Finite Group Databasebeta

← C109×#12255 #42130249

Subgroups — C109×#12255 #42130249

Order 27904.

378 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2807C211yesmaximal, characteristic
2C109×#2062C211yesmaximal, characteristic
3C109×#1614C211yesmaximal, characteristic
4C109×#1614C211yesmaximal, characteristic
5C109×#1352C211yesmaximal, characteristic
6C109×#1352C211yesmaximal, characteristic
7C109×#1326C211yesmaximal, characteristic
8#543×C109C2211yescharacteristic
9#484×C109—21no—
10#484×C109—21no—
11#400×C109—21no—
12#399×C109C2211yescharacteristic
13#399×C109C2211yescharacteristic
14#398×C109—21no—
15#389×C109—21no—
16#389×C109—21no—
17#388×C109—21no—
18#385×C109—21no—
19#385×C109—21no—
20#385×C109C2211yescharacteristic
21#342×C109—21no—
22#342×C109C2211yescharacteristic
23#328×C109—21no—
24#328×C109—21no—
25#323×C109C2211yescharacteristic
26#323×C109C2211yescharacteristic
27C23×C4×C109D411yescharacteristic
28C22⋊Q8×C109—21no—
29C22⋊Q8×C109—21no—
30C22⋊Q8×C109—41no—
31C22⋊Q8×C109—41no—
32C2×C4⋊C4×C109—21no—
33C2×C4⋊C4×C109—21no—
34C2×C4⋊C4×C109—21no—
35C2×C4⋊C4×C109—21no—
36C2×C4⋊C4×C109—21no—
37C2×C4⋊C4×C109—21no—
38C2×C4⋊C4×C109—41no—
39C2×C4⋊C4×C109D411yescharacteristic
40C2×C4⋊C4×C109D411yescharacteristic
41C2×C22⋊C4×C109—21no—
42C2×C22⋊C4×C109—21no—
43C2×C22⋊C4×C109—21no—
44C2×C22⋊C4×C109—21no—
45C2×C22⋊C4×C109D411yescharacteristic
46C2×C22⋊C4×C109C2311yescharacteristic
47C2×C42×C109—21no—
48#107×C109—41no—
49#107×C109—41no—
50#103×C109—41no—
51#103×C109—41no—
52C22⋊C8×C109—21no—
53C22⋊C8×C109—21no—
54#95×C109—21no—
55#95×C109—21no—
56#95×C109—21no—
57#95×C109—21no—
58#95×C109—21no—
59#95×C109—41no—
60#95×C109D411yescharacteristic
61#95×C109D411yescharacteristic
62C24×C109C2×D411yescharacteristic
63C2×Q8×C109—21no—
64C2×Q8×C109—21no—
65C22×C4×C109—21no—
66C22×C4×C109—21no—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109—21no—
70C22×C4×C109—21no—
71C22×C4×C109—21no—
72C22×C4×C109—21no—
73C22×C4×C109—41no—
74C22×C4×C109—41no—
75C22×C4×C109—81no—
76C22×C4×C109C2×D411yescharacteristic
77C22×C4×C109C2×D411yescharacteristic
78C2×C8×C109—41no—
79C2×C8×C109—41no—
80C4⋊C4×C109—21no—
81C4⋊C4×C109—21no—
82C4⋊C4×C109—41no—
83C4⋊C4×C109—41no—
84C4⋊C4×C109—42no—
85C4⋊C4×C109—42no—
86C4⋊C4×C109—41no—
87C4⋊C4×C109—41no—
88C4⋊C4×C109—81no—
89C4⋊C4×C109—81no—
90C4⋊C4×C109—81no—
91C4⋊C4×C109—81no—
92C4⋊C4×C109—82no—
93C22⋊C4×C109—21no—
94C22⋊C4×C109—22no—
95C22⋊C4×C109—41no—
96C22⋊C4×C109—42no—
97C22⋊C4×C109—41no—
98C22⋊C4×C109—41no—
99C22⋊C4×C109—41no—
100C22⋊C4×C109—81no—
101C22⋊C4×C109Q1612yes—
102C42×C109—81no—
103C23×C109—21no—
104C23×C109—21no—
105C23×C109—41no—
106C23×C109—41no—
107C23×C109#12011yescharacteristic
108C23×C109C2×Q1611yescharacteristic
109C23×C109#13711yescharacteristic
110Q8×C109—81no—
111Q8×C109—81no—
112C2×C4×C109—22no—
113C2×C4×C109—22no—
114C2×C4×C109—21no—
115C2×C4×C109—21no—
116C2×C4×C109—41no—
117C2×C4×C109—41no—
118C2×C4×C109—41no—
119C2×C4×C109—41no—
120C2×C4×C109—41no—
121C2×C4×C109—41no—
122C2×C4×C109—41no—
123C2×C4×C109—42no—
124C2×C4×C109—41no—
125C2×C4×C109—41no—
126C2×C4×C109—41no—
127C2×C4×C109—41no—
128C2×C4×C109—41no—
129C2×C4×C109—41no—
130C2×C4×C109—41no—
131C2×C4×C109—42no—
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—81no—
139C2×C4×C109—81no—
140C2×C4×C109—81no—
141C2×C4×C109—81no—
