Finite Group Databasebeta

← C109×#11798 #42129792

Subgroups — C109×#11798 #42129792

Order 27904.

486 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2392C211yesmaximal, characteristic
2C109×#1820C211yesmaximal, characteristic, complemented
3C109×#1464C211yesmaximal, characteristic, complemented
4C109×#1344C212yesmaximal, complemented
5C109×#1343C212yesmaximal
6C2×C4×D4×C109—21no—
7C2×C4×D4×C109C411yescharacteristic, complemented
8C2×#117×C109C2211yescharacteristic
9#432×C109—21no—
10#398×C109C411yescharacteristic, complemented
11#397×C109C2211yescharacteristic
12#396×C109—21no—
13#394×C109—21no—
14#394×C109C411yescharacteristic, complemented
15#392×C109C2211yescharacteristic
16#381×C109—21no—
17C4×C4⋊C4×C109C411yescharacteristic, complemented
18C4×C22⋊C4×C109—21no—
19#342×C109—21no—
20#342×C109C2212yes—
21#324×C109—21no—
22#323×C109C2212yes—
23C22×D4×C109—21no—
24C22×D4×C109C2×C411yescharacteristic
25C23×C4×C109—21no—
26C23×C4×C109D411yescharacteristic
27C4×D4×C109—41no—
28C4×D4×C109—41no—
29C4×D4×C109—44no—
30#117×C109—41no—
31#117×C109—41no—
32C2×C4⋊C4×C109—21no—
33C2×C4⋊C4×C109—21no—
34C2×C4⋊C4×C109C2×C411yescharacteristic
35C2×C4⋊C4×C109C2×C411yescharacteristic
36C2×C4⋊C4×C109D411yescharacteristic
37C2×C22⋊C4×C109—21no—
38C2×C22⋊C4×C109—21no—
39C2×C22⋊C4×C109—21no—
40C2×C22⋊C4×C109—21no—
41C2×C22⋊C4×C109—21no—
42C2×C22⋊C4×C109—22no—
43C2×C22⋊C4×C109D411yescharacteristic
44C2×C22⋊C4×C109C2311yescharacteristic
45C2×C42×C109—21no—
46C2×C42×C109C2×C411yescharacteristic
47C2×C42×C109D411yescharacteristic
48C4⋊C8×C109—41no—
49C22⋊C8×C109—22no—
50#97×C109—41no—
51#95×C109—21no—
52#95×C109—21no—
53#95×C109—41no—
54#95×C109—41no—
55#95×C109C2×C411yescharacteristic
56#95×C109C2×C411yescharacteristic
57C24×C109—21no—
58C24×C109C2×D411yescharacteristic
59C2×D4×C109—41no—
60C2×D4×C109—42no—
61C2×D4×C109—42no—
62C2×D4×C109—42no—
63C2×D4×C109—42no—
64C22×C4×C109—21no—
65C22×C4×C109—22no—
66C22×C4×C109—21no—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109—21no—
70C22×C4×C109—41no—
71C22×C4×C109—42no—
72C22×C4×C109—42no—
73C22×C4×C109—41no—
74C22×C4×C109—41no—
75C22×C4×C109—42no—
76C22×C4×C109C22⋊C411yescharacteristic
77C22×C4×C109C22⋊C411yescharacteristic
78C22×C4×C109C22⋊C411yescharacteristic
79C22×C4×C109C22⋊C411yescharacteristic
80C22×C4×C109C22×C411yescharacteristic
81C22×C4×C109C2×D411yescharacteristic
82C2×C8×C109—42no—
83C4⋊C4×C109—41no—
84C4⋊C4×C109—41no—
85C4⋊C4×C109—41no—
86C4⋊C4×C109—42no—
87C4⋊C4×C109—42no—
88C22⋊C4×C109—41no—
89C22⋊C4×C109—41no—
90C22⋊C4×C109—42no—
91C22⋊C4×C109—42no—
92C22⋊C4×C109—42no—
93C22⋊C4×C109—42no—
94C22⋊C4×C109—42no—
95C22⋊C4×C109—42no—
96C22⋊C4×C109—42no—
97C42×C109—21no—
98C42×C109—21no—
99C42×C109—21no—
100C42×C109—21no—
101C42×C109—42no—
102C23×C109—21no—
103C23×C109—21no—
104C23×C109—22no—
105C23×C109—41no—
106C23×C109—41no—
107C23×C109—41no—
108C23×C109—41no—
109C23×C109—41no—
110C23×C109—41no—
111C23×C109—41no—
112C23×C109—41no—
113C23×C109C2×C22⋊C411yescharacteristic
114C23×C109AGL(1,ℤ/8)12yes—
115D4×C109—42no—
116D4×C109—48no—
117D4×C109—42no—
118C2×C4×C109—21no—
119C2×C4×C109—21no—
120C2×C4×C109—22no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109—21no—
124C2×C4×C109—21no—
125C2×C4×C109—21no—
126C2×C4×C109—21no—
127C2×C4×C109—21no—
128C2×C4×C109—42no—
129C2×C4×C109—42no—
130C2×C4×C109—41no—
131C2×C4×C109—41no—
132C2×C4×C109—41no—
133C2×C4×C109—41no—
134C2×C4×C109—41no—
135C2×C4×C109—41no—
136C2×C4×C109—41no—
137C2×C4×C109—41no—
138C2×C4×C109—41no—
139C2×C4×C109—41no—
140C2×C4×C109—41no—
141C2×C4×C109—41no—
142C2×C4×C109—82no—
143C2×C4×C109—82no—
144C2×C4×C109—82no—
145C2×C4×C109—82no—
146C2×C4×C109—82no—
147C2×C4×C109#9911yescharacteristic
148C2×C4×C109#9911yescharacteristic
149C8×C109—82no—
150C22×C109—21no—
151C22×C109—21no—
152C22×C109—22no—
153C22×C109—42no—
154C22×C109—42no—
155C22×C109—42no—
156C22×C109—41no—
