Finite Group Databasebeta

← C109×#9172 #42127166

Subgroups — C109×#9172 #42127166

Order 27904.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3039C211yesmaximal, characteristic
2C109×#2925C211yesmaximal, characteristic
3C109×#1546C211yesmaximal, characteristic, complemented
4C2×#332×C109C211yesmaximal, characteristic, complemented
5C109×#1477C211yesmaximal, characteristic
6C109×#1301C211yesmaximal, characteristic, complemented
7C109×#1292C211yesmaximal, characteristic, complemented
8C2×#137×C109—21no—
9C2×C4⋊Q8×C109C2211yescharacteristic
10C4×#41×C109C2211yescharacteristic
11#462×C109—21no—
12#462×C109C412yes—
13#461×C109—21no—
14#461×C109C412yes—
15#433×C109—22no—
16#432×C109—21no—
17#431×C109—21no—
18#431×C109C2211yescharacteristic
19#423×C109C2211yescharacteristic
20#423×C109C2211yescharacteristic
21C2×C4⋊C8×C109C2211yescharacteristic
22C2×C4⋊C8×C109C2211yescharacteristic
23#419×C109—21no—
24#408×C109—21no—
25#405×C109—21no—
26#332×C109—21no—
27#332×C109—21no—
28#332×C109—22no—
29Q8⋊C8×C109—21no—
30D4⋊C8×C109—21no—
31C2×#41×C109D411yescharacteristic
32C22×Q8×C109—21no—
33#137×C109—42no—
34#137×C109—42no—
35C2×Q16×C109—22no—
36C2×SD16×C109—22no—
37C2×M16×C109—21no—
38C2×M16×C109—21no—
39C22×C8×C109—21no—
40C22×C8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109C2×C412yes—
45C4×Q8×C109C2×C411yescharacteristic
46C4×Q8×C109D411yescharacteristic
47C4×D4×C109—21no—
48C4×D4×C109—21no—
49C4×D4×C109C2×C411yescharacteristic
50C4×D4×C109D411yescharacteristic
51#117×C109—21no—
52#117×C109D411yescharacteristic
53C2×C4⋊C4×C109—21no—
54C2×C4⋊C4×C109—21no—
55C2×C42×C109—21no—
56C2×C42×C109C2311yescharacteristic
57C4⋊C8×C109—21no—
58C4⋊C8×C109—21no—
59C4⋊C8×C109—21no—
60C4⋊C8×C109—21no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109—21no—
63C4⋊C8×C109C2×C412yes—
64C4⋊C8×C109D411yescharacteristic
65C4⋊C8×C109D411yescharacteristic
66C4⋊C8×C109D412yes—
67#103×C109—22no—
68#103×C109—22no—
69C22⋊C8×C109—22no—
70C22⋊C8×C109—22no—
71C22⋊C8×C109—21no—
72#97×C109—22no—
73C4×C8×C109—22no—
74C4×C8×C109—21no—
75#41×C109—21no—
76#41×C109—21no—
77#41×C109—21no—
78#41×C109—21no—
79C2×Q8×C109—22no—
80C2×Q8×C109—41no—
81C2×Q8×C109—41no—
82C2×Q8×C109—41no—
83C2×Q8×C109—41no—
84C2×Q8×C109—41no—
85C2×Q8×C109C22⋊C411yescharacteristic
86C2×D4×C109—21no—
87C2×D4×C109C22⋊C411yescharacteristic
88C22×C4×C109—21no—
89C22×C4×C109—21no—
90C22×C4×C109—21no—
91C22×C4×C109—41no—
92C22×C4×C109C2×D411yescharacteristic
93C22×C4×C109#4111yescharacteristic
94C22×C4×C109#4111yescharacteristic
95Q16×C109—44no—
96SD16×C109—44no—
97M16×C109—41no—
98M16×C109—41no—
99M16×C109—41no—
100M16×C109—41no—
101C2×C8×C109—22no—
102C2×C8×C109—21no—
103C2×C8×C109—22no—
104C2×C8×C109—21no—
105C2×C8×C109—41no—
106C2×C8×C109—41no—
107C2×C8×C109—41no—
108C2×C8×C109—41no—
109C2×C8×C109—81no—
110C4⋊C4×C109—21no—
111C4⋊C4×C109—21no—
112C4⋊C4×C109—22no—
113C4⋊C4×C109—22no—
114C4⋊C4×C109—41no—
115C4⋊C4×C109—41no—
116C4⋊C4×C109C22⋊C411yescharacteristic
117C4⋊C4×C109C22⋊C411yescharacteristic
118C22⋊C4×C109—21no—
119C22⋊C4×C109—21no—
120C22⋊C4×C109—21no—
121C42×C109—21no—
