Finite Group Databasebeta

← C109×#12728 #42130722

Subgroups — C109×#12728 #42130722

Order 27904.

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

normal subgroups only

This group has 714 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×#3510C211yesmaximal, characteristic
2C109×#2127C212yesmaximal
3C2×#354×C109C212yesmaximal, complemented
4C109×#1828C212yesmaximal, complemented
5#559×C109—22no—
6#556×C109—22no—
7#551×C109—24no—
8#542×C109—22no—
9#536×C109C411yescharacteristic, complemented
10#535×C109C2211yescharacteristic
11#533×C109C411yescharacteristic, complemented
12#529×C109—22no—
13C2×#124×C109C2211yescharacteristic
14#519×C109C411yescharacteristic, complemented
15#518×C109C2211yescharacteristic
16C4×#41×C109C411yescharacteristic, complemented
17C2×#99×C109C2214yes—
18#354×C109—22no—
19#354×C109—22no—
20#354×C109—22no—
21#353×C109—22no—
22#342×C109—24no—
23C2×#41×C109C2×C411yescharacteristic
24C22×Q8×C109C2×C411yescharacteristic
25C22×D4×C109C2311yescharacteristic
26C4⋊Q8×C109—21no—
27C4⋊Q8×C109—21no—
28#127×C109D412yes—
29#126×C109—24no—
30#126×C109—24no—
31#125×C109—21no—
32#124×C109—24no—
33#124×C109—21no—
34#124×C109—22no—
35#124×C109D412yes—
36#124×C109D412yes—
37#124×C109D412yes—
38#123×C109—24no—
39#123×C109—24no—
40#123×C109—22no—
41C22⋊Q8×C109—24no—
42C22⋊Q8×C109—22no—
43C22⋊Q8×C109—22no—
44#121×C109—22no—
45#120×C109—22no—
46C4×Q8×C109—21no—
47C4×Q8×C109—22no—
48C4×Q8×C109—22no—
49C4×Q8×C109—22no—
50C4×D4×C109—24no—
51C4×D4×C109—21no—
52C4×D4×C109—22no—
53C4×D4×C109—22no—
54C4×D4×C109—22no—
55#117×C109—24no—
56#117×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×C109C2×D412yes—
69#41×C109—42no—
70C2×Q8×C109—22no—
71C2×Q8×C109—21no—
72C2×Q8×C109—42no—
73C2×Q8×C109C22⋊C412yes—
74C2×Q8×C109C22⋊C411yescharacteristic
75C2×D4×C109—21no—
76C2×D4×C109—24no—
77C2×D4×C109—42no—
78C2×D4×C109C22⋊C411yescharacteristic
79C2×D4×C109C2×D412yes—
80C2×D4×C109C2×D412yes—
81C22×C4×C109—21no—
82C22×C4×C109—22no—
83C22×C4×C109—21no—
84C22×C4×C109—24no—
85C22×C4×C109—21no—
86C22×C4×C109—44no—
87C22×C4×C109C22×C411yescharacteristic
88C4⋊C4×C109—22no—
89C4⋊C4×C109—22no—
90C4⋊C4×C109—24no—
91C4⋊C4×C109—24no—
92C4⋊C4×C109—22no—
93C4⋊C4×C109—22no—
94C4⋊C4×C109—21no—
95C4⋊C4×C109—21no—
96C4⋊C4×C109—21no—
97C22⋊C4×C109—22no—
98C22⋊C4×C109—24no—
99C22⋊C4×C109—21no—
100C22⋊C4×C109—24no—
101C22⋊C4×C109—24no—
102C22⋊C4×C109—24no—
103C22⋊C4×C109—24no—
104C22⋊C4×C109—44no—
105C22⋊C4×C109—44no—
106C22⋊C4×C109C22⋊C412yes—
107C22⋊C4×C109C22⋊C412yes—
108C42×C109—22no—
109C42×C109—24no—
110C42×C109—21no—
111C42×C109—21no—
112C42×C109—21no—
113C42×C109C22⋊C411yescharacteristic
114C42×C109C22⋊C411yescharacteristic
115C42×C109C22⋊C412yes—
116C23×C109—24no—
117C23×C109—21no—
118C23×C109—82no—
119C23×C109C2×C22⋊C411yescharacteristic
120C23×C109#12014yes—
121Q8×C109—42no—
122Q8×C109—41no—
123Q8×C109—42no—
124Q8×C109—82no—
125D4×C109—42no—
126D4×C109—42no—
127D4×C109—41no—
128D4×C109—42no—
129D4×C109—84no—
130C2×C4×C109—22no—
131C2×C4×C109—22no—
132C2×C4×C109—21no—
133C2×C4×C109—22no—
134C2×C4×C109—22no—
135C2×C4×C109—21no—
136C2×C4×C109—21no—
137C2×C4×C109—22no—
138C2×C4×C109—42no—
139C2×C4×C109—42no—
140C2×C4×C109—82no—
141C2×C4×C109—84no—
142C2×C4×C109—81no—
143C2×C4×C109—84no—
144C2×C4×C109—164no—
145C2×C4×C109C2×C22⋊C411yescharacteristic
146C2×C4×C109C2×C22⋊C411yescharacteristic
147C22×C109—24no—
148C22×C109—41no—
149C22×C109—44no—
150C22×C109—42no—
151C22×C109—41no—
152C22×C109—84no—
153C22×C109#37911yescharacteristic
154C22×C109#41012yes—
155C4×C109—21no—
156C4×C109—21no—
157C4×C109—42no—
158C4×C109—41no—
