Finite Group Databasebeta

← C67×#12997 #30692248

Subgroups — C67×#12997 #30692248

Order 17152.

492 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#473×C67C211yesmaximal, characteristic
2C67×#2913C211yesmaximal, characteristic
3C67×#2141C211yesmaximal, characteristic, complemented
4C2×#358×C67C211yesmaximal, characteristic, complemented
5C2×#350×C67C211yesmaximal, characteristic
6C67×#1387C211yesmaximal, characteristic, complemented
7C67×#1382C211yesmaximal, characteristic, complemented
8C2×#137×C67—21no—
9C22×Q16×C67C2211yescharacteristic
10C2×#131×C67C2211yescharacteristic
11#520×C67—21no—
12C2×M32×C67C2211yescharacteristic
13C22×C16×C67C2211yescharacteristic
14#473×C67—22no—
15#473×C67C414yescomplemented
16C2×#108×C67C2211yescharacteristic
17#428×C67—21no—
18#428×C67—22no—
19#427×C67—21no—
20#427×C67C2211yescharacteristic
21C2×#107×C67—21no—
22C2×#107×C67C2211yescharacteristic
23C2×#101×C67—21no—
24#358×C67—22no—
25#358×C67—21no—
26#358×C67—21no—
27#350×C67—22no—
28C2×#41×C67D411yescharacteristic
29C22×Q8×C67—21no—
30#137×C67—24no—
31#137×C67—42no—
32C2×Q16×C67—21no—
33C2×Q16×C67—21no—
34C2×Q16×C67—21no—
35C2×Q16×C67—21no—
36C2×Q16×C67—21no—
37C2×Q16×C67—41no—
38C2×Q16×C67C2×C412yescomplemented
39C2×SD16×C67—41no—
40#131×C67—21no—
41#131×C67—21no—
42#131×C67C2×C412yes—
43#131×C67D412yes—
44C2×M16×C67—21no—
45C2×M16×C67—21no—
46C2×M16×C67D411yescharacteristic
47C22×C8×C67—21no—
48C22×C8×C67C2311yescharacteristic
49C4×Q8×C67—41no—
50C4×D4×C67—42no—
51C4×D4×C67—41no—
52#117×C67—22no—
53#117×C67—21no—
54C2×C4⋊C4×C67—22no—
55C2×C4⋊C4×C67—21no—
56M32×C67—21no—
57M32×C67—21no—
58C2×C16×C67—21no—
59C2×C16×C67—21no—
60C2×C16×C67—21no—
61C2×C16×C67—21no—
62#108×C67—21no—
63#108×C67C2×C412yescomplemented
64#107×C67—22no—
65#107×C67—21no—
66#107×C67D411yescharacteristic
67#107×C67D412yes—
68#107×C67D411yescharacteristic
69#106×C67—22no—
70#106×C67—21no—
71#101×C67—24no—
72#41×C67—21no—
73#41×C67—21no—
74#41×C67C22⋊C412yes—
75#41×C67SD1612yes—
76C2×Q8×C67—22no—
77C2×Q8×C67—41no—
78C2×Q8×C67—41no—
79C2×Q8×C67—41no—
80C2×Q8×C67—41no—
81C2×Q8×C67—41no—
82C2×Q8×C67D811yescharacteristic
83C2×D4×C67—21no—
84C2×D4×C67D811yescharacteristic
85C22×C4×C67—21no—
86C22×C4×C67—41no—
87C22×C4×C67—42no—
88C22×C4×C67—41no—
89C22×C4×C67C2×D411yescharacteristic
90Q16×C67—22no—
91Q16×C67—22no—
92Q16×C67—44no—
93Q16×C67—41no—
94Q16×C67—41no—
95SD16×C67—44no—
96M16×C67—21no—
97M16×C67—22no—
98M16×C67—22no—
99M16×C67—22no—
100M16×C67—41no—
101M16×C67C22⋊C412yes—
102C2×C8×C67—21no—
103C2×C8×C67—22no—
104C2×C8×C67—22no—
105C2×C8×C67—41no—
106C2×C8×C67C22×C411yescharacteristic
107C2×C8×C67C2×D411yescharacteristic
108C2×C8×C67C2×D411yescharacteristic
109C2×C8×C67C2×D411yescharacteristic
110C2×C8×C67#4111yescharacteristic
111C2×C8×C67#4111yescharacteristic
112C4⋊C4×C67—21no—
113C4⋊C4×C67—22no—
114C4⋊C4×C67—21no—
115C4⋊C4×C67—42no—
116C4⋊C4×C67—41no—
117C22⋊C4×C67—42no—
118C22⋊C4×C67—41no—
119C42×C67—42no—
120C42×C67—41no—