142C2×C4×C109—81no—
143C8×C109—81no—
144C8×C109—81no—
145C22×C109—21no—
146C22×C109—21no—
147C22×C109—21no—
148C22×C109—21no—
149C22×C109—41no—
150C22×C109—41no—
151C22×C109—41no—
152C22×C109—41no—
153C22×C109—81no—
154C22×C109#45111yescharacteristic
155C22×C109#45611yescharacteristic
156C22×C109B(4,𝔽2)11yescharacteristic
157C4×C109—42no—
158C4×C109—42no—
159C4×C109—41no—
160C4×C109—41no—
161C4×C109—81no—
162C4×C109—82no—
163C4×C109—81no—
164C4×C109—81no—
165C4×C109—81no—
166C4×C109—81no—
167C4×C109—81no—
168#12255C10911yesmaximal, characteristic, direct product factor, Sylow
169C2×C109—21no—
170C2×C109—21no—
171C2×C109—41no—
172C2×C109—41no—
173C2×C109#160511yescharacteristic
174C2×C109#219811yescharacteristic
175C2×C109#220011yescharacteristic
176#2807C2×C10911yescharacteristic
177#2062C2×C10911yescharacteristic
178#1614C2×C10911yescharacteristic
179#1614C2×C10911yescharacteristic
180#1352C2×C10911yescharacteristic
181#1352C2×C10911yescharacteristic
182#1326C2×C10911yescharacteristic
183C109#1225511yescharacteristic, direct product factor, Sylow
184#543C22×C10911yescharacteristic
185#484—21no—
186#484—21no—
187#400—21no—
188#399C22×C10911yescharacteristic
189#399C22×C10911yescharacteristic
190#398—21no—
191#389—21no—
192#389—21no—
193#388—21no—
194#385—21no—
195#385—21no—
196#385C22×C10911yescharacteristic
197#342—21no—
198#342C22×C10911yescharacteristic
199#328—21no—
200#328—21no—
201#323C22×C10911yescharacteristic
202#323C22×C10911yescharacteristic
203C23×C4D4×C10911yescharacteristic
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4D4×C10911yescharacteristic
216C2×C4⋊C4D4×C10911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4D4×C10911yescharacteristic
222C2×C22⋊C4C23×C10911yescharacteristic
223C2×C42—21no—
224#107—41no—
225#107—41no—
226#103—41no—
227#103—41no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—41no—
236#95D4×C10911yescharacteristic
237#95D4×C10911yescharacteristic
238C24C2×D4×C10911yescharacteristic
239C2×Q8—21no—
240C2×Q8—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—81no—
252C22×C4C2×D4×C10911yescharacteristic
253C22×C4C2×D4×C10911yescharacteristic
254C2×C8—41no—
255C2×C8—41no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—82no—
269C22⋊C4—21no—
270C22⋊C4—22no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C22⋊C4Q16×C10912yes—
278C42—81no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23#120×C10911yescharacteristic
284C23C2×Q16×C10911yescharacteristic
285C23#137×C10911yescharacteristic
286Q8—81no—
287Q8—81no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C8—81no—
320C8—81no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22#451×C10911yescharacteristic
331C22#456×C10911yescharacteristic
332C22B(4,𝔽2)×C10911yescharacteristic
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—82no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C109×#160511yescharacteristic
349C2C109×#219811yescharacteristic
350C2C109×#220011yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C109
prime radicalC24×C109C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C22⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
168. #12255C109direct (N × H)
183. C109#12255direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#28072
C109×#20622
C109×#16142
C109×#13522
C109×#13262
#12255109