Finite Group Databasebeta

← C107×#7785 #41033749

Subgroups — C107×#7785 #41033749

Order 27392.

454 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C107×#2808C211yesmaximal, characteristic
2C107×#2058C211yesmaximal, characteristic, complemented
3C107×#2023C211yesmaximal, characteristic
4C107×#2016C211yesmaximal, characteristic, complemented
5C107×#1272C211yesmaximal, characteristic, complemented
6C107×#1271C211yesmaximal, characteristic, complemented
7C107×#1270C211yesmaximal, characteristic
8C22×Q16×C107—21no—
9C2×C4⋊Q8×C107C2211yescharacteristic
10C2×#106×C107—21no—
11C2×#103×C107—21no—
12C2×#102×C107—21no—
13#397×C107—21no—
14#394×C107—21no—
15#393×C107—21no—
16#391×C107C2211yescharacteristic
17#390×C107C2211yescharacteristic
18#389×C107C2211yescharacteristic
19#388×C107—21no—
20#388×C107—21no—
21#387×C107—21no—
22#340×C107—21no—
23#340×C107—21no—
24#336×C107C2211yescharacteristic
25#336×C107C2211yescharacteristic
26#336×C107C2211yescharacteristic
27C22×Q8×C107—21no—
28C22×Q8×C107D411yescharacteristic
29C22×D4×C107D411yescharacteristic
30C2×Q16×C107—42no—
31C2×Q16×C107—42no—
32C2×Q16×C107—42no—
33C22×C8×C107—21no—
34C22×C8×C107—21no—
35C22×C8×C107—21no—
36C4⋊Q8×C107—42no—
37C2×C4⋊C4×C107—21no—
38C2×C4⋊C4×C107—21no—
39C2×C4⋊C4×C107—21no—
40C2×C4⋊C4×C107D411yescharacteristic
41C2×C4⋊C4×C107D411yescharacteristic
42C2×C4⋊C4×C107D411yescharacteristic
43C2×C4⋊C4×C107D411yescharacteristic
44C2×C22⋊C4×C107—21no—
45C2×C22⋊C4×C107—21no—
46C2×C22⋊C4×C107—21no—
47C2×C22⋊C4×C107—21no—
48C2×C22⋊C4×C107—21no—
49C2×C22⋊C4×C107—21no—
50C2×C42×C107—21no—
51C2×C42×C107C2311yescharacteristic
52#106×C107—42no—
53#103×C107—42no—
54#102×C107—42no—
55#95×C107—21no—
56#95×C107—21no—
57#95×C107—21no—
58#95×C107—21no—
59#95×C107—21no—
60#95×C107—21no—
61C24×C107—21no—
62C2×Q8×C107—22no—
63C2×Q8×C107—21no—
64C2×Q8×C107—21no—
65C2×Q8×C107—41no—
66C2×Q8×C107—41no—
67C2×Q8×C107—41no—
68C2×Q8×C107—41no—
69C2×Q8×C107—41no—
70C2×Q8×C107—41no—
71C2×Q8×C107—41no—
72C2×D4×C107—21no—
73C2×D4×C107—21no—
74C2×D4×C107—41no—
75C2×D4×C107—41no—
76C22×C4×C107—21no—
77C22×C4×C107—21no—
78C22×C4×C107—21no—
79C22×C4×C107—21no—
80C22×C4×C107—21no—
81C22×C4×C107—41no—
82C22×C4×C107C2×D411yescharacteristic
83C22×C4×C107C2×D411yescharacteristic
84C22×C4×C107C2×D411yescharacteristic
85Q16×C107—48no—
86C2×C8×C107—22no—
87C2×C8×C107—22no—
88C2×C8×C107—41no—
89C2×C8×C107—41no—
90C2×C8×C107—41no—
91C2×C8×C107—41no—
92C2×C8×C107—41no—
93C2×C8×C107—41no—
94C2×C8×C107—41no—
95C4⋊C4×C107—22no—
96C4⋊C4×C107—22no—
97C4⋊C4×C107—21no—
98C4⋊C4×C107—21no—
99C4⋊C4×C107—41no—
100C4⋊C4×C107—41no—
101C4⋊C4×C107—41no—
102C4⋊C4×C107—41no—
103C4⋊C4×C107—41no—
104C4⋊C4×C107—41no—
105C4⋊C4×C107—41no—
106C22⋊C4×C107—41no—
107C22⋊C4×C107—41no—
108C22⋊C4×C107—41no—
109C22⋊C4×C107—41no—
110C22⋊C4×C107—81no—
111C22⋊C4×C107—81no—
112C22⋊C4×C107—81no—
113C22⋊C4×C107—81no—
114C42×C107—41no—
115C42×C107—81no—
116C23×C107—41no—
117C23×C107—41no—
118C23×C107—41no—
119C23×C107—41no—
120C23×C107—41no—
121C23×C107—41no—
122C23×C107—41no—
123C23×C107#12011yescharacteristic
124Q8×C107—42no—
125Q8×C107—42no—
126Q8×C107—42no—
127Q8×C107—81no—
128Q8×C107—81no—
129Q8×C107—81no—
130D4×C107—42no—
131D4×C107—81no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—41no—
139C2×C4×C107—41no—
140C2×C4×C107—41no—
141C2×C4×C107—41no—
142C2×C4×C107—41no—
143C2×C4×C107—41no—
144C2×C4×C107—41no—
145C2×C4×C107—41no—
146C2×C4×C107—41no—
147C2×C4×C107—41no—
148C2×C4×C107—41no—
149C2×C4×C107—41no—
150C2×C4×C107—41no—
151C2×C4×C107—41no—
152C2×C4×C107—41no—
153C2×C4×C107—42no—
154C2×C4×C107—81no—
155C2×C4×C107—81no—
156C2×C4×C107#13511yescharacteristic
157C2×C4×C107#13511yescharacteristic
158C2×C4×C107#13511yescharacteristic
159C2×C4×C107AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C107#13711yescharacteristic
161C2×C4×C107#13711yescharacteristic
162C8×C107—42no—
163C8×C107—42no—
164C8×C107—81no—
165C22×C107—81no—
166C22×C107—81no—
167C22×C107—81no—
168C22×C107—81no—
169C22×C107—81no—
170C22×C107—81no—
171C22×C107—81no—
172C22×C107#44911yescharacteristic
