Finite Group Databasebeta

← C109×#10170 #42128164

Subgroups — C109×#10170 #42128164

Order 27904.

574 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3206C211yesmaximal, characteristic
2C109×#3152C211yesmaximal, characteristic
3C2×#487×C109C211yesmaximal, characteristic, complemented
4C109×#1997C211yesmaximal, characteristic, complemented
5C109×#1546C211yesmaximal, characteristic, complemented
6C109×#1492C211yesmaximal, characteristic
7C109×#1278C211yesmaximal, characteristic, complemented
8C2×#137×C109—22no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×C4⋊Q8×C109C2211yescharacteristic
12#497×C109—22no—
13#494×C109—21no—
14#492×C109—21no—
15#492×C109—21no—
16#487×C109—21no—
17#487×C109—22no—
18#487×C109—22no—
19#461×C109—22no—
20#448×C109—22no—
21#431×C109C2211yescharacteristic
22#423×C109C2211yescharacteristic
23C2×#103×C109—22no—
24C4×M16×C109C2211yescharacteristic
25C2×C4×C8×C109C2211yescharacteristic
26#343×C109—22no—
27#332×C109—22no—
28D4⋊C8×C109—22no—
29C2×#41×C109—21no—
30C22×Q8×C109—21no—
31C22×Q8×C109D412yes—
32#137×C109—44no—
33#137×C109—44no—
34C2×Q16×C109—24no—
35C2×SD16×C109—24no—
36C2×SD16×C109—24no—
37C2×M16×C109—21no—
38C2×M16×C109D412yes—
39C22×C8×C109—21no—
40C4⋊Q8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109—21no—
45C4⋊Q8×C109—21no—
46C4⋊Q8×C109—21no—
47C4⋊Q8×C109D412yes—
48C4⋊Q8×C109D411yescharacteristic
49#127×C109D411yescharacteristic
50#124×C109—21no—
51#121×C109—21no—
52#121×C109—21no—
53C4×D4×C109—21no—
54C4×D4×C109—21no—
55C2×C4⋊C4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109—21no—
58C2×C4⋊C4×C109—21no—
59C2×C42×C109C2311yescharacteristic
60C4⋊C8×C109—21no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109D412yes—
63#103×C109—28no—
64#103×C109—24no—
65C22⋊C8×C109—21no—
66#97×C109—21no—
67C4×C8×C109—21no—
68C4×C8×C109—21no—
69C4×C8×C109—21no—
70C4×C8×C109D412yes—
71#41×C109—42no—
72#41×C109—42no—
73C2×Q8×C109—22no—
74C2×Q8×C109—22no—
75C2×Q8×C109—21no—
76C2×Q8×C109—22no—
77C2×Q8×C109—44no—
78C2×Q8×C109—41no—
79C2×Q8×C109—42no—
80C2×Q8×C109—42no—
81C2×D4×C109—21no—
82C2×D4×C109—21no—
83C2×D4×C109—21no—
84C2×D4×C109—41no—
85C22×C4×C109—22no—
86C22×C4×C109—41no—
87C22×C4×C109—41no—
88C22×C4×C109C2×D411yescharacteristic
89C22×C4×C109C2×D411yescharacteristic
90C22×C4×C109#4111yescharacteristic
91Q16×C109—48no—
92SD16×C109—48no—
93SD16×C109—84no—
94M16×C109—22no—
95M16×C109—42no—
96M16×C109SD1614yescomplemented
97C2×C8×C109—22no—
98C2×C8×C109—22no—
99C2×C8×C109—22no—
100C2×C8×C109—42no—
101C2×C8×C109—41no—
102C4⋊C4×C109—22no—
103C4⋊C4×C109—21no—
104C4⋊C4×C109—22no—
105C4⋊C4×C109—21no—
106C4⋊C4×C109—41no—
107C4⋊C4×C109—41no—
108C4⋊C4×C109—41no—
109C4⋊C4×C109—41no—
110C4⋊C4×C109—41no—
111C4⋊C4×C109—41no—
112C22⋊C4×C109—41no—
113C22⋊C4×C109—41no—
114C42×C109C2×D411yescharacteristic
115C42×C109C2×D411yescharacteristic
116C42×C109C2×D411yescharacteristic
117C42×C109C2×D411yescharacteristic
118C23×C109—41no—
119C23×C109#12111yescharacteristic
120Q8×C109—44no—
121Q8×C109—44no—
122Q8×C109—42no—
123Q8×C109—44no—
124Q8×C109—81no—
125Q8×C109—81no—
126D4×C109—42no—
127D4×C109—42no—
128D4×C109—42no—
129D4×C109—82no—
130C2×C4×C109—22no—
131C2×C4×C109—22no—
132C2×C4×C109—41no—
133C2×C4×C109—42no—
134C2×C4×C109—42no—
135C2×C4×C109—82no—
136C2×C4×C109—81no—
137C2×C4×C109—81no—
138C2×C4×C109—82no—
139C2×C4×C109#12011yescharacteristic
140C2×C4×C109#12011yescharacteristic
141C2×C4×C109#12011yescharacteristic
142C2×C4×C109#12111yescharacteristic
143C2×C4×C109#12111yescharacteristic
144C2×C4×C109#12711yescharacteristic
145C2×C4×C109C2×SD1612yes—
146C2×C4×C109AGL(1,ℤ/8)12yes—
147C8×C109—24no—
148C8×C109—42no—
149C8×C109—42no—
150C8×C109—81no—
151C22×C109—21no—
152C22×C109—21no—
153C22×C109—81no—
154C22×C109—81no—
155C22×C109—81no—
156C22×C109#39211yescharacteristic
157C22×C109#46012yes—
158C4×C109—21no—
159C4×C109—21no—
160C4×C109—82no—
161C4×C109—81no—
162C4×C109—82no—
163C4×C109#45012yes—
