Finite Group Databasebeta

← C67×#23896 #30703147

Subgroups — C67×#23896 #30703147

Order 17152.

1764 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 282 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×#3485C212yesmaximal, complemented
2C67×#3026C214yesmaximal, complemented
3C67×#2022C214yesmaximal, complemented
4C67×#1913C214yesmaximal, complemented
5C67×#1782C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C67C2212yescomplemented
7C22×#41×C67C2211yescharacteristic
8#547×C67—28no—
9#546×C67—28no—
10#536×C67—22no—
11#534×C67—22no—
12#534×C67C2214yes—
13#534×C67C2214yes—
14#525×C67—22no—
15C2×#123×C67—22no—
16C2×#123×C67C2214yescomplemented
17#518×C67C2212yescomplemented
18#458×C67—28no—
19B(4,𝔽2)×C67—28no—
20C2×#99×C67C2218yescomplemented
21C2×#99×C67C2214yescomplemented
22#381×C67C2212yescomplemented
23#342×C67C2214yescomplemented
24H(5,𝔽2)×C67—28no—
25H(5,𝔽2)×C67—44no—
26C2×#41×C67—28no—
27C2×#41×C67—22no—
28C2×#41×C67—24no—
29C2×#41×C67—42no—
30C22×Q8×C67D411yescharacteristic, complemented
31C22×D4×C67—22no—
32C22×D4×C67—24no—
33C22×D4×C67D411yescharacteristic, complemented
34C22×D4×C67C2314yes—
35C22×D4×C67C2312yes—
36C23×C4×C67D412yescomplemented
37C23×C4×C67C2311yescharacteristic
38C4⋊Q8×C67—24no—
39#124×C67—24no—
40#123×C67—28no—
41#123×C67—24no—
42#123×C67—24no—
43#123×C67—24no—
44#123×C67—24no—
45#123×C67—28no—
46#123×C67—28no—
47#123×C67D418yescomplemented
48C22⋊Q8×C67—24no—
49C22⋊Q8×C67—28no—
50#121×C67—24no—
51#121×C67—28no—
52#121×C67—28no—
53#121×C67—44no—
54#120×C67—24no—
55#120×C67—24no—
56#120×C67—28no—
57#120×C67D418yescomplemented
58C4×D4×C67—28no—
59C4×D4×C67—28no—
60#117×C67—22no—
61C2×C4⋊C4×C67—24no—
62C2×C4⋊C4×C67—24no—
63C2×C22⋊C4×C67—24no—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—24no—
66C2×C22⋊C4×C67C2314yes—
67C2×C22⋊C4×C67C2314yes—
68#99×C67—28no—
69#99×C67—216no—
70#95×C67—42no—
71C24×C67—22no—
72C24×C67C2×D414yes—
73C24×C67C2×D412yes—
74C24×C67C2411yescharacteristic
75#41×C67—416no—
76#41×C67—44no—
77#41×C67—416no—
78#41×C67—44no—
79C2×Q8×C67—22no—
80C2×Q8×C67—24no—
81C2×Q8×C67—41no—
82C2×D4×C67—24no—
83C2×D4×C67—24no—
84C2×D4×C67—24no—
85C2×D4×C67—22no—
86C2×D4×C67—24no—
87C2×D4×C67—24no—
88C2×D4×C67—28no—
89C2×D4×C67—24no—
90C2×D4×C67—28no—
91C2×D4×C67—28no—
92C2×D4×C67—41no—
93C2×D4×C67—416no—
94C2×D4×C67—44no—
95C2×D4×C67—44no—
96C2×D4×C67—44no—
97C2×D4×C67C2×D418yes—
98C22×C4×C67—22no—
99C22×C4×C67—24no—
100C22×C4×C67—24no—
101C22×C4×C67—24no—
102C22×C4×C67—22no—
103C22×C4×C67—24no—
104C22×C4×C67—24no—
105C22×C4×C67—24no—
106C22×C4×C67—24no—
107C22×C4×C67—42no—
108C22×C4×C67C2×D414yes—
109C22×C4×C67C2×D411yescharacteristic
110C22×C4×C67C2×D411yescharacteristic
111C22×C4×C67C2×D412yes—
112C4⋊C4×C67—28no—
113C4⋊C4×C67—28no—
114C4⋊C4×C67—24no—
115C4⋊C4×C67—28no—
116C4⋊C4×C67—44no—
117C22⋊C4×C67—28no—
118C22⋊C4×C67—24no—
119C22⋊C4×C67—28no—
120C22⋊C4×C67—28no—
