Finite Group Databasebeta

← C4×#16863189 #30770993

Subgroups — C4×#16863189 #30770993

Order 17152.

720 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 139 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
2C4×#14523958C212yesmaximal, complemented
3#22558334C212yesmaximal, complemented
4C4×#127×C67C211yesmaximal, characteristic
5C2×#14523958C2214yes—
6C4×C22⋊Dic67—22no—
7#16863189C418yesdirect product factor
8#16863163—22no—
9C2×#127×C67C2211yescharacteristic
10C2×C4×D4×C67—22no—
11C2×C4×D4×C67C411yescharacteristic, complemented
12C2×C4×D4×C67C2211yescharacteristic
13#390×C67—22no—
14#390×C67C411yescharacteristic, complemented
15#390×C67C2211yescharacteristic
16#384×C67C411yescharacteristic, complemented
17C43×C67C411yescharacteristic, complemented
18C2×C22⋊Dic67—24no—
19C2×C4×Dic67—42no—
20#14523958—24no—
21#14523958C2×C418yescomplemented
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×C67C2×C414yes—
30C4×D4×C67—41no—
31C4×D4×C67—41no—
32C4×D4×C67—48no—
33C4×D4×C67—41no—
34C4×D4×C67—41no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67C2×C411yescharacteristic
37C2×C4⋊C4×C67C2×C411yescharacteristic
38C2×C22⋊C4×C67—22no—
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67D412yes—
42C2×C42×C67—21no—
43C2×C42×C67—21no—
44C2×C42×C67C2×C411yescharacteristic
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67C2×C411yescharacteristic
47C22×Dic67—44no—
48C22⋊Dic67—28no—
49C22⋊Dic67—44no—
50C4×Dic67—84no—
51C24×C67—21no—
52C24×C67C2×D412yes—
53C2×D4×C67—28no—
54C2×D4×C67—21no—
55C2×D4×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67—41no—
58C2×D4×C67—41no—
59C2×D4×C67—44no—
60C2×D4×C67—44no—
61C2×D4×C67C4212yescomplemented
62C2×D4×C67C22×C412yes—
63C2×D4×C67#4112yes—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—22no—
67C22×C4×C67—22no—
68C22×C4×C67—21no—
69C22×C4×C67—44no—
70C22×C4×C67—44no—
71C22×C4×C67—41no—
72C22×C4×C67—41no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C412yes—
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—44no—
80C22⋊C4×C67—42no—
81C22⋊C4×C67—44no—
82C22⋊C4×C67—44no—
83C22⋊C4×C67—42no—
84C42×C67—24no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—21no—
88C42×C67—41no—
89C42×C67—42no—
90C42×C67C4212yescomplemented
91C23×C67—22no—
92C23×C67—22no—
93C23×C67—22no—
94C23×C67—42no—
95C23×C67—42no—
96C23×C67—42no—
97C23×C67—42no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67C2×C22⋊C411yescharacteristic
101C23×C67C4×D414yes—
102D4×C67—42no—
103D4×C67—42no—
104D4×C67—42no—
105D4×C67—416no—
106D4×C67—42no—
107C2×C4×C67—22no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—22no—
113C2×C4×C67—22no—
114C2×C4×C67—21no—
115C2×C4×C67—22no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—44no—
119C2×C4×C67—41no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67—41no—
123C2×C4×C67—82no—
124C2×C4×C67—82no—
125C2×C4×C67—82no—
126C2×C4×C67#9911yescharacteristic
127C2×C4×C67#9911yescharacteristic
128C2×C4×C67C2×C4211yescharacteristic
129C2×C4×C67#11711yescharacteristic
130C2×Dic67—48no—
131C2×Dic67—84no—
132C2×Dic67—84no—
133C22×C67—21no—
134C22×C67—21no—
135C22×C67—24no—
136C22×C67—44no—
137C22×C67—44no—
138C22×C67—42no—
139C22×C67—42no—
140C22×C67—42no—
141C22×C67—82no—
142C22×C67—82no—
143C22×C67—81no—
144C22×C67—81no—
145C22×C67C4×C22⋊C411yescharacteristic
146C22×C67C2×#9911yescharacteristic
147C22×C67#41011yescharacteristic
148C4×C67—24no—
149C4×C67—21no—
150C4×C67—21no—
151C4×C67—21no—
152C4×C67—42no—
153C4×C67—42no—
154C4×C67—41no—
155C4×C67—81no—
156C4×C67#35312yescomplemented
157Dic67—88no—
158C4×#353—671nomaximal, Sylow
159C2×C67—21no—
160C2×C67—21no—
