Finite Group Databasebeta

← C109×#12727 #42130721

Subgroups — C109×#12727 #42130721

Order 27904.

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

normal subgroups only

This group has 810 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
1C109×#3508C211yesmaximal, characteristic
2C109×#2127C212yesmaximal, complemented
3C2×#353×C109C212yesmaximal, complemented
4C109×#1828C212yesmaximal, complemented
5#559×C109—22no—
6#550×C109—22no—
7#546×C109—24no—
8#540×C109—22no—
9#535×C109C411yescharacteristic, complemented
10#535×C109C2211yescharacteristic
11#533×C109C411yescharacteristic, complemented
12#532×C109—22no—
13C2×#127×C109C2211yescharacteristic
14#518×C109C411yescharacteristic, complemented
15#518×C109C2211yescharacteristic
16C4×#41×C109C411yescharacteristic, complemented
17C2×#99×C109C2214yes—
18#354×C109—22no—
19#353×C109—22no—
20#353×C109—22no—
21#353×C109—22no—
22#342×C109—24no—
23C2×#41×C109—24no—
24C2×#41×C109C2×C411yescharacteristic
25C22×D4×C109—22no—
26C22×D4×C109C2×C411yescharacteristic
27C22×D4×C109C2311yescharacteristic
28C4⋊Q8×C109—21no—
29#127×C109—21no—
30#127×C109—22no—
31#127×C109D412yes—
32#127×C109D412yes—
33#127×C109D412yes—
34#125×C109—21no—
35#124×C109—24no—
36#124×C109—21no—
37#124×C109D412yes—
38#123×C109—24no—
39#123×C109—24no—
40C22⋊Q8×C109—22no—
41#121×C109—24no—
42#121×C109—24no—
43#121×C109—24no—
44#121×C109—22no—
45#121×C109—22no—
46#120×C109—22no—
47#120×C109—22no—
48C4×Q8×C109—21no—
49C4×Q8×C109—22no—
50C4×D4×C109—24no—
51C4×D4×C109—21no—
52C4×D4×C109—22no—
53C4×D4×C109—22no—
54C4×D4×C109—22no—
55C4×D4×C109—22no—
56C4×D4×C109—22no—
57#117×C109—21no—
58#117×C109C2×C411yescharacteristic
59#117×C109C2×C411yescharacteristic
60#117×C109C2×C411yescharacteristic
61C2×C22⋊C4×C109—22no—
62C2×C22⋊C4×C109—24no—
63C2×C22⋊C4×C109D414yes—
64C2×C42×C109—21no—
65C2×C42×C109C2×C411yescharacteristic
66#99×C109—24no—
67#99×C109—24no—
68C24×C109—21no—
69C24×C109C2×D412yes—
70#41×C109—42no—
71#41×C109—88no—
72C2×Q8×C109—22no—
73C2×Q8×C109C22⋊C411yescharacteristic
74C2×D4×C109—21no—
75C2×D4×C109—24no—
76C2×D4×C109—24no—
77C2×D4×C109—24no—
78C2×D4×C109—24no—
79C2×D4×C109—21no—
80C2×D4×C109—42no—
81C2×D4×C109—42no—
82C2×D4×C109—84no—
83C2×D4×C109C22⋊C412yes—
84C2×D4×C109C22⋊C411yescharacteristic
85C2×D4×C109C2×D412yes—
86C2×D4×C109C2×D412yes—
87C22×C4×C109—22no—
88C22×C4×C109—21no—
89C22×C4×C109—24no—
90C22×C4×C109—24no—
91C22×C4×C109—21no—
92C22×C4×C109—44no—
93C22×C4×C109C22×C411yescharacteristic
94C4⋊C4×C109—22no—
95C4⋊C4×C109—22no—
96C4⋊C4×C109—22no—
97C4⋊C4×C109—22no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109—21no—
100C4⋊C4×C109—21no—
101C22⋊C4×C109—24no—
102C22⋊C4×C109—22no—
103C22⋊C4×C109—24no—
104C22⋊C4×C109—21no—
105C22⋊C4×C109—24no—
106C22⋊C4×C109—24no—
107C22⋊C4×C109—44no—
108C22⋊C4×C109—44no—
109C22⋊C4×C109C22⋊C412yes—
110C22⋊C4×C109C22⋊C412yes—
111C42×C109—22no—
112C42×C109—21no—
113C42×C109—21no—
114C42×C109—21no—
115C42×C109C22⋊C411yescharacteristic
116C42×C109C22⋊C411yescharacteristic
117C42×C109C22⋊C412yes—
118C23×C109—24no—
119C23×C109—22no—
120C23×C109—21no—
121C23×C109—42no—
122C23×C109—82no—
123C23×C109—82no—
124C23×C109C2×C22⋊C411yescharacteristic
125C23×C109#12014yes—
126Q8×C109—41no—
127Q8×C109—44no—
128D4×C109—42no—
129D4×C109—42no—
130D4×C109—42no—
131D4×C109—41no—
132D4×C109—42no—
133D4×C109—48no—
134D4×C109—42no—
135D4×C109—84no—
136D4×C109—84no—
137D4×C109—84no—
138C2×C4×C109—21no—
139C2×C4×C109—22no—
140C2×C4×C109—21no—
141C2×C4×C109—22no—
142C2×C4×C109—21no—
143C2×C4×C109—22no—
144C2×C4×C109—22no—
145C2×C4×C109—42no—
146C2×C4×C109—42no—
147C2×C4×C109—44no—
148C2×C4×C109—84no—
149C2×C4×C109—84no—
150C2×C4×C109—84no—
151C2×C4×C109—164no—
152C2×C4×C109C2×C22⋊C411yescharacteristic
153C2×C4×C109C2×C22⋊C411yescharacteristic
154C22×C109—24no—
155C22×C109—41no—
156C22×C109—44no—
157C22×C109—42no—
158C22×C109—44no—
159C22×C109—41no—
160C22×C109—42no—
161C22×C109—84no—
162C22×C109—162no—
163C22×C109#37911yescharacteristic
164C22×C109#41012yes—
165C4×C109—22no—
