Finite Group Databasebeta

← C107×#10068 #41036032

Subgroups — C107×#10068 #41036032

Order 27392.

670 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#498×C107C211yesmaximal, characteristic, complemented
2C2×#461×C107C211yesmaximal, characteristic
3C107×#2542C211yesmaximal, characteristic
4C107×#1988C211yesmaximal, characteristic, complemented
5C2×#332×C107C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C107C211yesmaximal, characteristic
7C107×#1279C211yesmaximal, characteristic, complemented
8C22×SD16×C107—21no—
9C2×C4⋊Q8×C107C2211yescharacteristic
10C2×C4⋊Q8×C107C2211yescharacteristic
11C2×C22⋊Q8×C107—22no—
12C2×C4×D4×C107C2211yescharacteristic
13#498×C107—22no—
14#498×C107—22no—
15#461×C107—22no—
16#461×C107—22no—
17C2×#106×C107—22no—
18C2×C4⋊C8×C107C2211yescharacteristic
19C2×C4⋊C8×C107C2211yescharacteristic
20C2×#103×C107—21no—
21C2×C4×C8×C107C2211yescharacteristic
22#399×C107—22no—
23#390×C107—21no—
24#384×C107—21no—
25C4×C4⋊C4×C107C2211yescharacteristic
26#340×C107—22no—
27#332×C107—22no—
28D4⋊C8×C107—22no—
29C22×Q8×C107—21no—
30C22×Q8×C107—21no—
31C22×D4×C107D411yescharacteristic
32C23×C4×C107—21no—
33C2×SD16×C107—24no—
34C2×SD16×C107—42no—
35C2×SD16×C107—42no—
36C22×C8×C107—21no—
37C22×C8×C107—21no—
38C22×C8×C107—21no—
39C4⋊Q8×C107—21no—
40C4⋊Q8×C107—21no—
41C4⋊Q8×C107—21no—
42C4⋊Q8×C107—21no—
43C4⋊Q8×C107—21no—
44C4⋊Q8×C107—21no—
45C4⋊Q8×C107—21no—
46C4⋊Q8×C107D412yes—
47C22⋊Q8×C107—44no—
48C22⋊Q8×C107—44no—
49C4×D4×C107—21no—
50C4×D4×C107—21no—
51C4×D4×C107—21no—
52C4×D4×C107—21no—
53C2×C4⋊C4×C107—21no—
54C2×C4⋊C4×C107—21no—
55C2×C4⋊C4×C107—24no—
56C2×C4⋊C4×C107—21no—
57C2×C4⋊C4×C107D411yescharacteristic
58C2×C4⋊C4×C107D412yes—
59C2×C4⋊C4×C107D412yes—
60C2×C22⋊C4×C107—24no—
61C2×C22⋊C4×C107—21no—
62C2×C42×C107—21no—
63C2×C42×C107C2311yescharacteristic
64#106×C107—28no—
65C4⋊C8×C107—21no—
66C4⋊C8×C107—21no—
67C4⋊C8×C107—21no—
68C4⋊C8×C107D412yes—
69#103×C107—24no—
70C4×C8×C107—21no—
71C4×C8×C107D412yes—
72#95×C107—21no—
73C24×C107—21no—
74C2×Q8×C107—22no—
75C2×Q8×C107—41no—
76C2×Q8×C107—41no—
77C2×Q8×C107—41no—
78C2×Q8×C107—42no—
79C2×Q8×C107—42no—
80C2×Q8×C107—41no—
81C2×Q8×C107—41no—
82C2×Q8×C107—41no—
83C2×Q8×C107—41no—
84C2×D4×C107—21no—
85C2×D4×C107—21no—
86C2×D4×C107—21no—
87C2×D4×C107—21no—
88C2×D4×C107—21no—
89C2×D4×C107SD1612yes—
90C22×C4×C107—22no—
91C22×C4×C107—22no—
92C22×C4×C107—21no—
93C22×C4×C107—41no—
94C22×C4×C107—41no—
95C22×C4×C107—41no—
96C22×C4×C107—42no—
97C22×C4×C107—42no—
98C22×C4×C107C2×D411yescharacteristic
99C22×C4×C107C2×D411yescharacteristic
100C22×C4×C107C2×D411yescharacteristic
101SD16×C107—48no—
102C2×C8×C107—24no—
103C2×C8×C107—22no—
104C2×C8×C107—22no—
105C2×C8×C107—41no—
106C2×C8×C107—41no—
107C2×C8×C107—41no—
108C2×C8×C107—42no—
109C4⋊C4×C107—22no—
110C4⋊C4×C107—21no—
111C4⋊C4×C107—22no—
112C4⋊C4×C107—22no—
113C4⋊C4×C107—22no—
114C4⋊C4×C107—21no—
115C4⋊C4×C107—22no—
116C4⋊C4×C107—48no—
117C4⋊C4×C107—41no—
118C4⋊C4×C107—41no—
119C4⋊C4×C107—41no—
120C4⋊C4×C107—41no—
121C4⋊C4×C107SD1614yescomplemented
122C22⋊C4×C107—48no—
123C22⋊C4×C107—41no—
124C22⋊C4×C107—41no—
125C42×C107—41no—
126C42×C107—41no—
127C42×C107C2×D411yescharacteristic