161C2×C67—44no—
162C2×C67—82no—
163C2×C67C4×#9911yescharacteristic
164C2×C67C2×#35311yescharacteristic
165C2×C67#212611yescharacteristic
166C4×#127D6711yescharacteristic
167C2×#353—672no—
168C4×#99—672no—
169#1393—672no—
170C67C4×#35311yescharacteristic, complemented, Sylow
171C2×#127C2×D6711yescharacteristic
172C2×C4×D4—22no—
173C2×C4×D4Dic6711yescharacteristic, complemented
174C2×C4×D4C2×D6711yescharacteristic
175C2×#99—674no—
176#390—22no—
177#390Dic6711yescharacteristic, complemented
178#390C2×D6711yescharacteristic
179#384Dic6711yescharacteristic, complemented
180C4×C22⋊C4—1342no—
181C43Dic6711yescharacteristic, complemented
182#353—678no—
183#342—1342no—
184C22×D4—22no—
185C22×D4C2×Dic6711yescharacteristic
186C22×D4C22×D6711yescharacteristic
187C23×C4—21no—
188C23×C4C22⋊D6712yes—
189#127—22no—
190#127C4×D6714yes—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—48no—
194C4×D4—41no—
195C4×D4—41no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×Dic6711yescharacteristic
198C2×C4⋊C4C2×Dic6711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—1344no—
203C2×C22⋊C4C22⋊D6712yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—2682no—
207C2×C42C2×Dic6711yescharacteristic
208C2×C42C2×Dic6711yescharacteristic
209C2×C42C2×Dic6711yescharacteristic
210#99—678no—
211#99—1344no—
212#95—2682no—
213C24—21no—
214C24C2×C22⋊D6712yes—
215C2×D4—28no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4C4×Dic6712yescomplemented
224C2×D4C2×C4×D6712yes—
225C2×D4#1176162812yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—2684no—
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22⋊Dic6712yes—
238C22×C4C22⋊Dic6711yescharacteristic
239C22×C4C22×Dic6711yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—2684no—
248C22⋊C4—1348no—
249C42—24no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—42no—
255C42—5364no—
256C42C4×Dic6712yescomplemented
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic6711yescharacteristic
267C23C4×C22⋊D6714yes—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—416no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—2688no—
293C2×C4—5364no—
294C2×C4—5364no—
295C2×C4#1452395811yescharacteristic
296C2×C4#1452395811yescharacteristic
297C2×C4C2×C4×Dic6711yescharacteristic
298C2×C4#1452400211yescharacteristic
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22C4×C22⋊Dic6711yescharacteristic
312C22C2×#1452395811yescharacteristic
313C22#1686337011yescharacteristic
314C4—24no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—42no—
320C4—41no—
321C4—81no—
322C4—5368no—
323C4#1686318912yesdirect product factor
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2C4×#1452395811yescharacteristic
329C2C2×#1686318911yescharacteristic
330C2#2256187411yescharacteristic
subgroupG/H
centerC2×C4#14523958
commutatorC2×C4×C67C2×C42
FrattiniC22×D4C22×D67
FittingC4×#127×C67C2
socleC22×C67C2×#99
prime radicalC2×#127×C67C22
cosocleC22×D4×C67C23
hypercenterC4×#127D67
hypocenterC67C4×#353
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C4×#14523958C2semidirect
3. #22558334C2semidirect
7. #16863189C4direct (N × H)
11. C2×C4×D4×C67C4semidirect
14. #390×C67C4semidirect
16. #384×C67C4semidirect
17. C43×C67C4semidirect
21. #14523958C2×C4semidirect
61. C2×D4×C67C42semidirect
90. C42×C67C42semidirect
156. C4×C67#353semidirect
170. C67C4×#353semidirect
173. C2×C4×D4Dic67semidirect
177. #390Dic67semidirect
179. #384Dic67semidirect
181. C43Dic67semidirect
223. C2×D4C4×Dic67semidirect
256. C42C4×Dic67semidirect
323. C4#16863189direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168631892
C4×#145239582
#225583342
C4×#127×C672
C4×#35367