121C22⋊C4×C67—24no—
122C22⋊C4×C67—28no—
123C22⋊C4×C67—28no—
124C22⋊C4×C67—48no—
125C22⋊C4×C67C2×D418yes—
126C42×C67—24no—
127C23×C67—22no—
128C23×C67—22no—
129C23×C67—24no—
130C23×C67—28no—
131C23×C67—44no—
132C23×C67—81no—
133C23×C67—82no—
134C23×C67—84no—
135C23×C67—84no—
136C23×C67#12014yes—
137C23×C67C22×D411yescharacteristic
138C23×C67C22×D414yes—
139C23×C67H(5,𝔽2)12yes—
140Q8×C67—44no—
141Q8×C67—44no—
142D4×C67—216no—
143D4×C67—44no—
144D4×C67—44no—
145D4×C67—48no—
146D4×C67—44no—
147D4×C67—44no—
148D4×C67—48no—
149D4×C67—88no—
150D4×C67—164no—
151C2×C4×C67—22no—
152C2×C4×C67—22no—
153C2×C4×C67—28no—
154C2×C4×C67—22no—
155C2×C4×C67—22no—
156C2×C4×C67—24no—
157C2×C4×C67—21no—
158C2×C4×C67—21no—
159C2×C4×C67—28no—
160C2×C4×C67—48no—
161C2×C4×C67—48no—
162C2×C4×C67—44no—
163C2×C4×C67—44no—
164C2×C4×C67—44no—
165C2×C4×C67—84no—
166C2×C4×C67—84no—
167C2×C4×C67—84no—
168C2×C4×C67#12014yes—
169C22×C67—24no—
170C22×C67—42no—
171C22×C67—44no—
172C22×C67—41no—
173C22×C67—44no—
174C22×C67—44no—
175C22×C67—48no—
176C22×C67—44no—
177C22×C67—82no—
178C22×C67—88no—
179C22×C67—164no—
180C22×C67C2×#12012yes—
181C22×C67#53411yescharacteristic
182C22×C67D4214yes—
183C4×C67—24no—
184C4×C67—48no—
185C4×C67—41no—
186C4×C67—44no—
187C4×C67—41no—
188C4×C67—84no—
189#23896C6711yesmaximal, characteristic, direct product factor, Sylow
190C2×C67—22no—
191C2×C67—24no—
192C2×C67—21no—
193C2×C67—44no—
194C2×C67—84no—
195C2×C67#240311yescharacteristic
196#3485C2×C6712yescomplemented
197#3026C2×C6714yescomplemented
198#2022C2×C6714yescomplemented
199#1913C2×C6714yescomplemented
200#1782C2×C6711yescharacteristic, complemented
201C67#2389611yescharacteristic, direct product factor, Sylow
202C2×H(5,𝔽2)C22×C6712yescomplemented
203C22×#41C22×C6711yescharacteristic
204#547—28no—
205#546—28no—
206#536—22no—
207#534—22no—
208#534C22×C6714yes—
209#534C22×C6714yes—
210#525—22no—
211C2×#123—22no—
212C2×#123C22×C6714yescomplemented
213#518C22×C6712yescomplemented
214#458—28no—
215B(4,𝔽2)—28no—
216C2×#99C22×C6718yescomplemented
217C2×#99C22×C6714yescomplemented
218#381C22×C6712yescomplemented
219#342C22×C6714yescomplemented
220H(5,𝔽2)—28no—
221H(5,𝔽2)—44no—
222C2×#41—28no—
223C2×#41—22no—
224C2×#41—24no—
225C2×#41—42no—
226C22×Q8D4×C6711yescharacteristic, complemented
227C22×D4—22no—
228C22×D4—24no—
229C22×D4D4×C6711yescharacteristic, complemented
230C22×D4C23×C6714yes—
231C22×D4C23×C6712yes—
232C23×C4D4×C6712yescomplemented
233C23×C4C23×C6711yescharacteristic
234C4⋊Q8—24no—
235#124—24no—
236#123—28no—
237#123—24no—
238#123—24no—
239#123—24no—
240#123—24no—
241#123—28no—
242#123—28no—
243#123D4×C6718yescomplemented
244C22⋊Q8—24no—
245C22⋊Q8—28no—
246#121—24no—
247#121—28no—
248#121—28no—
249#121—44no—
250#120—24no—
251#120—24no—
252#120—28no—
253#120D4×C6718yescomplemented
254C4×D4—28no—
255C4×D4—28no—
256#117—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4C23×C6714yes—
