Finite Group Databasebeta

← C109×#12692 #42130686

Subgroups — C109×#12692 #42130686

Order 27904.

976 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 150 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×#536×C109C211yesmaximal, characteristic
2C2×#352×C109C214yesmaximal, complemented
3C109×#1785C211yesmaximal, characteristic
4C109×#1782C211yesmaximal, characteristic
5C22×#41×C109C2211yescharacteristic
6C23×Q8×C109C411yescharacteristic, complemented
7#536×C109—28no—
8C2×C4⋊Q8×C109—21no—
9C2×#124×C109—21no—
10C2×#123×C109C412yescomplemented
11C2×#123×C109C2212yes—
12C2×C22⋊Q8×C109—21no—
13C2×C22⋊Q8×C109—21no—
14C2×#117×C109C411yescharacteristic, complemented
15C2×#101×C109—21no—
16C2×#100×C109C2212yes—
17C2×#99×C109—21no—
18C2×#99×C109C2212yes—
19#407×C109—21no—
20#381×C109—21no—
21#352×C109—28no—
22C2×#41×C109—24no—
23C2×#41×C109—42no—
24C2×#41×C109—42no—
25C22×Q8×C109—21no—
26C22×Q8×C109—24no—
27C22×Q8×C109—21no—
28C22×Q8×C109—42no—
29C22×Q8×C109—42no—
30C22×Q8×C109D411yescharacteristic
31C22×D4×C109C2×C411yescharacteristic
32C22×D4×C109D411yescharacteristic
33C22×D4×C109C2311yescharacteristic
34C23×C4×C109C2×C411yescharacteristic
35C23×C4×C109C2×C412yes—
36C23×C4×C109D411yescharacteristic
37C23×C4×C109D411yescharacteristic
38C2×M16×C109—22no—
39C4⋊Q8×C109—28no—
40#124×C109—28no—
41#123×C109—24no—
42#123×C109—24no—
43#123×C109—24no—
44#123×C109D418yes—
45C22⋊Q8×C109—28no—
46C22⋊Q8×C109—28no—
47#117×C109—24no—
48C2×C4⋊C4×C109—22no—
49C2×C4⋊C4×C109—21no—
50C2×C4⋊C4×C109—22no—
51C2×C22⋊C4×C109—22no—
52C2×C22⋊C4×C109—22no—
53C2×C22⋊C4×C109—22no—
54C2×C22⋊C4×C109C2×C412yes—
55C2×C42×C109—21no—
56#101×C109—42no—
57#100×C109—24no—
58#99×C109—24no—
59#99×C109—42no—
60C22⋊C8×C109—42no—
61#95×C109—42no—
62C24×C109C22⋊C411yescharacteristic
63C24×C109C22×C411yescharacteristic
64C24×C109C2×D411yescharacteristic
65#41×C109—416no—
66C2×Q8×C109—24no—
67C2×Q8×C109—24no—
68C2×Q8×C109—22no—
69C2×Q8×C109—22no—
70C2×Q8×C109—41no—
71C2×Q8×C109—416no—
72C2×Q8×C109—42no—
73C2×Q8×C109—42no—
74C2×Q8×C109—42no—
75C2×Q8×C109—42no—
76C2×D4×C109—22no—
77C2×D4×C109—22no—
78C2×D4×C109—22no—
79C2×D4×C109—41no—
80C2×D4×C109—41no—
81C2×D4×C109—41no—
82C2×D4×C109—41no—
83C2×D4×C109—41no—
84C2×D4×C109C2×D414yes—
85C22×C4×C109—22no—
86C22×C4×C109—22no—
87C22×C4×C109—22no—
88C22×C4×C109—22no—
89C22×C4×C109—22no—
90C22×C4×C109—22no—
91C22×C4×C109—22no—
92C22×C4×C109—22no—
93C22×C4×C109—22no—
94C22×C4×C109—22no—
95C22×C4×C109—41no—
96C22×C4×C109—41no—
97C22×C4×C109—42no—
98C22×C4×C109—42no—
99C22×C4×C109—42no—
100C22×C4×C109C22⋊C411yescharacteristic
101C22×C4×C109C22⋊C411yescharacteristic
102C22×C4×C109C22⋊C414yes—
103C22×C4×C109C22⋊C411yescharacteristic
104C22×C4×C109C2×D411yescharacteristic
105M16×C109—42no—
106M16×C109—42no—
107C2×C8×C109—42no—
108C4⋊C4×C109—24no—
109C4⋊C4×C109—28no—
110C4⋊C4×C109—28no—
111C22⋊C4×C109—28no—
112C22⋊C4×C109—22no—
113C22⋊C4×C109—28no—
114C22⋊C4×C109—42no—
115C22⋊C4×C109—42no—
116C22⋊C4×C109C22⋊C414yes—
117C42×C109—24no—
118C23×C109—21no—
119C23×C109—21no—
120C23×C109—21no—
121C23×C109—21no—
122C23×C109—21no—
123C23×C109—21no—
124C23×C109—21no—
125C23×C109—42no—
126C23×C109—42no—
127C23×C109—42no—
128C23×C109C2×C22⋊C411yescharacteristic
129C23×C109C2×C22⋊C412yes—
130C23×C109#12012yes—
131Q8×C109—44no—
132Q8×C109—48no—
133Q8×C109—48no—
134D4×C109—44no—
135D4×C109—44no—
136D4×C109—44no—
137C2×C4×C109—28no—
138C2×C4×C109—21no—
139C2×C4×C109—21no—
140C2×C4×C109—21no—
141C2×C4×C109—21no—
142C2×C4×C109—21no—
143C2×C4×C109—21no—
144C2×C4×C109—24no—
145C2×C4×C109—24no—
146C2×C4×C109—21no—
147C2×C4×C109—21no—
148C2×C4×C109—21no—
149C2×C4×C109—44no—
150C2×C4×C109—44no—
151C2×C4×C109—44no—
152C2×C4×C109—44no—
153C2×C4×C109—42no—
154C2×C4×C109—42no—
155C2×C4×C109—44no—
156C2×C4×C109—82no—
157C2×C4×C109—84no—