122C42×C109—21no—
123C42×C109—21no—
124C42×C109C22×C411yescharacteristic
125C42×C109C2×D411yescharacteristic
126C42×C109C2×D411yescharacteristic
127C42×C109C2×D411yescharacteristic
128C23×C109—21no—
129C23×C109#12311yescharacteristic
130Q8×C109—21no—
131Q8×C109—21no—
132Q8×C109—42no—
133Q8×C109—42no—
134Q8×C109—81no—
135Q8×C109—81no—
136D4×C109—21no—
137D4×C109—21no—
138D4×C109—41no—
139D4×C109—41no—
140C2×C4×C109—21no—
141C2×C4×C109—21no—
142C2×C4×C109—21no—
143C2×C4×C109—21no—
144C2×C4×C109—21no—
145C2×C4×C109—41no—
146C2×C4×C109—41no—
147C2×C4×C109—41no—
148C2×C4×C109—41no—
149C2×C4×C109—41no—
150C2×C4×C109—42no—
151C2×C4×C109—81no—
152C2×C4×C109—81no—
153C2×C4×C109—81no—
154C2×C4×C109C2×C22⋊C411yescharacteristic
155C2×C4×C109C4×D411yescharacteristic
156C2×C4×C109C4×D411yescharacteristic
157C2×C4×C109#12011yescharacteristic
158C2×C4×C109#12111yescharacteristic
159C2×C4×C109#12111yescharacteristic
160C2×C4×C109#13511yescharacteristic
161C2×C4×C109#13511yescharacteristic
162C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
164C8×C109—42no—
165C8×C109—41no—
166C8×C109—41no—
167C8×C109—42no—
168C8×C109—81no—
169C8×C109—81no—
170C22×C109—21no—
171C22×C109—21no—
172C22×C109—41no—
173C22×C109—41no—
174C22×C109—41no—
175C22×C109#38611yescharacteristic
176C22×C109#48211yescharacteristic
177C22×C109#48211yescharacteristic
178C4×C109—21no—
179C4×C109—21no—
180C4×C109—21no—
181C4×C109—41no—
182C4×C109—41no—
183C4×C109—41no—
184C4×C109—82no—
185C4×C109#41611yescharacteristic
186C4×C109#41811yescharacteristic
187C4×C109#44911yescharacteristic
188C4×C109#44911yescharacteristic
189C4×C109#47111yescharacteristic
190C4×C109#47311yescharacteristic
191#9172C10911yesmaximal, characteristic, direct product factor, Sylow
192C2×C109—21no—
193C2×C109—21no—
194C2×C109—81no—
195C2×C109#187511yescharacteristic
196C2×C109#190211yescharacteristic
197C2×C109#190811yescharacteristic
198#3039C2×C10911yescharacteristic
199#2925C2×C10911yescharacteristic
200#1546C2×C10911yescharacteristic, complemented
201C2×#332C2×C10911yescharacteristic, complemented
202#1477C2×C10911yescharacteristic
203#1301C2×C10911yescharacteristic, complemented
204#1292C2×C10911yescharacteristic, complemented
205C109#917211yescharacteristic, direct product factor, Sylow
206C2×#137—21no—
207C2×C4⋊Q8C22×C10911yescharacteristic
208C4×#41C22×C10911yescharacteristic
209#462—21no—
210#462C4×C10912yes—
211#461—21no—
212#461C4×C10912yes—
213#433—22no—
214#432—21no—
215#431—21no—
216#431C22×C10911yescharacteristic
217#423C22×C10911yescharacteristic
218#423C22×C10911yescharacteristic
219C2×C4⋊C8C22×C10911yescharacteristic
220C2×C4⋊C8C22×C10911yescharacteristic
221#419—21no—
222#408—21no—
223#405—21no—
224#332—21no—
225#332—21no—
226#332—22no—
227Q8⋊C8—21no—
228D4⋊C8—21no—
229C2×#41D4×C10911yescharacteristic
230C22×Q8—21no—
231#137—42no—
232#137—42no—
233C2×Q16—22no—
234C2×SD16—22no—
235C2×M16—21no—
236C2×M16—21no—
237C22×C8—21no—
238C22×C8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8C2×C4×C10912yes—
243C4×Q8C2×C4×C10911yescharacteristic
244C4×Q8D4×C10911yescharacteristic
