Finite Group Databasebeta

← C67×#19210 #30698461

Subgroups — C67×#19210 #30698461

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3249C212yesmaximal
2C67×#3160C211yesmaximal, characteristic
3C67×#2960C211yesmaximal, characteristic
4C67×#1696C212yesmaximal, complemented
5C67×#1694C212yesmaximal, complemented
6C67×#1672C214yesmaximal, complemented
7C2×#334×C67C211yesmaximal, characteristic
8C67×#1550C212yesmaximal
9#533×C67C2212yes—
10#533×C67C2211yescharacteristic
11#500×C67C2211yescharacteristic
12#500×C67C2211yescharacteristic
13#499×C67—21no—
14#499×C67—21no—
15#498×C67C2212yes—
16#477×C67—22no—
17#477×C67C2214yes—
18#467×C67—24no—
19#462×C67—22no—
20#462×C67C2214yes—
21#446×C67—21no—
22#446×C67C2212yes—
23C8×Q8×C67—21no—
24C8×Q8×C67C2212yes—
25#427×C67—21no—
26#427×C67—21no—
27#424×C67—21no—
28#423×C67C2211yescharacteristic
29C2×C4⋊C8×C67C2212yes—
30#405×C67—21no—
31C2×C4×C8×C67C2211yescharacteristic
32#334×C67C2212yes—
33#334×C67C2212yes—
34#333×C67C2214yes—
35#332×C67C2214yes—
36C2×Q16×C67—24no—
37C2×M16×C67—21no—
38C22×C8×C67—22no—
39C22×C8×C67D411yescharacteristic
40C22×C8×C67D411yescharacteristic
41C4⋊Q8×C67D411yescharacteristic
42C4⋊Q8×C67D411yescharacteristic
43C4⋊Q8×C67C2312yes—
44C4⋊Q8×C67C2312yes—
45#125×C67—22no—
46#125×C67—21no—
47C22⋊Q8×C67—21no—
48C22⋊Q8×C67—22no—
49C22⋊Q8×C67—22no—
50C22⋊Q8×C67D412yes—
51C4×Q8×C67—21no—
52C4×Q8×C67—22no—
53C4×Q8×C67—22no—
54C4×Q8×C67C2312yes—
55#117×C67—21no—
56#117×C67—22no—
57C2×C42×C67C2311yescharacteristic
58#107×C67—21no—
59#107×C67—21no—
60#107×C67—21no—
61#107×C67—21no—
62#107×C67—24no—
63#106×C67—22no—
64#106×C67—22no—
65C4⋊C8×C67—22no—
66C4⋊C8×C67—22no—
67C4⋊C8×C67—22no—
68C4⋊C8×C67—21no—
69C4⋊C8×C67C2314yes—
70C4⋊C8×C67C2312yes—
71#103×C67—24no—
72#103×C67—24no—
73#97×C67—22no—
74C4×C8×C67—22no—
75C4×C8×C67D411yescharacteristic
76C4×C8×C67D411yescharacteristic
77C4×C8×C67C2311yescharacteristic
78C4×C8×C67C2311yescharacteristic
79C2×Q8×C67—21no—
80C2×Q8×C67—22no—
81C2×Q8×C67—22no—
82C2×Q8×C67C2×D412yes—
83C22×C4×C67C2×D411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C22×C4×C67#4111yescharacteristic
86Q16×C67—84no—
87M16×C67—42no—
88C2×C8×C67—22no—
89C2×C8×C67—24no—
90C2×C8×C67—22no—
91C2×C8×C67—82no—
92C2×C8×C67Q1614yescomplemented
93C2×C8×C67C2×D411yescharacteristic
94C2×C8×C67C2×D411yescharacteristic
95C2×C8×C67C2×D411yescharacteristic
96C2×C8×C67C2×D411yescharacteristic
97C4⋊C4×C67—21no—
98C4⋊C4×C67—21no—
99C4⋊C4×C67—21no—
100C4⋊C4×C67—22no—
101C4⋊C4×C67—22no—
102C4⋊C4×C67—22no—
103C4⋊C4×C67—22no—
104C4⋊C4×C67—22no—
105C4⋊C4×C67—22no—
106C4⋊C4×C67—42no—
107C4⋊C4×C67—42no—
108C4⋊C4×C67C2×D412yes—
109C22⋊C4×C67—22no—
110C22⋊C4×C67—42no—
111C42×C67—22no—
112C42×C67—42no—
113C42×C67C2×D411yescharacteristic
114C42×C67C2×D411yescharacteristic
115C42×C67#4111yescharacteristic
116C42×C67C2411yescharacteristic
117C23×C67#12111yescharacteristic
118Q8×C67—24no—
119Q8×C67—42no—
120Q8×C67—42no—
121Q8×C67—42no—
122Q8×C67—82no—
123C2×C4×C67—22no—
124C2×C4×C67—22no—
125C2×C4×C67—22no—
126C2×C4×C67—41no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67#12111yescharacteristic
130C2×C4×C67#12111yescharacteristic
131C2×C4×C67#12111yescharacteristic
132C2×C4×C67C2×Q1612yes—
133C2×C4×C67#13712yes—
134C2×C4×C67C22×D411yescharacteristic
135C2×C4×C67C22×D411yescharacteristic
136C2×C4×C67C2×#4111yescharacteristic
137C8×C67—21no—
138C8×C67—21no—
139C8×C67—42no—
140C8×C67—44no—
141C8×C67C2×Q1612yes—
142C8×C67C2×Q1612yes—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67#46212yes—
146C22×C67C2×#12111yescharacteristic
147C4×C67—21no—
148C4×C67—21no—
149C4×C67—21no—
150C4×C67—42no—
151C4×C67—42no—
152C4×C67—82no—
153C4×C67—82no—
154C4×C67#46212yes—
155C4×C67C22×Q1611yescharacteristic
156C4×C67C2×#13711yescharacteristic
157C4×C67#57712yes—
158#19210C6711yesmaximal, characteristic, direct product factor, Sylow
159C2×C67—22no—