121C16×C67—41no—
122C16×C67—41no—
123C23×C67—21no—
124C23×C67AGL(1,ℤ/8)11yescharacteristic
125Q8×C67—21no—
126Q8×C67—42no—
127Q8×C67—42no—
128Q8×C67—81no—
129Q8×C67—81no—
130Q8×C67#10212yes—
131D4×C67—22no—
132D4×C67—22no—
133D4×C67—21no—
134D4×C67#10212yes—
135C2×C4×C67—21no—
136C2×C4×C67—22no—
137C2×C4×C67—22no—
138C2×C4×C67—42no—
139C2×C4×C67—41no—
140C2×C4×C67—42no—
141C2×C4×C67—41no—
142C2×C4×C67—81no—
143C2×C4×C67—81no—
144C2×C4×C67—161no—
145C2×C4×C67—162no—
146C2×C4×C67C2×C22⋊C411yescharacteristic
147C2×C4×C67#12011yescharacteristic
148C2×C4×C67#12311yescharacteristic
149C2×C4×C67C2×D811yescharacteristic
150C2×C4×C67C2×SD1611yescharacteristic
151C2×C4×C67#13711yescharacteristic
152C8×C67—22no—
153C8×C67—42no—
154C8×C67C4×D411yescharacteristic
155C8×C67C4×D411yescharacteristic
156C8×C67#12111yescharacteristic
157C8×C67#12111yescharacteristic
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—22no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67#41711yescharacteristic
164C22×C67#44711yescharacteristic
165C22×C67#48111yescharacteristic
166C4×C67—22no—
167C4×C67—82no—
168C4×C67—82no—
169C4×C67—81no—
170C4×C67—81no—
171C4×C67#38611yescharacteristic
172C4×C67C2×#10211yescharacteristic
173C4×C67#44811yescharacteristic
174C4×C67#48011yescharacteristic
175#12997C6711yesmaximal, characteristic, direct product factor, Sylow
176C2×C67—21no—
177C2×C67—21no—
178C2×C67—42no—
179C2×C67#187411yescharacteristic
180C2×C67#219411yescharacteristic
181C2×C67#219511yescharacteristic
182C2×#473C2×C6711yescharacteristic
183#2913C2×C6711yescharacteristic
184#2141C2×C6711yescharacteristic, complemented
185C2×#358C2×C6711yescharacteristic, complemented
186C2×#350C2×C6711yescharacteristic
187#1387C2×C6711yescharacteristic, complemented
188#1382C2×C6711yescharacteristic, complemented
189C67#1299711yescharacteristic, direct product factor, Sylow
190C2×#137—21no—
191C22×Q16C22×C6711yescharacteristic
192C2×#131C22×C6711yescharacteristic
193#520—21no—
194C2×M32C22×C6711yescharacteristic
195C22×C16C22×C6711yescharacteristic
196#473—22no—
197#473C4×C6714yescomplemented
198C2×#108C22×C6711yescharacteristic
199#428—21no—
200#428—22no—
201#427—21no—
202#427C22×C6711yescharacteristic
203C2×#107—21no—
204C2×#107C22×C6711yescharacteristic
205C2×#101—21no—
206#358—22no—
207#358—21no—
208#358—21no—
209#350—22no—
210C2×#41D4×C6711yescharacteristic
211C22×Q8—21no—
212#137—24no—
213#137—42no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—41no—
220C2×Q16C2×C4×C6712yescomplemented
221C2×SD16—41no—
222#131—21no—
223#131—21no—
224#131C2×C4×C6712yes—
225#131D4×C6712yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16D4×C6711yescharacteristic
229C22×C8—21no—
230C22×C8C23×C6711yescharacteristic
231C4×Q8—41no—
232C4×D4—42no—
233C4×D4—41no—
234#117—22no—
235#117—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238M32—21no—
239M32—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244#108—21no—