245C4×D4—21no—
246C4×D4—21no—
247C4×D4C2×C4×C10911yescharacteristic
248C4×D4D4×C10911yescharacteristic
249#117—21no—
250#117D4×C10911yescharacteristic
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C42—21no—
254C2×C42C23×C10911yescharacteristic
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8C2×C4×C10912yes—
262C4⋊C8D4×C10911yescharacteristic
263C4⋊C8D4×C10911yescharacteristic
264C4⋊C8D4×C10912yes—
265#103—22no—
266#103—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—21no—
270#97—22no—
271C4×C8—22no—
272C4×C8—21no—
273#41—21no—
274#41—21no—
275#41—21no—
276#41—21no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C22⋊C4×C10911yescharacteristic
284C2×D4—21no—
285C2×D4C22⋊C4×C10911yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C10911yescharacteristic
291C22×C4#41×C10911yescharacteristic
292C22×C4#41×C10911yescharacteristic
293Q16—44no—
294SD16—44no—
295M16—41no—
296M16—41no—
297M16—41no—
298M16—41no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—81no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4C22⋊C4×C10911yescharacteristic
315C4⋊C4C22⋊C4×C10911yescharacteristic
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42C22×C4×C10911yescharacteristic
323C42C2×D4×C10911yescharacteristic
324C42C2×D4×C10911yescharacteristic
325C42C2×D4×C10911yescharacteristic
326C23—21no—
327C23#123×C10911yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—21no—
335D4—21no—
336D4—41no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C10911yescharacteristic
353C2×C4C4×D4×C10911yescharacteristic
354C2×C4C4×D4×C10911yescharacteristic
355C2×C4#120×C10911yescharacteristic
356C2×C4#121×C10911yescharacteristic
357C2×C4#121×C10911yescharacteristic
358C2×C4#135×C10911yescharacteristic
359C2×C4#135×C10911yescharacteristic
360C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
361C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
362C8—42no—
363C8—41no—
364C8—41no—
365C8—42no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#386×C10911yescharacteristic
374C22#482×C10911yescharacteristic
375C22#482×C10911yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—82no—
383C4#416×C10911yescharacteristic
384C4#418×C10911yescharacteristic
385C4#449×C10911yescharacteristic
386C4#449×C10911yescharacteristic
387C4#471×C10911yescharacteristic
388C4#473×C10911yescharacteristic
389C2—21no—
390C2—21no—
391C2—81no—
392C2C109×#187511yescharacteristic
393C2C109×#190211yescharacteristic
394C2C109×#190811yescharacteristic
subgroupG/H
center, socleC22×C109#386
commutatorC42C22×C4×C109
Frattini, cosocleC2×C42C23×C109
prime radicalC2×#41×C109D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
3. C109×#1546C2semidirect
4. C2×#332×C109C2semidirect
6. C109×#1301C2semidirect
7. C109×#1292C2semidirect
191. #9172C109direct (N × H)
200. #1546C2×C109semidirect
201. C2×#332C2×C109semidirect
203. #1301C2×C109semidirect
204. #1292C2×C109semidirect
205. C109#9172direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#30392
C109×#29252
C109×#15462
C2×#332×C1092
C109×#14772
C109×#13012
C109×#12922
#9172109