160C2×C67C2×#46211yescharacteristic
161C2×C67#306011yescharacteristic
162C2×C67#306711yescharacteristic
163#3249C2×C6712yes—
164#3160C2×C6711yescharacteristic
165#2960C2×C6711yescharacteristic
166#1696C2×C6712yescomplemented
167#1694C2×C6712yescomplemented
168#1672C2×C6714yescomplemented
169C2×#334C2×C6711yescharacteristic
170#1550C2×C6712yes—
171C67#1921011yescharacteristic, direct product factor, Sylow
172#533C22×C6712yes—
173#533C22×C6711yescharacteristic
174#500C22×C6711yescharacteristic
175#500C22×C6711yescharacteristic
176#499—21no—
177#499—21no—
178#498C22×C6712yes—
179#477—22no—
180#477C22×C6714yes—
181#467—24no—
182#462—22no—
183#462C22×C6714yes—
184#446—21no—
185#446C22×C6712yes—
186C8×Q8—21no—
187C8×Q8C22×C6712yes—
188#427—21no—
189#427—21no—
190#424—21no—
191#423C22×C6711yescharacteristic
192C2×C4⋊C8C22×C6712yes—
193#405—21no—
194C2×C4×C8C22×C6711yescharacteristic
195#334C22×C6712yes—
196#334C22×C6712yes—
197#333C22×C6714yes—
198#332C22×C6714yes—
199C2×Q16—24no—
200C2×M16—21no—
201C22×C8—22no—
202C22×C8D4×C6711yescharacteristic
203C22×C8D4×C6711yescharacteristic
204C4⋊Q8D4×C6711yescharacteristic
205C4⋊Q8D4×C6711yescharacteristic
206C4⋊Q8C23×C6712yes—
207C4⋊Q8C23×C6712yes—
208#125—22no—
209#125—21no—
210C22⋊Q8—21no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8D4×C6712yes—
214C4×Q8—21no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C23×C6712yes—
218#117—21no—
219#117—22no—
220C2×C42C23×C6711yescharacteristic
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—24no—
226#106—22no—
227#106—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8C23×C6712yes—
233C4⋊C8C23×C6714yes—
234#103—24no—
235#103—24no—
236#97—22no—
237C4×C8—22no—
238C4×C8D4×C6711yescharacteristic
239C4×C8D4×C6711yescharacteristic
240C4×C8C23×C6711yescharacteristic
241C4×C8C23×C6711yescharacteristic
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8C2×D4×C6712yes—
246C22×C4C2×D4×C6711yescharacteristic
247C22×C4C2×D4×C6711yescharacteristic
248C22×C4#41×C6711yescharacteristic
249Q16—84no—
250M16—42no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—82no—
255C2×C8Q16×C6714yescomplemented
256C2×C8C2×D4×C6711yescharacteristic
257C2×C8C2×D4×C6711yescharacteristic
258C2×C8C2×D4×C6711yescharacteristic
259C2×C8C2×D4×C6711yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4C2×D4×C6712yes—
272C22⋊C4—22no—
273C22⋊C4—42no—
274C42—22no—
275C42—42no—
276C42C2×D4×C6711yescharacteristic
277C42C2×D4×C6711yescharacteristic
278C42#41×C6711yescharacteristic
279C42C24×C6711yescharacteristic
280C23#121×C6711yescharacteristic
281Q8—24no—
282Q8—42no—
283Q8—42no—
284Q8—42no—
285Q8—82no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4#121×C6711yescharacteristic
293C2×C4#121×C6711yescharacteristic
294C2×C4#121×C6711yescharacteristic
295C2×C4C2×Q16×C6712yes—
296C2×C4#137×C6712yes—
297C2×C4C22×D4×C6711yescharacteristic
298C2×C4C22×D4×C6711yescharacteristic
299C2×C4C2×#41×C6711yescharacteristic
300C8—21no—
301C8—21no—
302C8—42no—
303C8—44no—
304C8C2×Q16×C6712yes—
305C8C2×Q16×C6712yes—
306C22—21no—
307C22—21no—
308C22#462×C6712yes—
309C22C2×#121×C6711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—82no—
316C4—82no—
317C4#462×C6712yes—
318C4C22×Q16×C6711yescharacteristic
319C4C2×#137×C6711yescharacteristic
320C4#577×C6712yes—
321C2—22no—
322C2C2×#462×C6711yescharacteristic
323C2C67×#306011yescharacteristic
324C2C67×#306711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radicalC23×C67#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1696C2semidirect
5. C67×#1694C2semidirect
6. C67×#1672C2semidirect
92. C2×C8×C67Q16semidirect
158. #19210C67direct (N × H)
166. #1696C2×C67semidirect
167. #1694C2×C67semidirect
168. #1672C2×C67semidirect
171. C67#19210direct (N × H)
255. C2×C8Q16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32492
C67×#31602
C67×#29602
C67×#16962
C67×#16942
C67×#16722
C2×#334×C672
C67×#15502
#1921067