dic6711yescharacteristic
238C22×C4C2×C22⋊D6711yescharacteristic
239C2×C8—2682no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—2682no—
252C22⋊C4—2682no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—42no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic6711yescharacteristic
265C23#1452407412yes—
266Q8—42no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—5362no—
296C2×C4—5362no—
297C2×C4—5362no—
298C2×C4#1452395811yescharacteristic
299C2×C4#1452395811yescharacteristic
300C8—5362no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1452395811yescharacteristic
309C22#1686340511yescharacteristic
310C22#1686341311yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—10722no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2255874211yescharacteristic
327C2#2256185011yescharacteristic
328C2#2256187411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2396C2
socleC22×C67C2×#99
prime radicalC24×C67C2×D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2396D67
hypocenterC67#11810
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C67C4semidirect
14. #400×C67C4semidirect
18. #395×C67C4semidirect
21. C4×C4⋊C4×C67C4semidirect
170. C67#11810semidirect
171. C2×C4×Q8Dic67semidirect
174. #400Dic67semidirect
178. #395Dic67semidirect
182. C4×C4⋊C4Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602862
#225585432
#225582332
#225582302
C67×#23962
#1181067