245#108C2×C4×C6712yescomplemented
246#107—22no—
247#107—21no—
248#107D4×C6711yescharacteristic
249#107D4×C6712yes—
250#107D4×C6711yescharacteristic
251#106—22no—
252#106—21no—
253#101—24no—
254#41—21no—
255#41—21no—
256#41C22⋊C4×C6712yes—
257#41SD16×C6712yes—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8D8×C6711yescharacteristic
265C2×D4—21no—
266C2×D4D8×C6711yescharacteristic
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4C2×D4×C6711yescharacteristic
272Q16—22no—
273Q16—22no—
274Q16—44no—
275Q16—41no—
276Q16—41no—
277SD16—44no—
278M16—21no—
279M16—22no—
280M16—22no—
281M16—22no—
282M16—41no—
283M16C22⋊C4×C6712yes—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8C22×C4×C6711yescharacteristic
289C2×C8C2×D4×C6711yescharacteristic
290C2×C8C2×D4×C6711yescharacteristic
291C2×C8C2×D4×C6711yescharacteristic
292C2×C8#41×C6711yescharacteristic
293C2×C8#41×C6711yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C42—42no—
302C42—41no—
303C16—41no—
304C16—41no—
305C23—21no—
306C23AGL(1,ℤ/8)×C6711yescharacteristic
307Q8—21no—
308Q8—42no—
309Q8—42no—
310Q8—81no—
311Q8—81no—
312Q8#102×C6712yes—
313D4—22no—
314D4—22no—
315D4—21no—
316D4#102×C6712yes—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—161no—
327C2×C4—162no—
328C2×C4C2×C22⋊C4×C6711yescharacteristic
329C2×C4#120×C6711yescharacteristic
330C2×C4#123×C6711yescharacteristic
331C2×C4C2×D8×C6711yescharacteristic
332C2×C4C2×SD16×C6711yescharacteristic
333C2×C4#137×C6711yescharacteristic
334C8—22no—
335C8—42no—
336C8C4×D4×C6711yescharacteristic
337C8C4×D4×C6711yescharacteristic
338C8#121×C6711yescharacteristic
339C8#121×C6711yescharacteristic
340C22—21no—
341C22—21no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22#417×C6711yescharacteristic
346C22#447×C6711yescharacteristic
347C22#481×C6711yescharacteristic
348C4—22no—
349C4—82no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4#386×C6711yescharacteristic
354C4C2×#102×C6711yescharacteristic
355C4#448×C6711yescharacteristic
356C4#480×C6711yescharacteristic
357C2—21no—
358C2—21no—
359C2—42no—
360C2C67×#187411yescharacteristic
361C2C67×#219411yescharacteristic
362C2C67×#219511yescharacteristic
subgroupG/H
center, socleC22×C67#447
commutatorC2×C8C22×C4×C67
Frattini, cosocleC22×C8C23×C67
prime radicalC2×#41×C67D4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4×C67 ◁ C2×#131×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#2141C2semidirect
4. C2×#358×C67C2semidirect
6. C67×#1387C2semidirect
7. C67×#1382C2semidirect
15. #473×C67C4semidirect
38. C2×Q16×C67C2×C4semidirect
63. #108×C67C2×C4semidirect
175. #12997C67direct (N × H)
184. #2141C2×C67semidirect
185. C2×#358C2×C67semidirect
187. #1387C2×C67semidirect
188. #1382C2×C67semidirect
189. C67#12997direct (N × H)
197. #473C4×C67semidirect
220. C2×Q16C2×C4×C67semidirect
245. #108C2×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#473×C672
C67×#29132
C67×#21412
C2×#358×C672
C2×#350×C672
C67×#13872
C67×#13822
#1299767