Finite Group Databasebeta

← C109×#11801 #42129795

Subgroups — C109×#11801 #42129795

Order 27904.

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

normal subgroups only

This group has 402 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×#2621C211yesmaximal, characteristic
2C109×#1820C211yesmaximal, characteristic
3C109×#1464C211yesmaximal, characteristic
4C109×#1344C214yesmaximal
5C2×C4⋊Q8×C109C411yescharacteristic, complemented
6C2×C22⋊Q8×C109—21no—
7C2×#117×C109C2211yescharacteristic
8#432×C109—21no—
9#398×C109—21no—
10#398×C109C412yescomplemented
11#397×C109—21no—
12#397×C109C2212yes—
13#391×C109—21no—
14#382×C109C411yescharacteristic, complemented
15#381×C109—21no—
16#342×C109—21no—
17#342×C109C2212yes—
18#324×C109—21no—
19#323×C109C2212yes—
20C22×Q8×C109—21no—
21C23×C4×C109D411yescharacteristic
22C4⋊Q8×C109—41no—
23C4⋊Q8×C109—41no—
24C22⋊Q8×C109—44no—
25#117×C109—41no—
26#117×C109—41no—
27C2×C4⋊C4×C109—22no—
28C2×C4⋊C4×C109—21no—
29C2×C4⋊C4×C109—21no—
30C2×C4⋊C4×C109C2×C412yes—
31C2×C4⋊C4×C109C2×C411yescharacteristic
32C2×C4⋊C4×C109D411yescharacteristic
33C2×C22⋊C4×C109—22no—
34C2×C22⋊C4×C109—22no—
35C2×C22⋊C4×C109D411yescharacteristic
36C2×C22⋊C4×C109C2311yescharacteristic
37C2×C42×C109C2×C411yescharacteristic
38C2×C42×C109D411yescharacteristic
39C4⋊C8×C109—41no—
40C22⋊C8×C109—22no—
41#97×C109—41no—
42#95×C109—21no—
43#95×C109—22no—
44#95×C109—22no—
45#95×C109—41no—
46#95×C109—41no—
47#95×C109C2×C412yes—
48C24×C109C2×D411yescharacteristic
49C2×Q8×C109—42no—
50C2×Q8×C109—42no—
51C2×Q8×C109—41no—
52C2×Q8×C109—41no—
53C22×C4×C109—22no—
54C22×C4×C109—21no—
55C22×C4×C109—21no—
56C22×C4×C109—22no—
57C22×C4×C109—22no—
58C22×C4×C109—41no—
59C22×C4×C109—42no—
60C22×C4×C109C22⋊C411yescharacteristic
61C22×C4×C109C22⋊C411yescharacteristic
62C22×C4×C109C22⋊C411yescharacteristic
63C22×C4×C109C22⋊C411yescharacteristic
64C22×C4×C109C22×C411yescharacteristic
65C22×C4×C109C2×D411yescharacteristic
66C2×C8×C109—42no—
67C4⋊C4×C109—41no—
68C4⋊C4×C109—41no—
69C4⋊C4×C109—42no—
70C4⋊C4×C109—42no—
71C4⋊C4×C109—44no—
72C4⋊C4×C109—42no—
73C22⋊C4×C109—41no—
74C22⋊C4×C109—41no—
75C22⋊C4×C109—44no—
76C22⋊C4×C109—42no—
77C22⋊C4×C109—42no—
78C42×C109—21no—
79C42×C109—21no—
80C42×C109—21no—
81C42×C109—21no—
82C23×C109—21no—
83C23×C109—21no—
84C23×C109—41no—
85C23×C109—41no—
86C23×C109C2×C22⋊C411yescharacteristic
87C23×C109AGL(1,ℤ/8)12yes—
88Q8×C109—82no—
89Q8×C109—82no—
90C2×C4×C109—21no—
91C2×C4×C109—21no—
92C2×C4×C109—21no—
93C2×C4×C109—21no—
94C2×C4×C109—21no—
95C2×C4×C109—21no—
96C2×C4×C109—21no—
97C2×C4×C109—21no—
98C2×C4×C109—21no—
99C2×C4×C109—42no—
100C2×C4×C109—42no—
101C2×C4×C109—41no—
102C2×C4×C109—41no—
103C2×C4×C109—41no—
104C2×C4×C109—41no—
105C2×C4×C109—42no—
106C2×C4×C109—42no—
107C2×C4×C109—42no—
108C2×C4×C109—42no—
109C2×C4×C109—42no—
110C2×C4×C109—42no—
111C2×C4×C109—82no—
112C2×C4×C109—82no—
113C2×C4×C109—82no—
114C2×C4×C109#9911yescharacteristic
115C2×C4×C109#9911yescharacteristic
116C8×C109—82no—
117C22×C109—21no—
118C22×C109—21no—
119C22×C109—22no—
120C22×C109—42no—
121C22×C109—41no—
122C22×C109—41no—