128C42×C107C2×D411yescharacteristic
129C42×C107C2×D411yescharacteristic
130C42×C107#4111yescharacteristic
131C23×C107—22no—
132C23×C107—41no—
133C23×C107—41no—
134C23×C107—42no—
135C23×C107—42no—
136C23×C107#12011yescharacteristic
137Q8×C107—42no—
138Q8×C107—42no—
139Q8×C107—82no—
140Q8×C107—81no—
141Q8×C107—81no—
142Q8×C107—81no—
143Q8×C107—81no—
144D4×C107—22no—
145D4×C107—22no—
146D4×C107—41no—
147D4×C107—41no—
148C2×C4×C107—22no—
149C2×C4×C107—24no—
150C2×C4×C107—42no—
151C2×C4×C107—42no—
152C2×C4×C107—41no—
153C2×C4×C107—42no—
154C2×C4×C107—42no—
155C2×C4×C107—44no—
156C2×C4×C107—42no—
157C2×C4×C107—81no—
158C2×C4×C107—81no—
159C2×C4×C107—82no—
160C2×C4×C107—81no—
161C2×C4×C107—81no—
162C2×C4×C107#12011yescharacteristic
163C2×C4×C107#12011yescharacteristic
164C2×C4×C107#12111yescharacteristic
165C2×C4×C107#12111yescharacteristic
166C2×C4×C107#12111yescharacteristic
167C2×C4×C107#12711yescharacteristic
168C2×C4×C107C2×SD1611yescharacteristic
169C2×C4×C107C2×SD1612yes—
170C2×C4×C107#13511yescharacteristic
171C2×C4×C107AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C107AGL(1,ℤ/8)12yes—
173C2×C4×C107#13711yescharacteristic
174C8×C107—24no—
175C8×C107—42no—
176C8×C107—81no—
177C22×C107—44no—
178C22×C107—42no—
179C22×C107—82no—
180C22×C107—81no—
181C22×C107—81no—
182C22×C107#39211yescharacteristic
183C22×C107#44811yescharacteristic
184C22×C107#44911yescharacteristic
185C22×C107#45012yes—
186C22×C107#45612yes—
187C4×C107—21no—
188C4×C107—21no—
189C4×C107—44no—
190C4×C107—82no—
191C4×C107—81no—
192C4×C107—82no—
193C4×C107#45011yescharacteristic
194C4×C107#45211yescharacteristic
195C4×C107#46012yes—
196C4×C107#46511yescharacteristic
197C4×C107#46811yescharacteristic
198C4×C107#49211yescharacteristic
199C4×C107#49611yescharacteristic
200#10068C10711yesmaximal, characteristic, direct product factor, Sylow
201C2×C107—82no—
202C2×C107#165112yes—
203C2×C107#165412yes—
204C2×C107#200011yescharacteristic
205C2×C107#200511yescharacteristic
206C2×C107#201411yescharacteristic
207C2×#498C2×C10711yescharacteristic, complemented
208C2×#461C2×C10711yescharacteristic
209#2542C2×C10711yescharacteristic
210#1988C2×C10711yescharacteristic, complemented
211C2×#332C2×C10711yescharacteristic, complemented
212C2×D4⋊C8C2×C10711yescharacteristic
213#1279C2×C10711yescharacteristic, complemented
214C107#1006811yescharacteristic, direct product factor, Sylow
215C22×SD16—21no—
216C2×C4⋊Q8C22×C10711yescharacteristic
217C2×C4⋊Q8C22×C10711yescharacteristic
218C2×C22⋊Q8—22no—
219C2×C4×D4C22×C10711yescharacteristic
220#498—22no—
221#498—22no—
222#461—22no—
223#461—22no—
224C2×#106—22no—
225C2×C4⋊C8C22×C10711yescharacteristic
226C2×C4⋊C8C22×C10711yescharacteristic
227C2×#103—21no—
228C2×C4×C8C22×C10711yescharacteristic
229#399—22no—
230#390—21no—
231#384—21no—
232C4×C4⋊C4C22×C10711yescharacteristic
233#340—22no—
234#332—22no—
235D4⋊C8—22no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×D4D4×C10711yescharacteristic
239C23×C4—21no—
240C2×SD16—24no—
241C2×SD16—42no—
242C2×SD16—42no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8D4×C10712yes—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4D4×C10711yescharacteristic
