Finite Group Databasebeta

← C67×#11555 #30690806

Subgroups — C67×#11555 #30690806

Order 17152.

414 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C67×#2502C211yesmaximal, characteristic
2C67×#1463C211yesmaximal, characteristic
3C2×#324×C67C211yesmaximal, characteristic
4C67×#1325C212yesmaximal, complemented
5C67×#1319C212yesmaximal, complemented
6C2×C22⋊Q8×C67C411yescharacteristic
7C22×C4⋊C4×C67C2211yescharacteristic
8#407×C67—21no—
9C2×C22⋊C8×C67—21no—
10#398×C67—21no—
11#398×C67—21no—
12#391×C67C411yescharacteristic
13#391×C67C2211yescharacteristic
14#385×C67—21no—
15#385×C67—21no—
16#384×C67C411yescharacteristic
17#384×C67C2211yescharacteristic
18C4×C22⋊C4×C67C411yescharacteristic
19#324×C67—21no—
20#324×C67—21no—
21#324×C67C2212yes—
22#324×C67C2212yes—
23C22×Q8×C67C2×C411yescharacteristic
24C23×C4×C67C2×C411yescharacteristic
25C23×C4×C67D411yescharacteristic
26C23×C4×C67D411yescharacteristic
27C2×M16×C67—41no—
28C22×C8×C67—41no—
29C22⋊Q8×C67—41no—
30C22⋊Q8×C67—41no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67C2311yescharacteristic
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67C2×C411yescharacteristic
47C22⋊C8×C67—22no—
48C22⋊C8×C67—22no—
49C22⋊C8×C67—41no—
50C22⋊C8×C67—41no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56C24×C67C22⋊C411yescharacteristic
57C2×Q8×C67—22no—
58C2×Q8×C67—41no—
59C2×Q8×C67—41no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—22no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—41no—
72C22×C4×C67—81no—
73C22×C4×C67—81no—
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22×C411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80M16×C67—82no—
81C2×C8×C67—42no—
82C2×C8×C67—42no—
83C2×C8×C67—82no—
84C4⋊C4×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—42no—
90C4⋊C4×C67—81no—
91C4⋊C4×C67—81no—
92C4⋊C4×C67—81no—
93C4⋊C4×C67—81no—
94C22⋊C4×C67—24no—
95C22⋊C4×C67—41no—
96C22⋊C4×C67—41no—
97C42×C67—22no—
98C42×C67—41no—
99C23×C67—22no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—41no—
103C23×C67#9911yescharacteristic
104C23×C67#9911yescharacteristic
105C23×C67C2×C22⋊C411yescharacteristic
106Q8×C67—82no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—42no—
123C2×C4×C67—42no—
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—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—81no—
137C2×C4×C67—81no—
138C2×C4×C67#13112yes—
139C8×C67—82no—
140C8×C67—82no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—42no—
149C22×C67—42no—
150C22×C67#35612yes—
151C22×C67#40811yescharacteristic
152C22×C67C2×#9911yescharacteristic
153C22×C67#41311yescharacteristic
154C4×C67—22no—
155C4×C67—42no—
156C4×C67—42no—
157C4×C67—41no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—81no—
163C4×C67—81no—
164#11555C6711yesmaximal, characteristic, direct product factor, Sylow
165C2×C67—21no—
166C2×C67—21no—
167C2×C67—22no—
168C2×C67—41no—
169C2×C67#146811yescharacteristic
170C2×C67#212711yescharacteristic
171C2×C67C2×#35611yescharacteristic
172#2502C2×C6711yescharacteristic
173#1463C2×C6711yescharacteristic
174C2×#324C2×C6711yescharacteristic
175#1325C2×C6712yescomplemented
176#1319C2×C6712yescomplemented
177C67#1155511yescharacteristic, direct product factor, Sylow
178C2×C22⋊Q8C4×C6711yescharacteristic
179C22×C4⋊C4C22×C6711yescharacteristic
180#407—21no—
181C2×C22⋊C8—21no—
182#398—21no—
183#398—21no—
184#391C4×C6711yescharacteristic
185#391C22×C6711yescharacteristic
186#385—21no—
187#385—21no—
188#384C4×C6711yescharacteristic
189#384C22×C6711yescharacteristic
190C4×C22⋊C4C4×C6711yescharacteristic
191#324—21no—
192#324—21no—
193#324C22×C6712yes—
194#324C22×C6712yes—
195C22×Q8C2×C4×C6711yescharacteristic
196C23×C4C2×C4×C6711yescharacteristic
197C23×C4D4×C6711yescharacteristic
198C23×C4D4×C6711yescharacteristic
199C2×M16—41no—
200C22×C8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×C4×C6711yescharacteristic
211C2×C4⋊C4C2×C4×C6711yescharacteristic
212C2×C4⋊C4D4×C6711yescharacteristic
213C2×C4⋊C4D4×C6711yescharacteristic
214C2×C4⋊C4C23×C6711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42C2×C4×C6711yescharacteristic
218C2×C42C2×C4×C6711yescharacteristic
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228C24C22⋊C4×C6711yescharacteristic
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C22⋊C4×C6711yescharacteristic
247C22×C4C22⋊C4×C6711yescharacteristic
248C22×C4C22⋊C4×C6711yescharacteristic
249C22×C4C22×C4×C6711yescharacteristic
250C22×C4C2×D4×C6711yescharacteristic
251C22×C4C2×D4×C6711yescharacteristic
252M16—82no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C22⋊C4—24no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—22no—
270C42—41no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23#99×C6711yescharacteristic
276C23#99×C6711yescharacteristic
277C23C2×C22⋊C4×C6711yescharacteristic
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
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—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#131×C6712yes—
311C8—82no—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22#356×C6712yes—
323C22#408×C6711yescharacteristic
324C22C2×#99×C6711yescharacteristic
325C22#413×C6711yescharacteristic
326C4—22no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2C67×#146811yescharacteristic
341C2C67×#212711yescharacteristic
342C2C2×#356×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#99
commutatorC22×C4C22×C4×C67
Frattini, cosocleC2×C4⋊C4C23×C67
prime radicalC24×C67C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C24×C67 ◁ C22×C4⋊C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1325C2semidirect
5. C67×#1319C2semidirect
164. #11555C67direct (N × H)
175. #1325C2×C67semidirect
176. #1319C2×C67semidirect
177. C67#11555direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#25022
C67×#14632
C2×#324×C672
C67×#13252
C67×#13192
#1155567