Finite Group Databasebeta

← C109×#7782 #42125776

Subgroups — C109×#7782 #42125776

Order 27904.

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

normal subgroups only

This group has 480 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×#2804C211yesmaximal, characteristic
2C109×#2024C211yesmaximal, characteristic, complemented
3C109×#2023C211yesmaximal, characteristic
4C109×#2016C211yesmaximal, characteristic, complemented
5C109×#1272C211yesmaximal, characteristic, complemented
6C109×#1272C211yesmaximal, characteristic, complemented
7C109×#1270C211yesmaximal, characteristic
8C22×Q16×C109—21no—
9C2×C4⋊Q8×C109C2211yescharacteristic
10C2×#103×C109—21no—
11C2×#103×C109—21no—
12C2×#102×C109—21no—
13#400×C109—21no—
14#393×C109—21no—
15#393×C109—21no—
16#391×C109C2211yescharacteristic
17#391×C109C2211yescharacteristic
18#390×C109C2211yescharacteristic
19#386×C109—21no—
20#385×C109—21no—
21#385×C109—21no—
22#340×C109—21no—
23#340×C109—21no—
24#336×C109C2211yescharacteristic
25#336×C109C2211yescharacteristic
26#336×C109C2211yescharacteristic
27C22×Q8×C109—21no—
28C22×Q8×C109D411yescharacteristic
29C22×Q8×C109D411yescharacteristic
30C22×D4×C109D411yescharacteristic
31C23×C4×C109—21no—
32C2×Q16×C109—42no—
33C2×Q16×C109—42no—
34C2×Q16×C109—42no—
35C22×C8×C109—21no—
36C22×C8×C109—21no—
37C22×C8×C109—21no—
38C4⋊Q8×C109—42no—
39C2×C4⋊C4×C109—21no—
40C2×C4⋊C4×C109—21no—
41C2×C4⋊C4×C109—21no—
42C2×C4⋊C4×C109D411yescharacteristic
43C2×C4⋊C4×C109D411yescharacteristic
44C2×C4⋊C4×C109D411yescharacteristic
45C2×C22⋊C4×C109—21no—
46C2×C22⋊C4×C109—21no—
47C2×C22⋊C4×C109—21no—
48C2×C22⋊C4×C109—21no—
49C2×C22⋊C4×C109—21no—
50C2×C42×C109C2311yescharacteristic
51#103×C109—42no—
52#103×C109—42no—
53#102×C109—42no—
54#95×C109—21no—
55#95×C109—21no—
56#95×C109—21no—
57#95×C109—21no—
58#95×C109—21no—
59#95×C109—21no—
60#95×C109—21no—
61C24×C109—21no—
62C2×Q8×C109—22no—
63C2×Q8×C109—21no—
64C2×Q8×C109—21no—
65C2×Q8×C109—21no—
66C2×Q8×C109—21no—
67C2×Q8×C109—41no—
68C2×Q8×C109—41no—
69C2×Q8×C109—41no—
70C2×Q8×C109—41no—
71C2×Q8×C109—41no—
72C2×Q8×C109—41no—
73C2×Q8×C109—41no—
74C2×Q8×C109—41no—
75C2×Q8×C109—41no—
76C2×D4×C109—21no—
77C2×D4×C109—41no—
78C2×D4×C109—41no—
79C2×D4×C109Q1612yescomplemented
80C22×C4×C109—21no—
81C22×C4×C109—21no—
82C22×C4×C109—21no—
83C22×C4×C109—21no—
84C22×C4×C109—21no—
85C22×C4×C109—41no—
86C22×C4×C109—81no—
87C22×C4×C109—81no—
88C22×C4×C109—81no—
89C22×C4×C109C2×D411yescharacteristic
90C22×C4×C109C2×D411yescharacteristic
91C22×C4×C109C2×D411yescharacteristic
92Q16×C109—48no—
93C2×C8×C109—22no—
94C2×C8×C109—41no—
95C2×C8×C109—41no—
96C2×C8×C109—41no—
97C2×C8×C109—41no—
98C2×C8×C109—41no—
99C2×C8×C109—41no—
100C2×C8×C109—41no—
101C2×C8×C109—41no—
102C4⋊C4×C109—22no—
103C4⋊C4×C109—22no—
104C4⋊C4×C109—41no—
105C4⋊C4×C109—41no—
106C4⋊C4×C109—41no—
107C4⋊C4×C109—41no—
108C4⋊C4×C109—41no—
109C4⋊C4×C109—81no—
110C22⋊C4×C109—41no—
111C22⋊C4×C109—41no—
112C22⋊C4×C109—41no—
113C22⋊C4×C109—41no—
114C22⋊C4×C109—81no—
115C22⋊C4×C109—81no—
116C22⋊C4×C109—81no—
117C42×C109—41no—
118C23×C109—22no—
119C23×C109—41no—
120C23×C109—41no—
121C23×C109—41no—
122C23×C109—41no—
123C23×C109—41no—
124C23×C109—41no—
125C23×C109#12011yescharacteristic
126Q8×C109—42no—
127Q8×C109—42no—
128Q8×C109—42no—
129Q8×C109—42no—
130Q8×C109—81no—
131Q8×C109—81no—
132Q8×C109—81no—
133Q8×C109—81no—
134D4×C109—42no—
135D4×C109—81no—
136C2×C4×C109—21no—
137C2×C4×C109—21no—
138C2×C4×C109—21no—
139C2×C4×C109—21no—
140C2×C4×C109—21no—
141C2×C4×C109—21no—
142C2×C4×C109—41no—
143C2×C4×C109—41no—
144C2×C4×C109—41no—
145C2×C4×C109—41no—
146C2×C4×C109—41no—
147C2×C4×C109—41no—
148C2×C4×C109—41no—
149C2×C4×C109—41no—
150C2×C4×C109—41no—
151C2×C4×C109—41no—
152C2×C4×C109—41no—
153C2×C4×C109—41no—
154C2×C4×C109—41no—
155C2×C4×C109—41no—
156C2×C4×C109—41no—
157C2×C4×C109—42no—
158C2×C4×C109—84no—
159C2×C4×C109—81no—
160C2×C4×C109—81no—
161C2×C4×C109C2×Q1611yescharacteristic
162C2×C4×C109#13511yescharacteristic
163C2×C4×C109#13511yescharacteristic
164C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C109#13711yescharacteristic
166C2×C4×C109#13711yescharacteristic
167C8×C109—42no—
168C8×C109—81no—
169C8×C109—81no—
170C22×C109—42no—
171C22×C109—42no—
172C22×C109—42no—
173C22×C109—81no—
174C22×C109—81no—
175C22×C109—81no—
176C22×C109—81no—
177C22×C109#44911yescharacteristic