158C2×C4×C109#9912yes—
159C2×C4×C109#9912yes—
160C2×C4×C109#12012yes—
161C8×C109—82no—
162C22×C109—22no—
163C22×C109—22no—
164C22×C109—22no—
165C22×C109—42no—
166C22×C109—42no—
167C22×C109—42no—
168C22×C109—42no—
169C22×C109—42no—
170C22×C109—42no—
171C22×C109—42no—
172C22×C109—41no—
173C22×C109—81no—
174C22×C109—81no—
175C22×C109#37911yescharacteristic
176C22×C109C2×#9911yescharacteristic
177C22×C109C2×#9911yescharacteristic
178C4×C109—22no—
179C4×C109—22no—
180C4×C109—22no—
181C4×C109—44no—
182C4×C109—41no—
183C4×C109—44no—
184C4×C109—162no—
185#12692C10911yesmaximal, characteristic, direct product factor, Sylow
186C2×C109—21no—
187C2×C109—21no—
188C2×C109—42no—
189C2×C109—42no—
190C2×C109—81no—
191C2×C109#178111yescharacteristic
192C2×C109#212312yes—
193C2×#536C2×C10911yescharacteristic
194C2×#352C2×C10914yescomplemented
195#1785C2×C10911yescharacteristic
196#1782C2×C10911yescharacteristic
197C109#1269211yescharacteristic, direct product factor, Sylow
198C22×#41C22×C10911yescharacteristic
199C23×Q8C4×C10911yescharacteristic, complemented
200#536—28no—
201C2×C4⋊Q8—21no—
202C2×#124—21no—
203C2×#123C4×C10912yescomplemented
204C2×#123C22×C10912yes—
205C2×C22⋊Q8—21no—
206C2×C22⋊Q8—21no—
207C2×#117C4×C10911yescharacteristic, complemented
208C2×#101—21no—
209C2×#100C22×C10912yes—
210C2×#99—21no—
211C2×#99C22×C10912yes—
212#407—21no—
213#381—21no—
214#352—28no—
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8D4×C10911yescharacteristic
224C22×D4C2×C4×C10911yescharacteristic
225C22×D4D4×C10911yescharacteristic
226C22×D4C23×C10911yescharacteristic
227C23×C4C2×C4×C10911yescharacteristic
228C23×C4C2×C4×C10912yes—
229C23×C4D4×C10911yescharacteristic
230C23×C4D4×C10911yescharacteristic
231C2×M16—22no—
232C4⋊Q8—28no—
233#124—28no—
234#123—24no—
235#123—24no—
236#123—24no—
237#123D4×C10918yes—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240#117—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C2×C4×C10912yes—
248C2×C42—21no—
249#101—42no—
250#100—24no—
251#99—24no—
252#99—42no—
253C22⋊C8—42no—
254#95—42no—
255C24C22⋊C4×C10911yescharacteristic
256C24C22×C4×C10911yescharacteristic
257C24C2×D4×C10911yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C2×D4×C10914yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C22⋊C4×C10911yescharacteristic
294C22×C4C22⋊C4×C10911yescharacteristic
295C22×C4C22⋊C4×C10911yescharacteristic
296C22×C4C22⋊C4×C10914yes—
297C22×C4C2×D4×C10911yescharacteristic
298M16—42no—
299M16—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—22no—
306C22⋊C4—28no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4C22⋊C4×C10914yes—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C2×C22⋊C4×C10911yescharacteristic
322C23C2×C22⋊C4×C10912yes—
323C23#120×C10912yes—
324Q8—44no—
325Q8—48no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4#99×C10912yes—
352C2×C4#99×C10912yes—
353C2×C4#120×C10912yes—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22#379×C10911yescharacteristic
369C22C2×#99×C10911yescharacteristic
370C22C2×#99×C10911yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—162no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2C109×#178111yescharacteristic
384C2C109×#212312yes—
subgroupG/H
center, socleC22×C109C2×#99
commutatorC24C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC22×#41×C109C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C22×#41×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#352×C109C2semidirect
6. C23×Q8×C109C4semidirect
10. C2×#123×C109C4semidirect
14. C2×#117×C109C4semidirect
185. #12692C109direct (N × H)
194. C2×#352C2×C109semidirect
197. C109#12692direct (N × H)
199. C23×Q8C4×C109semidirect
203. C2×#123C4×C109semidirect
207. C2×#117C4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1092
C2×#352×C1092
C109×#17852
C109×#17822
#12692109