263C2×C22⋊C4C23×C6714yes—
264#99—28no—
265#99—216no—
266#95—42no—
267C24—22no—
268C24C2×D4×C6714yes—
269C24C2×D4×C6712yes—
270C24C24×C6711yescharacteristic
271#41—416no—
272#41—44no—
273#41—416no—
274#41—44no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—41no—
289C2×D4—416no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4C2×D4×C6718yes—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4C2×D4×C6714yes—
305C22×C4C2×D4×C6711yescharacteristic
306C22×C4C2×D4×C6711yescharacteristic
307C22×C4C2×D4×C6712yes—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—44no—
313C22⋊C4—28no—
314C22⋊C4—24no—
315C22⋊C4—28no—
316C22⋊C4—28no—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—48no—
321C22⋊C4C2×D4×C6718yes—
322C42—24no—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—28no—
327C23—44no—
328C23—81no—
329C23—82no—
330C23—84no—
331C23—84no—
332C23#120×C6714yes—
333C23C22×D4×C6711yescharacteristic
334C23C22×D4×C6714yes—
335C23H(5,𝔽2)×C6712yes—
336Q8—44no—
337Q8—44no—
338D4—216no—
339D4—44no—
340D4—44no—
341D4—48no—
342D4—44no—
343D4—44no—
344D4—48no—
345D4—88no—
346D4—164no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—28no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4#120×C6714yes—
365C22—24no—
366C22—42no—
367C22—44no—
368C22—41no—
369C22—44no—
370C22—44no—
371C22—48no—
372C22—44no—
373C22—82no—
374C22—88no—
375C22—164no—
376C22C2×#120×C6712yes—
377C22#534×C6711yescharacteristic
378C22D42×C6714yes—
379C4—24no—
380C4—48no—
381C4—41no—
382C4—44no—
383C4—41no—
384C4—84no—
385C2—22no—
386C2—24no—
387C2—21no—
388C2—44no—
389C2—84no—
390C2C67×#240311yescharacteristic
subgroupG/H
center, socleC2×C67#2403
commutator, Frattini, cosocleC24C24×C67
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C24×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3485C2semidirect
2. C67×#3026C2semidirect
3. C67×#2022C2semidirect
4. C67×#1913C2semidirect
5. C67×#1782C2semidirect
6. C2×H(5,𝔽2)×C67C22semidirect
16. C2×#123×C67C22semidirect
17. #518×C67C22semidirect
20. C2×#99×C67C22semidirect
21. C2×#99×C67C22semidirect
22. #381×C67C22semidirect
23. #342×C67C22semidirect
30. C22×Q8×C67D4semidirect
33. C22×D4×C67D4semidirect
36. C23×C4×C67D4semidirect
47. #123×C67D4semidirect
57. #120×C67D4semidirect
189. #23896C67direct (N × H)
196. #3485C2×C67semidirect
197. #3026C2×C67semidirect
198. #2022C2×C67semidirect
199. #1913C2×C67semidirect
200. #1782C2×C67semidirect
201. C67#23896direct (N × H)
202. C2×H(5,𝔽2)C22×C67semidirect
212. C2×#123C22×C67semidirect
213. #518C22×C67semidirect
216. C2×#99C22×C67semidirect
217. C2×#99C22×C67semidirect
218. #381C22×C67semidirect
219. #342C22×C67semidirect
226. C22×Q8D4×C67semidirect
229. C22×D4D4×C67semidirect
232. C23×C4D4×C67semidirect
243. #123D4×C67semidirect
253. #120D4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34852
C67×#30262
C67×#20222
C67×#19132
C67×#17822
#2389667