159C4×C109—42no—
160C4×C109—41no—
161C4×C109—42no—
162C4×C109—41no—
163C4×C109—42no—
164C4×C109—164no—
165#12728C10911yesmaximal, characteristic, direct product factor, Sylow
166C2×C109—21no—
167C2×C109—22no—
168C2×C109—44no—
169C2×C109—81no—
170C2×C109#178211yescharacteristic
171#3510C2×C10911yescharacteristic
172#2127C2×C10912yes—
173C2×#354C2×C10912yescomplemented
174#1828C2×C10912yescomplemented
175C109#1272811yescharacteristic, direct product factor, Sylow
176#559—22no—
177#556—22no—
178#551—24no—
179#542—22no—
180#536C4×C10911yescharacteristic, complemented
181#535C22×C10911yescharacteristic
182#533C4×C10911yescharacteristic, complemented
183#529—22no—
184C2×#124C22×C10911yescharacteristic
185#519C4×C10911yescharacteristic, complemented
186#518C22×C10911yescharacteristic
187C4×#41C4×C10911yescharacteristic, complemented
188C2×#99C22×C10914yes—
189#354—22no—
190#354—22no—
191#354—22no—
192#353—22no—
193#342—24no—
194C2×#41C2×C4×C10911yescharacteristic
195C22×Q8C2×C4×C10911yescharacteristic
196C22×D4C23×C10911yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199#127D4×C10912yes—
200#126—24no—
201#126—24no—
202#125—21no—
203#124—24no—
204#124—21no—
205#124—22no—
206#124D4×C10912yes—
207#124D4×C10912yes—
208#124D4×C10912yes—
209#123—24no—
210#123—24no—
211#123—22no—
212C22⋊Q8—24no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215#121—22no—
216#120—22no—
217C4×Q8—21no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×D4—24no—
222C4×D4—21no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226#117—24no—
227#117—22no—
228#117—21no—
229#117C2×C4×C10911yescharacteristic
230#117C2×C4×C10911yescharacteristic
231#117C2×C4×C10911yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4D4×C10914yes—
235C2×C42—21no—
236C2×C42C2×C4×C10911yescharacteristic
237#99—24no—
238#99—24no—
239C24C2×D4×C10912yes—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊C4×C10912yes—
245C2×Q8C22⋊C4×C10911yescharacteristic
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4C22⋊C4×C10911yescharacteristic
250C2×D4C2×D4×C10912yes—
251C2×D4C2×D4×C10912yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—44no—
258C22×C4C22×C4×C10911yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4C22⋊C4×C10912yes—
278C22⋊C4C22⋊C4×C10912yes—
279C42—22no—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C22⋊C4×C10911yescharacteristic
285C42C22⋊C4×C10911yescharacteristic
286C42C22⋊C4×C10912yes—
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊C4×C10911yescharacteristic
291C23#120×C10914yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—84no—
315C2×C4—164no—
316C2×C4C2×C22⋊C4×C10911yescharacteristic
317C2×C4C2×C22⋊C4×C10911yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22#379×C10911yescharacteristic
325C22#410×C10912yes—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—164no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2C109×#178211yescharacteristic
subgroupG/H
center, socleC2×C109#1782
commutatorC22×C4C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radical#535×C109C22
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
3. C2×#354×C109C2semidirect
4. C109×#1828C2semidirect
9. #536×C109C4semidirect
11. #533×C109C4semidirect
14. #519×C109C4semidirect
16. C4×#41×C109C4semidirect
165. #12728C109direct (N × H)
173. C2×#354C2×C109semidirect
174. #1828C2×C109semidirect
175. C109#12728direct (N × H)
180. #536C4×C109semidirect
182. #533C4×C109semidirect
185. #519C4×C109semidirect
187. C4×#41C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35102
C109×#21272
C2×#354×C1092
C109×#18282
#12728109