157C22×C109—41no—
158C22×C109—42no—
159C22×C109—81no—
160C22×C109—81no—
161C22×C109—81no—
162C22×C109—81no—
163C22×C109—81no—
164C22×C109C2×#9911yescharacteristic
165C22×C109#41811yescharacteristic
166C22×C109#42111yescharacteristic
167C4×C109—21no—
168C4×C109—21no—
169C4×C109—42no—
170C4×C109—41no—
171C4×C109—41no—
172C4×C109—41no—
173C4×C109—41no—
174C4×C109—41no—
175C4×C109—81no—
176C4×C109—81no—
177C4×C109—162no—
178#11798C10911yesmaximal, characteristic, direct product factor, Sylow
179C2×C109—21no—
180C2×C109—21no—
181C2×C109—42no—
182C2×C109—82no—
183C2×C109#151011yescharacteristic
184C2×C109#212011yescharacteristic
185C2×C109#212611yescharacteristic
186#2392C2×C10911yescharacteristic
187#1820C2×C10911yescharacteristic, complemented
188#1464C2×C10911yescharacteristic, complemented
189#1344C2×C10912yescomplemented
190#1343C2×C10912yes—
191C109#1179811yescharacteristic, direct product factor, Sylow
192C2×C4×D4—21no—
193C2×C4×D4C4×C10911yescharacteristic, complemented
194C2×#117C22×C10911yescharacteristic
195#432—21no—
196#398C4×C10911yescharacteristic, complemented
197#397C22×C10911yescharacteristic
198#396—21no—
199#394—21no—
200#394C4×C10911yescharacteristic, complemented
201#392C22×C10911yescharacteristic
202#381—21no—
203C4×C4⋊C4C4×C10911yescharacteristic, complemented
204C4×C22⋊C4—21no—
205#342—21no—
206#342C22×C10912yes—
207#324—21no—
208#323C22×C10912yes—
209C22×D4—21no—
210C22×D4C2×C4×C10911yescharacteristic
211C23×C4—21no—
212C23×C4D4×C10911yescharacteristic
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—44no—
216#117—41no—
217#117—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×C4×C10911yescharacteristic
221C2×C4⋊C4C2×C4×C10911yescharacteristic
222C2×C4⋊C4D4×C10911yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4D4×C10911yescharacteristic
230C2×C22⋊C4C23×C10911yescharacteristic
231C2×C42—21no—
232C2×C42C2×C4×C10911yescharacteristic
233C2×C42D4×C10911yescharacteristic
234C4⋊C8—41no—
235C22⋊C8—22no—
236#97—41no—
237#95—21no—
238#95—21no—
239#95—41no—
240#95—41no—
241#95C2×C4×C10911yescharacteristic
242#95C2×C4×C10911yescharacteristic
243C24—21no—
244C24C2×D4×C10911yescharacteristic
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4C22⋊C4×C10911yescharacteristic
263C22×C4C22⋊C4×C10911yescharacteristic
264C22×C4C22⋊C4×C10911yescharacteristic
265C22×C4C22⋊C4×C10911yescharacteristic
266C22×C4C22×C4×C10911yescharacteristic
267C22×C4C2×D4×C10911yescharacteristic
268C2×C8—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C10911yescharacteristic
300C23AGL(1,ℤ/8)×C10912yes—
301D4—42no—
302D4—48no—
303D4—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#99×C10911yescharacteristic
334C2×C4#99×C10911yescharacteristic
335C8—82no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×#99×C10911yescharacteristic
351C22#418×C10911yescharacteristic
352C22#421×C10911yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—162no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—82no—
368C2C109×#151011yescharacteristic
369C2C109×#212011yescharacteristic
370C2C109×#212611yescharacteristic
subgroupG/H
center, socleC22×C109C2×#99
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×C22⋊C4C23×C109
prime radical#392×C109C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×#117×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#1820C2semidirect
3. C109×#1464C2semidirect
4. C109×#1344C2semidirect
7. C2×C4×D4×C109C4semidirect
10. #398×C109C4semidirect
14. #394×C109C4semidirect
17. C4×C4⋊C4×C109C4semidirect
178. #11798C109direct (N × H)
187. #1820C2×C109semidirect
188. #1464C2×C109semidirect
189. #1344C2×C109semidirect
191. C109#11798direct (N × H)
193. C2×C4×D4C4×C109semidirect
196. #398C4×C109semidirect
200. #394C4×C109semidirect
203. C4×C4⋊C4C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#23922
C109×#18202
C109×#14642
C109×#13442
C109×#13432
#11798109