Finite Group Databasebeta

← C109×#12775 #42130769

Subgroups — C109×#12775 #42130769

Order 27904.

740 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#563×C109C211yesmaximal, characteristic
2C2×#356×C109C212yesmaximal, complemented
3C2×#355×C109C211yesmaximal, characteristic
4C109×#1888C211yesmaximal, characteristic
5C109×#1886C212yesmaximal, complemented
6#563×C109—24no—
7#563×C109—22no—
8#563×C109C414yescomplemented
9C2×C4⋊Q8×C109C2212yes—
10C2×#126×C109—22no—
11C2×#125×C109—21no—
12C2×#124×C109C2211yescharacteristic
13C2×#123×C109—21no—
14C2×C22⋊Q8×C109—22no—
15#519×C109C2211yescharacteristic
16#419×C109—22no—
17#418×C109—21no—
18C2×#101×C109C2212yes—
19C2×#101×C109C2211yescharacteristic
20#356×C109—24no—
21#355×C109—22no—
22#339×C109—22no—
23#339×C109—21no—
24C22×Q8×C109D412yes—
25C22×Q8×C109C2311yescharacteristic
26C22×D4×C109D411yescharacteristic
27C23×C4×C109—21no—
28C2×M16×C109—21no—
29C2×M16×C109—22no—
30C4⋊Q8×C109—22no—
31C4⋊Q8×C109—22no—
32C4⋊Q8×C109C2×C414yes—
33C4⋊Q8×C109D414yes—
34#126×C109—28no—
35#126×C109—28no—
36#125×C109—24no—
37#125×C109—24no—
38#124×C109—22no—
39#124×C109C2×C412yes—
40#124×C109D412yes—
41#123×C109—24no—
42#123×C109—24no—
43C22⋊Q8×C109—28no—
44C22⋊Q8×C109—28no—
45C4×Q8×C109—42no—
46#117×C109—21no—
47#117×C109—22no—
48C2×C4⋊C4×C109—21no—
49C2×C4⋊C4×C109—22no—
50C2×C4⋊C4×C109—21no—
51C2×C4⋊C4×C109—22no—
52C2×C4⋊C4×C109—22no—
53C2×C22⋊C4×C109—22no—
54C2×C22⋊C4×C109—21no—
55C2×C22⋊C4×C109—22no—
56C2×C42×C109D411yescharacteristic
57C2×C42×C109D412yes—
58#103×C109—44no—
59#102×C109—42no—
60#101×C109—22no—
61#101×C109—22no—
62#101×C109—22no—
63C24×C109—21no—
64C2×Q8×C109—22no—
65C2×Q8×C109—41no—
66C2×Q8×C109—41no—
67C2×Q8×C109—42no—
68C2×Q8×C109—42no—
69C2×Q8×C109C22⋊C414yes—
70C2×Q8×C109C22×C411yescharacteristic
71C2×Q8×C109C2×D412yes—
72C2×Q8×C109C2×D411yescharacteristic
73C2×D4×C109—21no—
74C2×D4×C109—41no—
75C2×D4×C109—41no—
76C2×D4×C109C22⋊C412yes—
77C22×C4×C109—21no—
78C22×C4×C109—22no—
79C22×C4×C109—22no—
80C22×C4×C109—24no—
81C22×C4×C109—81no—
82C22×C4×C109—81no—
83C22×C4×C109C2×D412yes—
84C22×C4×C109C2×D411yescharacteristic
85M16×C109—41no—
86M16×C109—42no—
87M16×C109—42no—
88M16×C109—41no—
89C2×C8×C109—41no—
90C2×C8×C109—42no—
91C4⋊C4×C109—24no—
92C4⋊C4×C109—28no—
93C4⋊C4×C109—24no—
94C4⋊C4×C109—24no—
95C4⋊C4×C109—24no—
96C4⋊C4×C109—24no—
97C4⋊C4×C109—24no—
98C4⋊C4×C109—22no—
99C4⋊C4×C109—24no—
100C22⋊C4×C109—28no—
101C22⋊C4×C109—24no—
102C22⋊C4×C109—24no—
103C22⋊C4×C109—24no—
104C22⋊C4×C109—41no—
105C42×C109—21no—
106C42×C109—22no—
107C42×C109—41no—
108C42×C109—42no—
109C42×C109C22⋊C412yes—
110C42×C109C22⋊C414yes—
111C23×C109—22no—
112C23×C109—44no—
113C23×C109—41no—
114C23×C109—41no—
115C23×C109#12011yescharacteristic
116Q8×C109—41no—
117Q8×C109—42no—
118Q8×C109—41no—
119Q8×C109—44no—
120Q8×C109—82no—
121D4×C109—42no—
122D4×C109—81no—
123C2×C4×C109—21no—
124C2×C4×C109—22no—
125C2×C4×C109—21no—
126C2×C4×C109—22no—
127C2×C4×C109—22no—
128C2×C4×C109—24no—
129C2×C4×C109—24no—
130C2×C4×C109—42no—
131C2×C4×C109—44no—
132C2×C4×C109—42no—
133C2×C4×C109—42no—
134C2×C4×C109—42no—
135C2×C4×C109—84no—
136C2×C4×C109C2×C22⋊C412yes—
137C2×C4×C109C2×C22⋊C411yescharacteristic
138C2×C4×C109#12012yes—
139C2×C4×C109#12011yescharacteristic
140C8×C109—81no—
141C8×C109—82no—
142C22×C109—22no—
143C22×C109—44no—
144C22×C109—42no—
145C22×C109—82no—
146C22×C109—82no—
147C22×C109#37911yescharacteristic
148C22×C109B(4,𝔽2)11yescharacteristic
149C22×C109#45811yescharacteristic
