Finite Group Databasebeta

← C67×#12730 #30691981

Subgroups — C67×#12730 #30691981

Order 17152.

666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3509C211yesmaximal, characteristic
2C67×#2133C212yesmaximal
3C2×#356×C67C212yesmaximal, complemented
4C67×#1830C212yesmaximal, complemented
5#557×C67—22no—
6#556×C67—22no—
7#552×C67—24no—
8#541×C67—22no—
9#536×C67C411yescharacteristic
10#536×C67C2211yescharacteristic
11#533×C67—22no—
12#533×C67C411yescharacteristic
13C2×C4⋊Q8×C67C2211yescharacteristic
14#519×C67C411yescharacteristic
15#519×C67C2211yescharacteristic
16C4×#41×C67C411yescharacteristic
17C2×#101×C67C2214yes—
18#356×C67—22no—
19#356×C67—22no—
20#356×C67—22no—
21#355×C67—22no—
22#344×C67—24no—
23C2×#41×C67C2×C411yescharacteristic
24C22×Q8×C67C2×C411yescharacteristic
25C22×Q8×C67C2311yescharacteristic
26C2×M16×C67—24no—
27C4⋊Q8×C67—21no—
28C4⋊Q8×C67—22no—
29C4⋊Q8×C67—21no—
30C4⋊Q8×C67D412yes—
31C4⋊Q8×C67D412yes—
32C4⋊Q8×C67D412yes—
33#126×C67—24no—
34#126×C67—24no—
35#125×C67—24no—
36#125×C67—24no—
37#125×C67—24no—
38#125×C67—21no—
39#124×C67—21no—
40#124×C67D412yes—
41#123×C67—22no—
42#123×C67—22no—
43C22⋊Q8×C67—24no—
44C22⋊Q8×C67—22no—
45C22⋊Q8×C67—22no—
46C22⋊Q8×C67—22no—
47C4×Q8×C67—24no—
48C4×Q8×C67—21no—
49C4×Q8×C67—22no—
50C4×Q8×C67—22no—
51C4×Q8×C67—22no—
52C4×Q8×C67—22no—
53C4×Q8×C67—22no—
54C4×D4×C67—21no—
55C4×D4×C67—22no—
56#117×C67—24no—
57#117×C67—22no—
58#117×C67—21no—
59#117×C67C2×C411yescharacteristic
60#117×C67C2×C411yescharacteristic
61#117×C67C2×C411yescharacteristic
62#117×C67D414yes—
63C2×C4⋊C4×C67—22no—
64C2×C42×C67—21no—
65C2×C42×C67C2×C411yescharacteristic
66#101×C67—24no—
67#101×C67—24no—
68#41×C67—42no—
69C2×Q8×C67—22no—
70C2×Q8×C67—21no—
71C2×Q8×C67—42no—
72C2×Q8×C67—42no—
73C2×Q8×C67C22⋊C412yes—
74C2×Q8×C67C22⋊C411yescharacteristic
75C2×Q8×C67C2×D412yes—
76C2×Q8×C67C2×D412yes—
77C2×D4×C67—21no—
78C2×D4×C67C22⋊C411yescharacteristic
79C22×C4×C67—22no—
80C22×C4×C67—21no—
81C22×C4×C67—21no—
82C22×C4×C67—21no—
83C22×C4×C67C22×C411yescharacteristic
84C22×C4×C67C2×D412yes—
85M16×C67—44no—
86M16×C67—44no—
87C2×C8×C67—44no—
88C4⋊C4×C67—22no—
89C4⋊C4×C67—22no—
90C4⋊C4×C67—24no—
91C4⋊C4×C67—24no—
92C4⋊C4×C67—22no—
93C4⋊C4×C67—22no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67—21no—
97C4⋊C4×C67—24no—
98C4⋊C4×C67—24no—
99C4⋊C4×C67—24no—
100C4⋊C4×C67—24no—
101C22⋊C4×C67—22no—
102C22⋊C4×C67—24no—
103C22⋊C4×C67—21no—
104C22⋊C4×C67—24no—
105C22⋊C4×C67C22⋊C412yes—
106C22⋊C4×C67C22⋊C412yes—
107C42×C67—22no—
108C42×C67—24no—
109C42×C67—21no—
110C42×C67—21no—
111C42×C67—21no—
112C42×C67—24no—
113C42×C67C22⋊C411yescharacteristic
114C42×C67C22⋊C411yescharacteristic
115C42×C67C22⋊C412yes—
116C23×C67—21no—
117C23×C67C2×C22⋊C411yescharacteristic
118Q8×C67—42no—
119Q8×C67—42no—
120Q8×C67—42no—
121Q8×C67—41no—
122Q8×C67—44no—
123Q8×C67—42no—
124D4×C67—41no—
125D4×C67—42no—
126C2×C4×C67—21no—
127C2×C4×C67—24no—
128C2×C4×C67—22no—
129C2×C4×C67—22no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—22no—
133C2×C4×C67—21no—
134C2×C4×C67—21no—
135C2×C4×C67—42no—
136C2×C4×C67—44no—
137C2×C4×C67—42no—
138C2×C4×C67—42no—
139C2×C4×C67C2×C22⋊C411yescharacteristic
140C2×C4×C67C2×C22⋊C411yescharacteristic
141C2×C4×C67#12014yes—
142C8×C67—84no—
143C22×C67—41no—
144C22×C67—42no—
145C22×C67—41no—
146C22×C67#37911yescharacteristic
147C22×C67#41012yes—
148C4×C67—22no—
149C4×C67—24no—
150C4×C67—21no—
151C4×C67—21no—
152C4×C67—42no—
153C4×C67—41no—
154C4×C67—42no—
155C4×C67—42no—
