Finite Group Databasebeta

← C109×#12707 #42130701

Subgroups — C109×#12707 #42130701

Order 27904.

1216 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3484C211yesmaximal, characteristic
2C109×#2127C214yesmaximal, complemented
3C109×#1781C212yesmaximal, complemented
4C2×H(5,𝔽2)×C109C411yescharacteristic, complemented
5C23×D4×C109C2211yescharacteristic
6#546×C109—24no—
7D42×C109—24no—
8#535×C109C2212yes—
9#534×C109—22no—
10#525×C109C412yescomplemented
11C2×#120×C109—22no—
12#518×C109C411yescharacteristic, complemented
13C2×#99×C109—22no—
14C2×#99×C109C2214yescomplemented
15#379×C109—22no—
16#354×C109—24no—
17#353×C109—24no—
18C25×C109D412yes—
19H(5,𝔽2)×C109—44no—
20C2×#41×C109—22no—
21C2×#41×C109C2×C412yes—
22C22×D4×C109—22no—
23C22×D4×C109—24no—
24C22×D4×C109—22no—
25C22×D4×C109—24no—
26C22×D4×C109—44no—
27C22×D4×C109C2×C411yescharacteristic
28C22×D4×C109D412yes—
29C22×D4×C109C2311yescharacteristic
30C23×C4×C109C2×C411yescharacteristic
31#127×C109D414yes—
32#124×C109D414yes—
33#123×C109—24no—
34#123×C109—22no—
35C22⋊Q8×C109—22no—
36#121×C109—28no—
37#121×C109—22no—
38#121×C109—24no—
39#120×C109—28no—
40#120×C109—24no—
41#120×C109—22no—
42#120×C109—24no—
43C4×D4×C109—24no—
44C4×D4×C109—24no—
45#117×C109C2×C412yes—
46C2×C22⋊C4×C109—24no—
47C2×C22⋊C4×C109—22no—
48C2×C22⋊C4×C109—24no—
49C2×C22⋊C4×C109—44no—
50#99×C109—28no—
51#99×C109—44no—
52C24×C109—22no—
53C24×C109—24no—
54C24×C109—24no—
55C24×C109—42no—
56C24×C109—84no—
57C24×C109C22⋊C411yescharacteristic
58C24×C109C22⋊C412yes—
59C24×C109C2×D412yes—
60#41×C109—44no—
61#41×C109—48no—
62C2×Q8×C109C22⋊C412yescomplemented
63C2×D4×C109—24no—
64C2×D4×C109—24no—
65C2×D4×C109—24no—
66C2×D4×C109—24no—
67C2×D4×C109—22no—
68C2×D4×C109—24no—
69C2×D4×C109—28no—
70C2×D4×C109—22no—
71C2×D4×C109—42no—
72C2×D4×C109—42no—
73C2×D4×C109—416no—
74C2×D4×C109—48no—
75C2×D4×C109—44no—
76C2×D4×C109—44no—
77C2×D4×C109—88no—
78C2×D4×C109C22⋊C412yescomplemented
79C2×D4×C109C2×D414yes—
80C22×C4×C109—22no—
81C22×C4×C109—24no—
82C22×C4×C109—22no—
83C22×C4×C109—22no—
84C22×C4×C109—42no—
85C22×C4×C109—44no—
86C22×C4×C109C22⋊C411yescharacteristic
87C22×C4×C109C22×C411yescharacteristic
88C4⋊C4×C109—24no—
89C4⋊C4×C109—22no—
90C22⋊C4×C109—24no—
91C22⋊C4×C109—24no—
92C22⋊C4×C109—24no—
93C22⋊C4×C109—22no—
94C22⋊C4×C109—28no—
95C22⋊C4×C109—44no—
96C22⋊C4×C109—44no—
97C22⋊C4×C109—88no—
98C42×C109C22⋊C414yescomplemented
99C23×C109—21no—
100C23×C109—22no—
101C23×C109—22no—
102C23×C109—22no—
103C23×C109—22no—
104C23×C109—22no—
105C23×C109—22no—
106C23×C109—42no—
107C23×C109—48no—
108C23×C109—48no—
109C23×C109—48no—
110C23×C109—44no—
111C23×C109—84no—
112C23×C109—82no—
113C23×C109—82no—
114C23×C109—84no—
115C23×C109—84no—
116C23×C109—84no—
117C23×C109—84no—
118C23×C109#9914yes—
119C23×C109C2×C22⋊C411yescharacteristic
120C23×C109#12014yes—
121Q8×C109—42no—
122D4×C109—28no—
123D4×C109—44no—
124D4×C109—42no—
125D4×C109—48no—
126D4×C109—48no—
127D4×C109—48no—
128D4×C109—44no—
129D4×C109—88no—
130C2×C4×C109—22no—
131C2×C4×C109—24no—
132C2×C4×C109—22no—
133C2×C4×C109—21no—
134C2×C4×C109—22no—
135C2×C4×C109—44no—
136C2×C4×C109—44no—
137C2×C4×C109—44no—
138C2×C4×C109—84no—
139C2×C4×C109—84no—
140C2×C4×C109—88no—
141C2×C4×C109C2×C22⋊C412yes—
142C22×C109—22no—
143C22×C109—24no—
144C22×C109—24no—
145C22×C109—24no—
146C22×C109—41no—
147C22×C109—44no—
148C22×C109—44no—
149C22×C109—44no—
150C22×C109—44no—
151C22×C109—41no—
152C22×C109—42no—
153C22×C109—44no—
154C22×C109—41no—
155C22×C109—42no—
156C22×C109—82no—
157C22×C109—82no—
158C22×C109—816no—
159C22×C109—164no—
160C22×C109#37911yescharacteristic
161C22×C109C2×#9912yes—
162C4×C109—22no—
163C4×C109—44no—
164C4×C109—41no—
165C4×C109—42no—
166C4×C109—164no—
167#12707C10911yesmaximal, characteristic, direct product factor, Sylow
168C2×C109—21no—
169C2×C109—22no—
170C2×C109—44no—
171C2×C109—42no—