150C4×C109—21no—
151C4×C109—22no—
152C4×C109—21no—
153C4×C109—22no—
154C4×C109—42no—
155C4×C109—44no—
156C4×C109—44no—
157C4×C109—82no—
158#12775C10911yesmaximal, characteristic, direct product factor, Sylow
159C2×C109—21no—
160C2×C109—21no—
161C2×C109—82no—
162C2×C109#179011yescharacteristic
163C2×C109#220312yes—
164C2×#563C2×C10911yescharacteristic
165C2×#356C2×C10912yescomplemented
166C2×#355C2×C10911yescharacteristic
167#1888C2×C10911yescharacteristic
168#1886C2×C10912yescomplemented
169C109#1277511yescharacteristic, direct product factor, Sylow
170#563—24no—
171#563—22no—
172#563C4×C10914yescomplemented
173C2×C4⋊Q8C22×C10912yes—
174C2×#126—22no—
175C2×#125—21no—
176C2×#124C22×C10911yescharacteristic
177C2×#123—21no—
178C2×C22⋊Q8—22no—
179#519C22×C10911yescharacteristic
180#419—22no—
181#418—21no—
182C2×#101C22×C10912yes—
183C2×#101C22×C10911yescharacteristic
184#356—24no—
185#355—22no—
186#339—22no—
187#339—21no—
188C22×Q8D4×C10912yes—
189C22×Q8C23×C10911yescharacteristic
190C22×D4D4×C10911yescharacteristic
191C23×C4—21no—
192C2×M16—21no—
193C2×M16—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8C2×C4×C10914yes—
197C4⋊Q8D4×C10914yes—
198#126—28no—
199#126—28no—
200#125—24no—
201#125—24no—
202#124—22no—
203#124C2×C4×C10912yes—
204#124D4×C10912yes—
205#123—24no—
206#123—24no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209C4×Q8—42no—
210#117—21no—
211#117—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C42D4×C10911yescharacteristic
221C2×C42D4×C10912yes—
222#103—44no—
223#102—42no—
224#101—22no—
225#101—22no—
226#101—22no—
227C24—21no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8C22⋊C4×C10914yes—
234C2×Q8C22×C4×C10911yescharacteristic
235C2×Q8C2×D4×C10912yes—
236C2×Q8C2×D4×C10911yescharacteristic
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊C4×C10912yes—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×D4×C10912yes—
248C22×C4C2×D4×C10911yescharacteristic
249M16—41no—
250M16—42no—
251M16—42no—
252M16—41no—
253C2×C8—41no—
254C2×C8—42no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C22⋊C4—28no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—41no—
269C42—21no—
270C42—22no—
271C42—41no—
272C42—42no—
273C42C22⋊C4×C10912yes—
274C42C22⋊C4×C10914yes—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23#120×C10911yescharacteristic
280Q8—41no—
281Q8—42no—
282Q8—41no—
283Q8—44no—
284Q8—82no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4C2×C22⋊C4×C10912yes—
301C2×C4C2×C22⋊C4×C10911yescharacteristic
302C2×C4#120×C10912yes—
303C2×C4#120×C10911yescharacteristic
304C8—81no—
305C8—82no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22#379×C10911yescharacteristic
312C22B(4,𝔽2)×C10911yescharacteristic
313C22#458×C10911yescharacteristic
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—82no—
322C2—21no—
323C2—21no—
324C2—82no—
325C2C109×#179011yescharacteristic
326C2C109×#220312yes—
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutatorC2×Q8C22×C4×C109
Frattini, cosocleC22×Q8C23×C109
prime radicalC22×D4×C109D4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C22×Q8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#356×C109C2semidirect
5. C109×#1886C2semidirect
8. #563×C109C4semidirect
158. #12775C109direct (N × H)
165. C2×#356C2×C109semidirect
168. #1886C2×C109semidirect
169. C109#12775direct (N × H)
172. #563C4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#563×C1092
C2×#356×C1092
C2×#355×C1092
C109×#18882
C109×#18862
#12775109