178C22×C109#45111yescharacteristic
179C22×C109#45211yescharacteristic
180C22×C109#45311yescharacteristic
181C22×C109#45511yescharacteristic
182C22×C109#45511yescharacteristic
183C22×C109B(4,𝔽2)11yescharacteristic
184C4×C109—21no—
185C4×C109—21no—
186C4×C109—21no—
187C4×C109—21no—
188C4×C109—21no—
189C4×C109—21no—
190C4×C109—81no—
191C4×C109—82no—
192C4×C109—81no—
193C4×C109—81no—
194C4×C109—81no—
195C4×C109—81no—
196#7782C10911yesmaximal, characteristic, direct product factor, Sylow
197C2×C109—82no—
198C2×C109#160011yescharacteristic
199C2×C109#160211yescharacteristic
200C2×C109#160611yescharacteristic
201C2×C109#163211yescharacteristic
202C2×C109#164011yescharacteristic
203C2×C109#164411yescharacteristic
204C2×C109#165911yescharacteristic
205#2804C2×C10911yescharacteristic
206#2024C2×C10911yescharacteristic, complemented
207#2023C2×C10911yescharacteristic
208#2016C2×C10911yescharacteristic, complemented
209#1272C2×C10911yescharacteristic, complemented
210#1272C2×C10911yescharacteristic, complemented
211#1270C2×C10911yescharacteristic
212C109#778211yescharacteristic, direct product factor, Sylow
213C22×Q16—21no—
214C2×C4⋊Q8C22×C10911yescharacteristic
215C2×#103—21no—
216C2×#103—21no—
217C2×#102—21no—
218#400—21no—
219#393—21no—
220#393—21no—
221#391C22×C10911yescharacteristic
222#391C22×C10911yescharacteristic
223#390C22×C10911yescharacteristic
224#386—21no—
225#385—21no—
226#385—21no—
227#340—21no—
228#340—21no—
229#336C22×C10911yescharacteristic
230#336C22×C10911yescharacteristic
231#336C22×C10911yescharacteristic
232C22×Q8—21no—
233C22×Q8D4×C10911yescharacteristic
234C22×Q8D4×C10911yescharacteristic
235C22×D4D4×C10911yescharacteristic
236C23×C4—21no—
237C2×Q16—42no—
238C2×Q16—42no—
239C2×Q16—42no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C4⋊Q8—42no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4D4×C10911yescharacteristic
248C2×C4⋊C4D4×C10911yescharacteristic
249C2×C4⋊C4D4×C10911yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42C23×C10911yescharacteristic
256#103—42no—
257#103—42no—
258#102—42no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266C24—21no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4Q16×C10912yescomplemented
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4C2×D4×C10911yescharacteristic
295C22×C4C2×D4×C10911yescharacteristic
296C22×C4C2×D4×C10911yescharacteristic
297Q16—48no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—81no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—81no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—41no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#120×C10911yescharacteristic
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—42no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—84no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4C2×Q16×C10911yescharacteristic
367C2×C4#135×C10911yescharacteristic
368C2×C4#135×C10911yescharacteristic
369C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
370C2×C4#137×C10911yescharacteristic
371C2×C4#137×C10911yescharacteristic
372C8—42no—
373C8—81no—
374C8—81no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22#449×C10911yescharacteristic
383C22#451×C10911yescharacteristic
384C22#452×C10911yescharacteristic
385C22#453×C10911yescharacteristic
386C22#455×C10911yescharacteristic
387C22#455×C10911yescharacteristic
388C22B(4,𝔽2)×C10911yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—82no—
399C4—81no—
400C4—81no—
401C2—82no—
402C2C109×#160011yescharacteristic
403C2C109×#160211yescharacteristic
404C2C109×#160611yescharacteristic
405C2C109×#163211yescharacteristic
406C2C109×#164011yescharacteristic
407C2C109×#164411yescharacteristic
408C2C109×#165911yescharacteristic
subgroupG/H
center, socleC23×C109#120
commutator, Frattini, cosocleC2×C42C23×C109
prime radicalC22×D4×C109D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2024C2semidirect
4. C109×#2016C2semidirect
5. C109×#1272C2semidirect
6. C109×#1272C2semidirect
79. C2×D4×C109Q16semidirect
196. #7782C109direct (N × H)
206. #2024C2×C109semidirect
208. #2016C2×C109semidirect
209. #1272C2×C109semidirect
210. #1272C2×C109semidirect
212. C109#7782direct (N × H)
284. C2×D4Q16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#28042
C109×#20242
C109×#20232
C109×#20162
C109×#12722
C109×#12702
#7782109