Finite Group Databasebeta

← #30771822

Subgroups — #30771822

Order 17152.

588 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16863189C212yesmaximal
2#22560197C212yesmaximal, complemented
3#22558334C212yesmaximal, complemented
4C67×#2328C211yesmaximal, characteristic
5C2×#14523958C2214yes—
6#16863256—22no—
7#16863189—24no—
8#16863163—22no—
9C2×#127×C67C2211yescharacteristic
10C2×C4×D4×C67C411yescharacteristic, complemented
11C2×C4×D4×C67C2211yescharacteristic
12#390×C67C411yescharacteristic, complemented
13#390×C67C2211yescharacteristic
14#386×C67—22no—
15#386×C67—22no—
16#382×C67C411yescharacteristic, complemented
17#376×C67C411yescharacteristic, complemented
18C2×C4⋊Dic67—42no—
19C2×C22⋊Dic67—24no—
20#14523958—24no—
21#14523958—24no—
22C67⋊#95—42no—
23C22×D4×C67—22no—
24C22×D4×C67C2×C411yescharacteristic
25C22×D4×C67C2311yescharacteristic
26C23×C4×C67—21no—
27C23×C4×C67D412yes—
28#127×C67—22no—
29#127×C67D412yes—
30#127×C67Q812yes—
31C4×D4×C67—42no—
32C4×D4×C67—42no—
33C2×C4⋊C4×C67C2×C411yescharacteristic
34C2×C4⋊C4×C67C2×C411yescharacteristic
35C2×C22⋊C4×C67—21no—
36C2×C22⋊C4×C67—22no—
37C2×C22⋊C4×C67—22no—
38C2×C22⋊C4×C67D412yes—
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67C2×C411yescharacteristic
41C2×C42×C67C2×C411yescharacteristic
42#95×C67—21no—
43#95×C67—21no—
44#95×C67—22no—
45C22×Dic67—44no—
46C4⋊Dic67—84no—
47C22⋊Dic67—44no—
48C22⋊Dic67—44no—
49C24×C67—21no—
50C24×C67C2×D412yes—
51C2×D4×C67—24no—
52C2×D4×C67—24no—
53C2×D4×C67—21no—
54C2×D4×C67—41no—
55C2×D4×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67—41no—
58C2×D4×C67—82no—
59C2×D4×C67—82no—
60C2×D4×C67C4⋊C412yescomplemented
61C2×D4×C67C2×D411yescharacteristic
62C2×D4×C67C2×Q811yescharacteristic
63C2×D4×C67#4112yes—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—42no—
70C22×C4×C67—82no—
71C22×C4×C67—82no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C412yes—
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22×C411yescharacteristic
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C22⋊C4×C67—42no—
79C22⋊C4×C67—42no—
80C22⋊C4×C67—82no—
81C22⋊C4×C67—82no—
82C42×C67—22no—
83C42×C67—21no—
84C42×C67—41no—
85C42×C67C4⋊C412yescomplemented
86C23×C67—22no—
87C23×C67—22no—
88C23×C67—22no—
89C23×C67—42no—
90C23×C67—42no—
91C23×C67—42no—
92C23×C67—42no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67C2×C22⋊C411yescharacteristic
96C23×C67#12112yes—
97C23×C67C22⋊Q812yes—
98D4×C67—41no—
99D4×C67—41no—
100D4×C67—42no—
101D4×C67—42no—
102D4×C67—42no—
103D4×C67—84no—
104D4×C67—84no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—21no—
109C2×C4×C67—22no—
110C2×C4×C67—41no—
111C2×C4×C67—44no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—42no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—42no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—82no—
122C2×C4×C67—84no—
123C2×C4×C67#9912yes—
124C2×C4×C67C2×C4⋊C411yescharacteristic
125C2×C4×C67#11711yescharacteristic
126C2×Dic67—84no—
127C2×Dic67—84no—
128C2×Dic67—84no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—22no—
132C22×C67—22no—
133C22×C67—42no—
134C22×C67—42no—
135C22×C67—44no—
136C22×C67—42no—
137C22×C67—42no—
138C22×C67—42no—
139C22×C67—82no—
140C22×C67—82no—
141C22×C67—81no—
142C22×C67—81no—
143C22×C67#38011yescharacteristic
144C22×C67C2×#9911yescharacteristic
145C22×C67#41011yescharacteristic
146C4×C67—21no—
147C4×C67—21no—
148C4×C67—22no—
149C4×C67—42no—
150C4×C67—42no—