123C22×C109—81no—
124C22×C109C2×#9911yescharacteristic
125C22×C109#41811yescharacteristic
126C22×C109#42111yescharacteristic
127C4×C109—21no—
128C4×C109—21no—
129C4×C109—41no—
130C4×C109—41no—
131C4×C109—41no—
132C4×C109—41no—
133C4×C109—41no—
134C4×C109—82no—
135C4×C109—82no—
136C4×C109—162no—
137#11801C10911yesmaximal, characteristic, direct product factor, Sylow
138C2×C109—21no—
139C2×C109—21no—
140C2×C109—42no—
141C2×C109#151011yescharacteristic
142C2×C109#212311yescharacteristic
143C2×C109#212911yescharacteristic
144#2621C2×C10911yescharacteristic
145#1820C2×C10911yescharacteristic
146#1464C2×C10911yescharacteristic
147#1344C2×C10914yes—
148C109#1180111yescharacteristic, direct product factor, Sylow
149C2×C4⋊Q8C4×C10911yescharacteristic, complemented
150C2×C22⋊Q8—21no—
151C2×#117C22×C10911yescharacteristic
152#432—21no—
153#398—21no—
154#398C4×C10912yescomplemented
155#397—21no—
156#397C22×C10912yes—
157#391—21no—
158#382C4×C10911yescharacteristic, complemented
159#381—21no—
160#342—21no—
161#342C22×C10912yes—
162#324—21no—
163#323C22×C10912yes—
164C22×Q8—21no—
165C23×C4D4×C10911yescharacteristic
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C22⋊Q8—44no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×C4×C10912yes—
175C2×C4⋊C4C2×C4×C10911yescharacteristic
176C2×C4⋊C4D4×C10911yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4D4×C10911yescharacteristic
180C2×C22⋊C4C23×C10911yescharacteristic
181C2×C42C2×C4×C10911yescharacteristic
182C2×C42D4×C10911yescharacteristic
183C4⋊C8—41no—
184C22⋊C8—22no—
185#97—41no—
186#95—21no—
187#95—22no—
188#95—22no—
189#95—41no—
190#95—41no—
191#95C2×C4×C10912yes—
192C24C2×D4×C10911yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4C22⋊C4×C10911yescharacteristic
205C22×C4C22⋊C4×C10911yescharacteristic
206C22×C4C22⋊C4×C10911yescharacteristic
207C22×C4C22⋊C4×C10911yescharacteristic
208C22×C4C22×C4×C10911yescharacteristic
209C22×C4C2×D4×C10911yescharacteristic
210C2×C8—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×C22⋊C4×C10911yescharacteristic
231C23AGL(1,ℤ/8)×C10912yes—
232Q8—82no—
233Q8—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#99×C10911yescharacteristic
259C2×C4#99×C10911yescharacteristic
260C8—82no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#99×C10911yescharacteristic
269C22#418×C10911yescharacteristic
270C22#421×C10911yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4—162no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2C109×#151011yescharacteristic
285C2C109×#212311yescharacteristic
286C2C109×#212911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#99
commutatorC22×C4C22×C4×C109
Frattini, cosocleC2×C22⋊C4C23×C109
prime radicalC24×C109C2×D4
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
5. C2×C4⋊Q8×C109C4semidirect
10. #398×C109C4semidirect
14. #382×C109C4semidirect
137. #11801C109direct (N × H)
148. C109#11801direct (N × H)
149. C2×C4⋊Q8C4×C109semidirect
154. #398C4×C109semidirect
158. #382C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#26212
C109×#18202
C109×#14642
C109×#13442
#11801109