265C2×C4⋊C4D4×C10712yes—
266C2×C4⋊C4D4×C10712yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—21no—
269C2×C42—21no—
270C2×C42C23×C10711yescharacteristic
271#106—28no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8D4×C10712yes—
276#103—24no—
277C4×C8—21no—
278C4×C8D4×C10712yes—
279#95—21no—
280C24—21no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4SD16×C10712yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4C2×D4×C10711yescharacteristic
306C22×C4C2×D4×C10711yescharacteristic
307C22×C4C2×D4×C10711yescharacteristic
308SD16—48no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—48no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4SD16×C10714yescomplemented
329C22⋊C4—48no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42—41no—
333C42—41no—
334C42C2×D4×C10711yescharacteristic
335C42C2×D4×C10711yescharacteristic
336C42C2×D4×C10711yescharacteristic
337C42#41×C10711yescharacteristic
338C23—22no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—42no—
343C23#120×C10711yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—82no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351D4—22no—
352D4—22no—
353D4—41no—
354D4—41no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4#120×C10711yescharacteristic
370C2×C4#120×C10711yescharacteristic
371C2×C4#121×C10711yescharacteristic
372C2×C4#121×C10711yescharacteristic
373C2×C4#121×C10711yescharacteristic
374C2×C4#127×C10711yescharacteristic
375C2×C4C2×SD16×C10711yescharacteristic
376C2×C4C2×SD16×C10712yes—
377C2×C4#135×C10711yescharacteristic
378C2×C4AGL(1,ℤ/8)×C10711yescharacteristic
379C2×C4AGL(1,ℤ/8)×C10712yes—
380C2×C4#137×C10711yescharacteristic
381C8—24no—
382C8—42no—
383C8—81no—
384C22—44no—
385C22—42no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22#392×C10711yescharacteristic
390C22#448×C10711yescharacteristic
391C22#449×C10711yescharacteristic
392C22#450×C10712yes—
393C22#456×C10712yes—
394C4—21no—
395C4—21no—
396C4—44no—
397C4—82no—
398C4—81no—
399C4—82no—
400C4#450×C10711yescharacteristic
401C4#452×C10711yescharacteristic
402C4#460×C10712yes—
403C4#465×C10711yescharacteristic
404C4#468×C10711yescharacteristic
405C4#492×C10711yescharacteristic
406C4#496×C10711yescharacteristic
407C2—82no—
408C2C107×#165112yes—
409C2C107×#165412yes—
410C2C107×#200011yescharacteristic
411C2C107×#200511yescharacteristic
412C2C107×#201411yescharacteristic
subgroupG/H
center, socleC23×C107#120
commutator, Frattini, cosocleC2×C42C23×C107
prime radicalC22×D4×C107D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C107 ◁ C2×C42×C107 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#498×C107C2semidirect
4. C107×#1988C2semidirect
5. C2×#332×C107C2semidirect
7. C107×#1279C2semidirect
121. C4⋊C4×C107SD16semidirect
200. #10068C107direct (N × H)
207. C2×#498C2×C107semidirect
210. #1988C2×C107semidirect
211. C2×#332C2×C107semidirect
213. #1279C2×C107semidirect
214. C107#10068direct (N × H)
328. C4⋊C4SD16×C107semidirect

Maximal subgroups

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

subgroupindex
C2×#498×C1072
C2×#461×C1072
C107×#25422
C107×#19882
C2×#332×C1072
C2×D4⋊C8×C1072
C107×#12792
#10068107