Finite Group Databasebeta

← C109×#19277 #42137271

Subgroups — C109×#19277 #42137271

Order 27904.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#487×C109C211yesmaximal, characteristic, complemented
2C109×#3079C211yesmaximal, characteristic
3C109×#3079C211yesmaximal, characteristic
4C8×#41×C109C211yesmaximal, characteristic
5C109×#1684C214yesmaximal, complemented
6C109×#1683C214yesmaximal, complemented
7C109×#1567C211yesmaximal, characteristic, complemented
8C109×#1549C211yesmaximal, characteristic, complemented
9C109×#1548C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×C4⋊Q8×C109C2211yescharacteristic
12C4×#41×C109C2211yescharacteristic
13#487×C109—21no—
14#487×C109—21no—
15#487×C109C2214yes—
16#477×C109—21no—
17#477×C109C2212yes—
18#476×C109—21no—
19#476×C109C2212yes—
20#475×C109—21no—
21#475×C109C2212yes—
22#474×C109—21no—
23#474×C109C2212yes—
24C8×Q8×C109C2212yes—
25C8×D4×C109—22no—
26C8×D4×C109C2212yes—
27#431×C109—21no—
28#431×C109C2211yescharacteristic
29#428×C109—21no—
30#428×C109—21no—
31#423×C109C2211yescharacteristic
32#423×C109C2211yescharacteristic
33#417×C109—21no—
34#417×C109—21no—
35C2×#103×C109—22no—
36C2×C4×C8×C109—21no—
37C2×C4×C8×C109C2211yescharacteristic
38#335×C109C2214yes—
39#333×C109C2214yes—
40#332×C109C2214yes—
41C2×#41×C109D411yescharacteristic
42C22×Q8×C109—21no—
43C22×Q8×C109—21no—
44C2×M16×C109—21no—
45C2×M16×C109—21no—
46C22×C8×C109—22no—
47C22×C8×C109D412yes—
48C4⋊Q8×C109—21no—
49C4⋊Q8×C109—21no—
50C4⋊Q8×C109—21no—
51C4⋊Q8×C109—21no—
52C4⋊Q8×C109—21no—
53C4⋊Q8×C109—21no—
54C4⋊Q8×C109C2312yes—
55C4⋊Q8×C109C2312yes—
56C4×Q8×C109D411yescharacteristic
57C4×Q8×C109C2311yescharacteristic
58C4×D4×C109—21no—
59C4×D4×C109—21no—
60C4×D4×C109D411yescharacteristic
61C4×D4×C109C2311yescharacteristic
62#117×C109—21no—
63#117×C109D411yescharacteristic
64C2×C4⋊C4×C109—21no—
65C2×C4⋊C4×C109—21no—
66C2×C4⋊C4×C109—21no—
67C2×C4⋊C4×C109—21no—
68C2×C42×C109—21no—
69C2×C42×C109C2311yescharacteristic
70#107×C109—22no—
71#107×C109—22no—
72#106×C109—22no—
73#106×C109—22no—
74C4⋊C8×C109—22no—
75C4⋊C8×C109—21no—
76C4⋊C8×C109—21no—
77C4⋊C8×C109C2312yes—
78C4⋊C8×C109C2312yes—
79C4⋊C8×C109C2312yes—
80#103×C109—22no—
81#103×C109—22no—
82#103×C109—24no—
83#103×C109—24no—
84#102×C109—22no—
85#102×C109—22no—
86C22⋊C8×C109—21no—
87C22⋊C8×C109—22no—
88C4×C8×C109—22no—
89C4×C8×C109—21no—
90C4×C8×C109D411yescharacteristic
91C4×C8×C109D411yescharacteristic
92C4×C8×C109C2312yes—
93#41×C109—22no—
94#41×C109Q1614yes—
95C2×Q8×C109—22no—
96C2×Q8×C109—22no—
97C2×Q8×C109—44no—
98C2×Q8×C109—41no—
99C2×Q8×C109—41no—
100C2×Q8×C109—82no—
101C2×Q8×C109C2×D411yescharacteristic
102C2×D4×C109—21no—
103C2×D4×C109C2×D411yescharacteristic
104C22×C4×C109—21no—
105C22×C4×C109—21no—
106C22×C4×C109—21no—
107C22×C4×C109—41no—
108C22×C4×C109—41no—
109C22×C4×C109C2×D411yescharacteristic
110C22×C4×C109C2×D411yescharacteristic
111C22×C4×C109#4111yescharacteristic
112M16×C109—42no—
113M16×C109—42no—
114C2×C8×C109—22no—
115C2×C8×C109—22no—
116C2×C8×C109—22no—
117C2×C8×C109—42no—
118C2×C8×C109—44no—
119C2×C8×C109C2×D412yes—
120C2×C8×C109C2×D412yes—
121C4⋊C4×C109—21no—
122C4⋊C4×C109—21no—
123C4⋊C4×C109—22no—
124C4⋊C4×C109—22no—
125C4⋊C4×C109—22no—