151C4×C67—41no—
152C4×C67—81no—
153C4×C67—81no—
154Dic67—164no—
155#12799—671nomaximal, Sylow
156C2×C67—21no—
157C2×C67—21no—
158C2×C67—42no—
159C2×C67—42no—
160C2×C67—82no—
161C2×C67#179911yescharacteristic
162C2×C67#212711yescharacteristic
163C2×C67#212811yescharacteristic
164#2328D6711yescharacteristic
165C2×#353—672no—
166#1799—672no—
167#1393—672no—
168C67#1279911yescharacteristic, complemented, Sylow
169C2×#127C2×D6711yescharacteristic
170C2×C4×D4Dic6711yescharacteristic, complemented
171C2×C4×D4C2×D6711yescharacteristic
172C2×#99—674no—
173#390Dic6711yescharacteristic, complemented
174#390C2×D6711yescharacteristic
175#386—22no—
176#386—22no—
177#382Dic6711yescharacteristic, complemented
178#380—1342no—
179#376Dic6711yescharacteristic, complemented
180#353—1344no—
181#342—1342no—
182C22×D4—22no—
183C22×D4C2×Dic6711yescharacteristic
184C22×D4C22×D6711yescharacteristic
185C23×C4—21no—
186C23×C4C22⋊D6712yes—
187#127—22no—
188#127Dic13412yes—
189#127D26812yes—
190C4×D4—42no—
191C4×D4—42no—
192C2×C4⋊C4—2682no—
193C2×C4⋊C4C2×Dic6711yescharacteristic
194C2×C4⋊C4C2×Dic6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1344no—
199C2×C22⋊C4C22⋊D6712yes—
200C2×C42C2×Dic6711yescharacteristic
201C2×C42C2×Dic6711yescharacteristic
202C2×C42C2×Dic6711yescharacteristic
203#99—1344no—
204#99—1344no—
205#95—21no—
206#95—22no—
207#95—21no—
208#95—2682no—
209C24—21no—
210C24C2×C22⋊D6712yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C4⋊Dic6712yescomplemented
221C2×D4C2×D26811yescharacteristic
222C2×D4C2×Dic13411yescharacteristic
223C2×D4#1176162812yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—2684no—
233C22×C4C22⋊Dic6711yescharacteristic
234C22×C4C22⋊Dic6712yes—
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22×Dic6711yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—5364no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C22⋊C4—2684no—
245C22⋊C4—2684no—
246C42—22no—
247C42—21no—
248C42—41no—
249C42C4⋊Dic6712yescomplemented
250C23—22no—
251C23—22no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊Dic6711yescharacteristic
260C23#1452401712yes—
261C23#1452402212yes—
262D4—41no—
263D4—41no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—5364no—
288C2×C4—5364no—
289C2×C4—5364no—
290C2×C4#1452395812yes—
291C2×C4C2×C4⋊Dic6711yescharacteristic
292C2×C4#1452400211yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22#1686325611yescharacteristic
308C22C2×#1452395811yescharacteristic
309C22#1686337011yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—10724no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2—42no—
323C2—82no—
324C2#2256019711yescharacteristic
325C2#2256187811yescharacteristic
326C2#2256188111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×D4C22×D67
FittingC67×#2328C2
socleC22×C67C2×#99
prime radicalC2×#127×C67C22
cosocleC22×D4×C67C23
hypercenter#2328D67
hypocenterC67#12799
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #22560197C2semidirect
3. #22558334C2semidirect
10. C2×C4×D4×C67C4semidirect
12. #390×C67C4semidirect
16. #382×C67C4semidirect
17. #376×C67C4semidirect
60. C2×D4×C67C4⋊C4semidirect
85. C42×C67C4⋊C4semidirect
168. C67#12799semidirect
170. C2×C4×D4Dic67semidirect
173. #390Dic67semidirect
177. #382Dic67semidirect
179. #376Dic67semidirect
220. C2×D4C4⋊Dic67semidirect
249. C42C4⋊Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168631892
#225601972
#225583342
C67×#23282
#1279967