173C22×C107#45211yescharacteristic
174C22×C107#45211yescharacteristic
175C22×C107#45311yescharacteristic
176C22×C107#45511yescharacteristic
177C22×C107#45611yescharacteristic
178C22×C107B(4,𝔽2)11yescharacteristic
179C4×C107—21no—
180C4×C107—21no—
181C4×C107—21no—
182C4×C107—21no—
183C4×C107—21no—
184C4×C107—21no—
185C4×C107—81no—
186C4×C107—82no—
187C4×C107—81no—
188C4×C107—81no—
189C4×C107—81no—
190C4×C107—81no—
191#7785C10711yesmaximal, characteristic, direct product factor, Sylow
192C2×C107—161no—
193C2×C107#160011yescharacteristic
194C2×C107#160311yescharacteristic
195C2×C107#160611yescharacteristic
196C2×C107#163411yescharacteristic
197C2×C107#164011yescharacteristic
198C2×C107#165211yescharacteristic
199C2×C107#166111yescharacteristic
200#2808C2×C10711yescharacteristic
201#2058C2×C10711yescharacteristic, complemented
202#2023C2×C10711yescharacteristic
203#2016C2×C10711yescharacteristic, complemented
204#1272C2×C10711yescharacteristic, complemented
205#1271C2×C10711yescharacteristic, complemented
206#1270C2×C10711yescharacteristic
207C107#778511yescharacteristic, direct product factor, Sylow
208C22×Q16—21no—
209C2×C4⋊Q8C22×C10711yescharacteristic
210C2×#106—21no—
211C2×#103—21no—
212C2×#102—21no—
213#397—21no—
214#394—21no—
215#393—21no—
216#391C22×C10711yescharacteristic
217#390C22×C10711yescharacteristic
218#389C22×C10711yescharacteristic
219#388—21no—
220#388—21no—
221#387—21no—
222#340—21no—
223#340—21no—
224#336C22×C10711yescharacteristic
225#336C22×C10711yescharacteristic
226#336C22×C10711yescharacteristic
227C22×Q8—21no—
228C22×Q8D4×C10711yescharacteristic
229C22×D4D4×C10711yescharacteristic
230C2×Q16—42no—
231C2×Q16—42no—
232C2×Q16—42no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C4⋊Q8—42no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4D4×C10711yescharacteristic
241C2×C4⋊C4D4×C10711yescharacteristic
242C2×C4⋊C4D4×C10711yescharacteristic
243C2×C4⋊C4D4×C10711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42C23×C10711yescharacteristic
252#106—42no—
253#103—42no—
254#102—42no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4C2×D4×C10711yescharacteristic
283C22×C4C2×D4×C10711yescharacteristic
284C22×C4C2×D4×C10711yescharacteristic
285Q16—48no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—81no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#120×C10711yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329Q8—81no—
330D4—42no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4#135×C10711yescharacteristic
357C2×C4#135×C10711yescharacteristic
358C2×C4#135×C10711yescharacteristic
359C2×C4AGL(1,ℤ/8)×C10711yescharacteristic
360C2×C4#137×C10711yescharacteristic
361C2×C4#137×C10711yescharacteristic
362C8—42no—
363C8—42no—
364C8—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#449×C10711yescharacteristic
373C22#452×C10711yescharacteristic
374C22#452×C10711yescharacteristic
375C22#453×C10711yescharacteristic
376C22#455×C10711yescharacteristic
377C22#456×C10711yescharacteristic
378C22B(4,𝔽2)×C10711yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—82no—
391C2—161no—
392C2C107×#160011yescharacteristic
393C2C107×#160311yescharacteristic
394C2C107×#160611yescharacteristic
395C2C107×#163411yescharacteristic
396C2C107×#164011yescharacteristic
397C2C107×#165211yescharacteristic
398C2C107×#166111yescharacteristic
subgroupG/H
center, socleC23×C107#120
commutator, Frattini, cosocleC2×C42C23×C107
prime radicalC22×D4×C107D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C107 ◁ C2×C42×C107 ◁ G

Complements

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

NH ≅ G/Ntype
2. C107×#2058C2semidirect
4. C107×#2016C2semidirect
5. C107×#1272C2semidirect
6. C107×#1271C2semidirect
191. #7785C107direct (N × H)
201. #2058C2×C107semidirect
203. #2016C2×C107semidirect
204. #1272C2×C107semidirect
205. #1271C2×C107semidirect
207. C107#7785direct (N × H)

Maximal subgroups

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

subgroupindex
C107×#28082
C107×#20582
C107×#20232
C107×#20162
C107×#12722
C107×#12712
C107×#12702
#7785107