164C4×C109#45512yes—
165C4×C109#47312yes—
166C4×C109#49211yescharacteristic
167C4×C109#49611yescharacteristic
168#10170C10911yesmaximal, characteristic, direct product factor, Sylow
169C2×C109—22no—
170C2×C109—161no—
171C2×C109#200411yescharacteristic
172C2×C109#201411yescharacteristic
173C2×C109#202011yescharacteristic
174#3206C2×C10911yescharacteristic
175#3152C2×C10911yescharacteristic
176C2×#487C2×C10911yescharacteristic, complemented
177#1997C2×C10911yescharacteristic, complemented
178#1546C2×C10911yescharacteristic, complemented
179#1492C2×C10911yescharacteristic
180#1278C2×C10911yescharacteristic, complemented
181C109#1017011yescharacteristic, direct product factor, Sylow
182C2×#137—22no—
183#532C22×C10911yescharacteristic
184C2×C4⋊Q8C22×C10911yescharacteristic
185C2×C4⋊Q8C22×C10911yescharacteristic
186#497—22no—
187#494—21no—
188#492—21no—
189#492—21no—
190#487—22no—
191#487—21no—
192#487—22no—
193#461—22no—
194#448—22no—
195#431C22×C10911yescharacteristic
196#423C22×C10911yescharacteristic
197C2×#103—22no—
198C4×M16C22×C10911yescharacteristic
199C2×C4×C8C22×C10911yescharacteristic
200#343—22no—
201#332—22no—
202D4⋊C8—22no—
203C2×#41—21no—
204C22×Q8—21no—
205C22×Q8D4×C10912yes—
206#137—44no—
207#137—44no—
208C2×Q16—24no—
209C2×SD16—24no—
210C2×SD16—24no—
211C2×M16—21no—
212C2×M16D4×C10912yes—
213C22×C8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8D4×C10912yes—
222C4⋊Q8D4×C10911yescharacteristic
223#127D4×C10911yescharacteristic
224#124—21no—
225#121—21no—
226#121—21no—
227C4×D4—21no—
228C4×D4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C42C23×C10911yescharacteristic
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8D4×C10912yes—
237#103—28no—
238#103—24no—
239C22⋊C8—21no—
240#97—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8D4×C10912yes—
245#41—42no—
246#41—42no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×D4×C10911yescharacteristic
263C22×C4C2×D4×C10911yescharacteristic
264C22×C4#41×C10911yescharacteristic
265Q16—48no—
266SD16—48no—
267SD16—84no—
268M16—22no—
269M16—42no—
270M16SD16×C10914yescomplemented
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—42no—
275C2×C8—41no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42C2×D4×C10911yescharacteristic
289C42C2×D4×C10911yescharacteristic
290C42C2×D4×C10911yescharacteristic
291C42C2×D4×C10911yescharacteristic
292C23—41no—
293C23#121×C10911yescharacteristic
294Q8—44no—
295Q8—44no—
296Q8—42no—
297Q8—44no—
298Q8—81no—
299Q8—81no—
300D4—42no—
301D4—42no—
302D4—42no—
303D4—82no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4#120×C10911yescharacteristic
314C2×C4#120×C10911yescharacteristic
315C2×C4#120×C10911yescharacteristic
316C2×C4#121×C10911yescharacteristic
317C2×C4#121×C10911yescharacteristic
318C2×C4#127×C10911yescharacteristic
319C2×C4C2×SD16×C10912yes—
320C2×C4AGL(1,ℤ/8)×C10912yes—
321C8—24no—
322C8—42no—
323C8—42no—
324C8—81no—
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#392×C10911yescharacteristic
331C22#460×C10912yes—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—81no—
336C4—82no—
337C4#450×C10912yes—
338C4#455×C10912yes—
339C4#473×C10912yes—
340C4#492×C10911yescharacteristic
341C4#496×C10911yescharacteristic
342C2—22no—
343C2—161no—
344C2C109×#200411yescharacteristic
345C2C109×#201411yescharacteristic
346C2C109×#202011yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radical#532×C109C22
LCSG ▷ C2×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. C2×#487×C109C2semidirect
4. C109×#1997C2semidirect
5. C109×#1546C2semidirect
7. C109×#1278C2semidirect
96. M16×C109SD16semidirect
168. #10170C109direct (N × H)
176. C2×#487C2×C109semidirect
177. #1997C2×C109semidirect
178. #1546C2×C109semidirect
180. #1278C2×C109semidirect
181. C109#10170direct (N × H)
270. M16SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32062
C109×#31522
C2×#487×C1092
C109×#19972
C109×#15462
C109×#14922
C109×#12782
#10170109