172C2×C109—44no—
173C2×C109—82no—
174C2×C109#178111yescharacteristic
175#3484C2×C10911yescharacteristic
176#2127C2×C10914yescomplemented
177#1781C2×C10912yescomplemented
178C109#1270711yescharacteristic, direct product factor, Sylow
179C2×H(5,𝔽2)C4×C10911yescharacteristic, complemented
180C23×D4C22×C10911yescharacteristic
181#546—24no—
182D42—24no—
183#535C22×C10912yes—
184#534—22no—
185#525C4×C10912yescomplemented
186C2×#120—22no—
187#518C4×C10911yescharacteristic, complemented
188C2×#99—22no—
189C2×#99C22×C10914yescomplemented
190#379—22no—
191#354—24no—
192#353—24no—
193C25D4×C10912yes—
194H(5,𝔽2)—44no—
195C2×#41—22no—
196C2×#41C2×C4×C10912yes—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—44no—
202C22×D4C2×C4×C10911yescharacteristic
203C22×D4D4×C10912yes—
204C22×D4C23×C10911yescharacteristic
205C23×C4C2×C4×C10911yescharacteristic
206#127D4×C10914yes—
207#124D4×C10914yes—
208#123—24no—
209#123—22no—
210C22⋊Q8—22no—
211#121—28no—
212#121—22no—
213#121—24no—
214#120—28no—
215#120—24no—
216#120—22no—
217#120—24no—
218C4×D4—24no—
219C4×D4—24no—
220#117C2×C4×C10912yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—44no—
225#99—28no—
226#99—44no—
227C24—22no—
228C24—24no—
229C24—24no—
230C24—42no—
231C24—84no—
232C24C22⋊C4×C10911yescharacteristic
233C24C22⋊C4×C10912yes—
234C24C2×D4×C10912yes—
235#41—44no—
236#41—48no—
237C2×Q8C22⋊C4×C10912yescomplemented
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—28no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—416no—
249C2×D4—48no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—88no—
253C2×D4C22⋊C4×C10912yescomplemented
254C2×D4C2×D4×C10914yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4C22⋊C4×C10911yescharacteristic
262C22×C4C22×C4×C10911yescharacteristic
263C4⋊C4—24no—
264C4⋊C4—22no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—22no—
269C22⋊C4—28no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—88no—
273C42C22⋊C4×C10914yescomplemented
274C23—21no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—42no—
282C23—48no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—84no—
287C23—82no—
288C23—82no—
289C23—84no—
290C23—84no—
291C23—84no—
292C23—84no—
293C23#99×C10914yes—
294C23C2×C22⋊C4×C10911yescharacteristic
295C23#120×C10914yes—
296Q8—42no—
297D4—28no—
298D4—44no—
299D4—42no—
300D4—48no—
301D4—48no—
302D4—48no—
303D4—44no—
304D4—88no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—88no—
316C2×C4C2×C22⋊C4×C10912yes—
317C22—22no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—41no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—41no—
327C22—42no—
328C22—44no—
329C22—41no—
330C22—42no—
331C22—82no—
332C22—82no—
333C22—816no—
334C22—164no—
335C22#379×C10911yescharacteristic
336C22C2×#99×C10912yes—
337C4—22no—
338C4—44no—
339C4—41no—
340C4—42no—
341C4—164no—
342C2—21no—
343C2—22no—
344C2—44no—
345C2—42no—
346C2—44no—
347C2—82no—
348C2C109×#178111yescharacteristic
subgroupG/H
center, socleC2×C109#1781
commutatorC22×C4C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC109×#3484C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C23×C109 ◁ C23×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2127C2semidirect
3. C109×#1781C2semidirect
4. C2×H(5,𝔽2)×C109C4semidirect
10. #525×C109C4semidirect
12. #518×C109C4semidirect
14. C2×#99×C109C22semidirect
62. C2×Q8×C109C22⋊C4semidirect
78. C2×D4×C109C22⋊C4semidirect
98. C42×C109C22⋊C4semidirect
167. #12707C109direct (N × H)
176. #2127C2×C109semidirect
177. #1781C2×C109semidirect
178. C109#12707direct (N × H)
179. C2×H(5,𝔽2)C4×C109semidirect
185. #525C4×C109semidirect
187. #518C4×C109semidirect
189. C2×#99C22×C109semidirect
237. C2×Q8C22⋊C4×C109semidirect
253. C2×D4C22⋊C4×C109semidirect
273. C42C22⋊C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34842
C109×#21272
C109×#17812
#12707109