126C4⋊C4×C109—22no—
127C4⋊C4×C109—41no—
128C4⋊C4×C109—41no—
129C4⋊C4×C109—41no—
130C4⋊C4×C109—41no—
131C4⋊C4×C109C2×D411yescharacteristic
132C4⋊C4×C109C2×D411yescharacteristic
133C22⋊C4×C109—21no—
134C22⋊C4×C109—21no—
135C22⋊C4×C109—21no—
136C42×C109—21no—
137C42×C109—21no—
138C42×C109—21no—
139C42×C109C2×D411yescharacteristic
140C42×C109C2×D411yescharacteristic
141C42×C109#4111yescharacteristic
142C42×C109C2411yescharacteristic
143C23×C109—21no—
144C23×C109#12111yescharacteristic
145Q8×C109—21no—
146Q8×C109—44no—
147Q8×C109—82no—
148Q8×C109—82no—
149Q8×C109—82no—
150Q8×C109C2×Q1612yes—
151D4×C109—24no—
152D4×C109—21no—
153D4×C109C2×Q1612yes—
154C2×C4×C109—21no—
155C2×C4×C109—22no—
156C2×C4×C109—24no—
157C2×C4×C109—22no—
158C2×C4×C109—21no—
159C2×C4×C109—21no—
160C2×C4×C109—42no—
161C2×C4×C109—42no—
162C2×C4×C109—42no—
163C2×C4×C109—42no—
164C2×C4×C109—82no—
165C2×C4×C109—82no—
166C2×C4×C109#12111yescharacteristic
167C2×C4×C109#12111yescharacteristic
168C2×C4×C109#12111yescharacteristic
169C2×C4×C109C2×Q1612yes—
170C2×C4×C109#13512yes—
171C2×C4×C109C22×D411yescharacteristic
172C2×C4×C109C22×D411yescharacteristic
173C2×C4×C109C2×#4111yescharacteristic
174C8×C109—22no—
175C8×C109—22no—
176C8×C109—42no—
177C8×C109—42no—
178C8×C109—42no—
179C22×C109—21no—
180C22×C109—21no—
181C22×C109—22no—
182C22×C109—41no—
183C22×C109—41no—
184C22×C109#46712yes—
185C22×C109C2×#12111yescharacteristic
186C4×C109—21no—
187C4×C109—22no—
188C4×C109—21no—
189C4×C109—41no—
190C4×C109—41no—
191C4×C109—82no—
192C4×C109—82no—
193C4×C109#46712yes—
194C4×C109#47112yes—
195C4×C109C22×Q1611yescharacteristic
196C4×C109C2×#13511yescharacteristic
197C4×C109#57512yes—
198#19277C10911yesmaximal, characteristic, direct product factor, Sylow
199C2×C109—22no—
200C2×C109—42no—
201C2×C109#304011yescharacteristic
202C2×C109C2×#46711yescharacteristic
203C2×C109C2×#47111yescharacteristic
204C2×#487C2×C10911yescharacteristic, complemented
205#3079C2×C10911yescharacteristic
206#3079C2×C10911yescharacteristic
207C8×#41C2×C10911yescharacteristic
208#1684C2×C10914yescomplemented
209#1683C2×C10914yescomplemented
210#1567C2×C10911yescharacteristic, complemented
211#1549C2×C10911yescharacteristic, complemented
212#1548C2×C10911yescharacteristic, complemented
213C109#1927711yescharacteristic, direct product factor, Sylow
214C2×C4⋊Q8C22×C10911yescharacteristic
215C2×C4⋊Q8C22×C10911yescharacteristic
216C4×#41C22×C10911yescharacteristic
217#487—21no—
218#487—21no—
219#487C22×C10914yes—
220#477—21no—
221#477C22×C10912yes—
222#476—21no—
223#476C22×C10912yes—
224#475—21no—
225#475C22×C10912yes—
226#474—21no—
227#474C22×C10912yes—
228C8×Q8C22×C10912yes—
229C8×D4—22no—
230C8×D4C22×C10912yes—
231#431—21no—
232#431C22×C10911yescharacteristic
233#428—21no—
234#428—21no—
235#423C22×C10911yescharacteristic
236#423C22×C10911yescharacteristic
237#417—21no—
238#417—21no—
239C2×#103—22no—
240C2×C4×C8—21no—
241C2×C4×C8C22×C10911yescharacteristic
242#335C22×C10914yes—
243#333C22×C10914yes—
244#332C22×C10914yes—
245C2×#41D4×C10911yescharacteristic
246C22×Q8—21no—
247C22×Q8—21no—
248C2×M16—21no—
249C2×M16—21no—
250C22×C8—22no—
251C22×C8D4×C10912yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C23×C10912yes—