166C4×C109—21no—
167C4×C109—21no—
168C4×C109—42no—
169C4×C109—41no—
170C4×C109—42no—
171C4×C109—42no—
172C4×C109—41no—
173C4×C109—164no—
174#12727C10911yesmaximal, characteristic, direct product factor, Sylow
175C2×C109—21no—
176C2×C109—22no—
177C2×C109—44no—
178C2×C109—81no—
179C2×C109—82no—
180C2×C109#178211yescharacteristic
181#3508C2×C10911yescharacteristic
182#2127C2×C10912yescomplemented
183C2×#353C2×C10912yescomplemented
184#1828C2×C10912yescomplemented
185C109#1272711yescharacteristic, direct product factor, Sylow
186#559—22no—
187#550—22no—
188#546—24no—
189#540—22no—
190#535C4×C10911yescharacteristic, complemented
191#535C22×C10911yescharacteristic
192#533C4×C10911yescharacteristic, complemented
193#532—22no—
194C2×#127C22×C10911yescharacteristic
195#518C4×C10911yescharacteristic, complemented
196#518C22×C10911yescharacteristic
197C4×#41C4×C10911yescharacteristic, complemented
198C2×#99C22×C10914yes—
199#354—22no—
200#353—22no—
201#353—22no—
202#353—22no—
203#342—24no—
204C2×#41—24no—
205C2×#41C2×C4×C10911yescharacteristic
206C22×D4—22no—
207C22×D4C2×C4×C10911yescharacteristic
208C22×D4C23×C10911yescharacteristic
209C4⋊Q8—21no—
210#127—21no—
211#127—22no—
212#127D4×C10912yes—
213#127D4×C10912yes—
214#127D4×C10912yes—
215#125—21no—
216#124—24no—
217#124—21no—
218#124D4×C10912yes—
219#123—24no—
220#123—24no—
221C22⋊Q8—22no—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—22no—
226#121—22no—
227#120—22no—
228#120—22no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×D4—24no—
232C4×D4—21no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238#117—21no—
239#117C2×C4×C10911yescharacteristic
240#117C2×C4×C10911yescharacteristic
241#117C2×C4×C10911yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4D4×C10914yes—
245C2×C42—21no—
246C2×C42C2×C4×C10911yescharacteristic
247#99—24no—
248#99—24no—
249C24—21no—
250C24C2×D4×C10912yes—
251#41—42no—
252#41—88no—
253C2×Q8—22no—
254C2×Q8C22⋊C4×C10911yescharacteristic
255C2×D4—21no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4C22⋊C4×C10912yes—
265C2×D4C22⋊C4×C10911yescharacteristic
266C2×D4C2×D4×C10912yes—
267C2×D4C2×D4×C10912yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—44no—
274C22×C4C22×C4×C10911yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—21no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4C22⋊C4×C10912yes—
291C22⋊C4C22⋊C4×C10912yes—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22⋊C4×C10911yescharacteristic
297C42C22⋊C4×C10911yescharacteristic
298C42C22⋊C4×C10912yes—
299C23—24no—
300C23—22no—
301C23—21no—
302C23—42no—
303C23—82no—
304C23—82no—
305C23C2×C22⋊C4×C10911yescharacteristic
306C23#120×C10914yes—
307Q8—41no—
308Q8—44no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—41no—
313D4—42no—
314D4—48no—
315D4—42no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—164no—
333C2×C4C2×C22⋊C4×C10911yescharacteristic
334C2×C4C2×C22⋊C4×C10911yescharacteristic
335C22—24no—
336C22—41no—
337C22—44no—
338C22—42no—
339C22—44no—
340C22—41no—
341C22—42no—
342C22—84no—
343C22—162no—
344C22#379×C10911yescharacteristic
345C22#410×C10912yes—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—164no—
355C2—21no—
356C2—22no—
357C2—44no—
358C2—82no—
359C2—81no—
360C2C109×#178211yescharacteristic
subgroupG/H
center, socleC2×C109#1782
commutatorC22×C4C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC109×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C23×C109 ◁ #518×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2127C2semidirect
3. C2×#353×C109C2semidirect
4. C109×#1828C2semidirect
9. #535×C109C4semidirect
11. #533×C109C4semidirect
14. #518×C109C4semidirect
16. C4×#41×C109C4semidirect
174. #12727C109direct (N × H)
182. #2127C2×C109semidirect
183. C2×#353C2×C109semidirect
184. #1828C2×C109semidirect
185. C109#12727direct (N × H)
190. #535C4×C109semidirect
192. #533C4×C109semidirect
195. #518C4×C109semidirect
197. C4×#41C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35082
C109×#21272
C2×#353×C1092
C109×#18282
#12727109