156C4×C67—41no—
157C4×C67—42no—
158#12730C6711yesmaximal, characteristic, direct product factor, Sylow
159C2×C67—21no—
160C2×C67—22no—
161C2×C67—81no—
162C2×C67#178211yescharacteristic
163#3509C2×C6711yescharacteristic
164#2133C2×C6712yes—
165C2×#356C2×C6712yescomplemented
166#1830C2×C6712yescomplemented
167C67#1273011yescharacteristic, direct product factor, Sylow
168#557—22no—
169#556—22no—
170#552—24no—
171#541—22no—
172#536C4×C6711yescharacteristic
173#536C22×C6711yescharacteristic
174#533—22no—
175#533C4×C6711yescharacteristic
176C2×C4⋊Q8C22×C6711yescharacteristic
177#519C4×C6711yescharacteristic
178#519C22×C6711yescharacteristic
179C4×#41C4×C6711yescharacteristic
180C2×#101C22×C6714yes—
181#356—22no—
182#356—22no—
183#356—22no—
184#355—22no—
185#344—24no—
186C2×#41C2×C4×C6711yescharacteristic
187C22×Q8C2×C4×C6711yescharacteristic
188C22×Q8C23×C6711yescharacteristic
189C2×M16—24no—
190C4⋊Q8—21no—
191C4⋊Q8—22no—
192C4⋊Q8—21no—
193C4⋊Q8D4×C6712yes—
194C4⋊Q8D4×C6712yes—
195C4⋊Q8D4×C6712yes—
196#126—24no—
197#126—24no—
198#125—24no—
199#125—24no—
200#125—24no—
201#125—21no—
202#124—21no—
203#124D4×C6712yes—
204#123—22no—
205#123—22no—
206C22⋊Q8—24no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C4×Q8—24no—
211C4×Q8—21no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×D4—21no—
218C4×D4—22no—
219#117—24no—
220#117—22no—
221#117—21no—
222#117C2×C4×C6711yescharacteristic
223#117C2×C4×C6711yescharacteristic
224#117C2×C4×C6711yescharacteristic
225#117D4×C6714yes—
226C2×C4⋊C4—22no—
227C2×C42—21no—
228C2×C42C2×C4×C6711yescharacteristic
229#101—24no—
230#101—24no—
231#41—42no—
232C2×Q8—22no—
233C2×Q8—21no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8C22⋊C4×C6712yes—
237C2×Q8C22⋊C4×C6711yescharacteristic
238C2×Q8C2×D4×C6712yes—
239C2×Q8C2×D4×C6712yes—
240C2×D4—21no—
241C2×D4C22⋊C4×C6711yescharacteristic
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C22×C4×C6711yescharacteristic
247C22×C4C2×D4×C6712yes—
248M16—44no—
249M16—44no—
250C2×C8—44no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C22⋊C4—22no—
265C22⋊C4—24no—
266C22⋊C4—21no—
267C22⋊C4—24no—
268C22⋊C4C22⋊C4×C6712yes—
269C22⋊C4C22⋊C4×C6712yes—
270C42—22no—
271C42—24no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—24no—
276C42C22⋊C4×C6711yescharacteristic
277C42C22⋊C4×C6711yescharacteristic
278C42C22⋊C4×C6712yes—
279C23—21no—
280C23C2×C22⋊C4×C6711yescharacteristic
281Q8—42no—
282Q8—42no—
283Q8—42no—
284Q8—41no—
285Q8—42no—
286Q8—44no—
287D4—41no—
288D4—42no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4C2×C22⋊C4×C6711yescharacteristic
303C2×C4C2×C22⋊C4×C6711yescharacteristic
304C2×C4#120×C6714yes—
305C8—84no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22#379×C6711yescharacteristic
310C22#410×C6712yes—
311C4—22no—
312C4—24no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—41no—
317C4—42no—
318C4—42no—
319C4—41no—
320C4—42no—
321C2—21no—
322C2—22no—
323C2—81no—
324C2C67×#178211yescharacteristic
subgroupG/H
center, socleC2×C67#1782
commutatorC22×C4C22×C4×C67
Frattini, cosocleC22×Q8C23×C67
prime radicalC2×#41×C67C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C23×C67 ◁ #519×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#356×C67C2semidirect
4. C67×#1830C2semidirect
158. #12730C67direct (N × H)
165. C2×#356C2×C67semidirect
166. #1830C2×C67semidirect
167. C67#12730direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35092
C67×#21332
C2×#356×C672
C67×#18302
#1273067