259C4⋊Q8C23×C10912yes—
260C4×Q8D4×C10911yescharacteristic
261C4×Q8C23×C10911yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4D4×C10911yescharacteristic
265C4×D4C23×C10911yescharacteristic
266#117—21no—
267#117D4×C10911yescharacteristic
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C42—21no—
273C2×C42C23×C10911yescharacteristic
274#107—22no—
275#107—22no—
276#106—22no—
277#106—22no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8C23×C10912yes—
282C4⋊C8C23×C10912yes—
283C4⋊C8C23×C10912yes—
284#103—22no—
285#103—22no—
286#103—24no—
287#103—24no—
288#102—22no—
289#102—22no—
290C22⋊C8—21no—
291C22⋊C8—22no—
292C4×C8—22no—
293C4×C8—21no—
294C4×C8D4×C10911yescharacteristic
295C4×C8D4×C10911yescharacteristic
296C4×C8C23×C10912yes—
297#41—22no—
298#41Q16×C10914yes—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8C2×D4×C10911yescharacteristic
306C2×D4—21no—
307C2×D4C2×D4×C10911yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C2×D4×C10911yescharacteristic
314C22×C4C2×D4×C10911yescharacteristic
315C22×C4#41×C10911yescharacteristic
316M16—42no—
317M16—42no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—42no—
322C2×C8—44no—
323C2×C8C2×D4×C10912yes—
324C2×C8C2×D4×C10912yes—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4C2×D4×C10911yescharacteristic
336C4⋊C4C2×D4×C10911yescharacteristic
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42C2×D4×C10911yescharacteristic
344C42C2×D4×C10911yescharacteristic
345C42#41×C10911yescharacteristic
346C42C24×C10911yescharacteristic
347C23—21no—
348C23#121×C10911yescharacteristic
349Q8—21no—
350Q8—44no—
351Q8—82no—
352Q8—82no—
353Q8—82no—
354Q8C2×Q16×C10912yes—
355D4—24no—
356D4—21no—
357D4C2×Q16×C10912yes—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#121×C10911yescharacteristic
371C2×C4#121×C10911yescharacteristic
372C2×C4#121×C10911yescharacteristic
373C2×C4C2×Q16×C10912yes—
374C2×C4#135×C10912yes—
375C2×C4C22×D4×C10911yescharacteristic
376C2×C4C22×D4×C10911yescharacteristic
377C2×C4C2×#41×C10911yescharacteristic
378C8—22no—
379C8—22no—
380C8—42no—
381C8—42no—
382C8—42no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22#467×C10912yes—
389C22C2×#121×C10911yescharacteristic
390C4—21no—
391C4—22no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—82no—
397C4#467×C10912yes—
398C4#471×C10912yes—
399C4C22×Q16×C10911yescharacteristic
400C4C2×#135×C10911yescharacteristic
401C4#575×C10912yes—
402C2—22no—
403C2—42no—
404C2C109×#304011yescharacteristic
405C2C2×#467×C10911yescharacteristic
406C2C2×#471×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#121
commutator, Frattini, cosocleC42C24×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
1. C2×#487×C109C2semidirect
5. C109×#1684C2semidirect
6. C109×#1683C2semidirect
7. C109×#1567C2semidirect
8. C109×#1549C2semidirect
9. C109×#1548C2semidirect
198. #19277C109direct (N × H)
204. C2×#487C2×C109semidirect
208. #1684C2×C109semidirect
209. #1683C2×C109semidirect
210. #1567C2×C109semidirect
211. #1549C2×C109semidirect
212. #1548C2×C109semidirect
213. C109#19277direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#487×C1092
C109×#30792
C8×#41×C1092
C109×#16842
C109×#16832
C109×#15672
C109×#15492
C109×#15482
#19277109