Finite Group Databasebeta

← C109×#10174 #42128168

Subgroups — C109×#10174 #42128168

Order 27904.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3226C211yesmaximal, characteristic
2C2×#500×C109C211yesmaximal, characteristic, complemented
3C109×#3152C211yesmaximal, characteristic
4C109×#1966C211yesmaximal, 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#500×C109—21no—
13#500×C109—22no—
14#500×C109—22no—
15#497×C109—22no—
16#494×C109—21no—
17#492×C109—21no—
18#486×C109—21no—
19#480×C109—22no—
20#474×C109—22no—
21#431×C109C2211yescharacteristic
22C2×#107×C109—22no—
23#423×C109C2211yescharacteristic
24C2×#101×C109—22no—
25C4×M16×C109C2211yescharacteristic
26C2×C4×C8×C109C2211yescharacteristic
27#332×C109—22no—
28D4⋊C8×C109—22no—
29C2×#41×C109—21no—
30C22×Q8×C109—21no—
31C22×Q8×C109—21no—
32#137×C109—44no—
33#137×C109—44no—
34C2×Q16×C109—24no—
35C2×SD16×C109—24no—
36C2×M16×C109—21no—
37C2×M16×C109D412yes—
38C22×C8×C109—21no—
39C4⋊Q8×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×C109D412yes—
47C4⋊Q8×C109D411yescharacteristic
48#127×C109D411yescharacteristic
49#124×C109—21no—
50#121×C109—21no—
51#121×C109—21no—
52C4×D4×C109—21no—
53C4×D4×C109—21no—
54C2×C4⋊C4×C109—21no—
55C2×C4⋊C4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109D412yes—
58C2×C42×C109C2311yescharacteristic
59#107×C109—28no—
60#107×C109—24no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109—21no—
63C4⋊C8×C109D412yes—
64#102×C109—24no—
65#101×C109—44no—
66C22⋊C8×C109—21no—
67#97×C109—21no—
68C4×C8×C109—21no—
69C4×C8×C109—21no—
70C4×C8×C109—21no—
71C4×C8×C109D412yes—
72#41×C109—42no—
73#41×C109—42no—
74C2×Q8×C109—21no—
75C2×Q8×C109—22no—
76C2×Q8×C109—41no—
77C2×Q8×C109—41no—
78C2×Q8×C109—41no—
79C2×Q8×C109—42no—
80C2×Q8×C109—42no—
81C2×Q8×C109—42no—
82C2×Q8×C109—42no—
83C2×D4×C109—21no—
84C2×D4×C109—21no—
85C2×D4×C109—21no—
86C2×D4×C109—41no—
87C22×C4×C109—22no—
88C22×C4×C109—41no—
89C22×C4×C109—41no—
90C22×C4×C109C2×D411yescharacteristic
91C22×C4×C109C2×D411yescharacteristic
92C22×C4×C109#4111yescharacteristic
93Q16×C109—48no—
94SD16×C109—48no—
95M16×C109—22no—
96M16×C109—42no—
97M16×C109D814yes—
98C2×C8×C109—24no—
99C2×C8×C109—22no—
100C2×C8×C109—22no—
101C2×C8×C109—22no—
102C2×C8×C109—41no—
103C4⋊C4×C109—22no—
104C4⋊C4×C109—22no—
105C4⋊C4×C109—22no—
106C4⋊C4×C109—21no—
107C4⋊C4×C109—22no—
108C4⋊C4×C109—21no—
109C4⋊C4×C109—41no—
110C4⋊C4×C109—41no—
111C4⋊C4×C109—41no—
112C4⋊C4×C109—41no—
113C22⋊C4×C109—41no—
114C22⋊C4×C109—41no—
115C42×C109C2×D411yescharacteristic
116C42×C109C2×D411yescharacteristic
117C42×C109C2×D411yescharacteristic
118C42×C109C2×D411yescharacteristic
119C23×C109—41no—
120C23×C109#12111yescharacteristic
121Q8×C109—44no—
122Q8×C109—42no—
123Q8×C109—82no—
124Q8×C109—81no—
125Q8×C109—81no—
126Q8×C109—81no—
127Q8×C109—81no—
128D4×C109—42no—
129D4×C109—42no—
130D4×C109—42no—
131D4×C109—82no—
132C2×C4×C109—22no—
133C2×C4×C109—22no—
134C2×C4×C109—41no—
135C2×C4×C109—42no—
136C2×C4×C109—42no—
137C2×C4×C109—44no—
138C2×C4×C109—82no—
139C2×C4×C109—81no—
140C2×C4×C109—81no—
141C2×C4×C109#12011yescharacteristic
142C2×C4×C109#12011yescharacteristic
143C2×C4×C109#12011yescharacteristic
144C2×C4×C109#12111yescharacteristic
145C2×C4×C109#12111yescharacteristic
146C2×C4×C109#12711yescharacteristic
147C2×C4×C109C2×D812yes—
148C2×C4×C109AGL(1,ℤ/8)12yes—
149C8×C109—24no—
150C8×C109—24no—
151C8×C109—42no—
152C8×C109—81no—
153C22×C109—21no—
154C22×C109—21no—
155C22×C109—81no—
156C22×C109—81no—
157C22×C109—81no—
158C22×C109#39211yescharacteristic
159C22×C109#45912yes—
160C4×C109—21no—
161C4×C109—21no—
162C4×C109—82no—
163C4×C109—81no—
164C4×C109—82no—
165C4×C109#44712yes—
166C4×C109#45612yes—
167C4×C109#47312yes—
168C4×C109#49311yescharacteristic
169C4×C109#49611yescharacteristic
170#10174C10911yesmaximal, characteristic, direct product factor, Sylow
171C2×C109—22no—
172C2×C109—161no—
173C2×C109#200511yescharacteristic
174C2×C109#201111yescharacteristic
175C2×C109#202011yescharacteristic
176#3226C2×C10911yescharacteristic
177C2×#500C2×C10911yescharacteristic, complemented
178#3152C2×C10911yescharacteristic
179#1966C2×C10911yescharacteristic, complemented
180#1546C2×C10911yescharacteristic, complemented
181#1492C2×C10911yescharacteristic
182#1278C2×C10911yescharacteristic, complemented
183C109#1017411yescharacteristic, direct product factor, Sylow
184C2×#137—22no—
185#532C22×C10911yescharacteristic
186C2×C4⋊Q8C22×C10911yescharacteristic
187C2×C4⋊Q8C22×C10911yescharacteristic
188#500—22no—
189#500—21no—
190#500—22no—
191#497—22no—
192#494—21no—
193#492—21no—
194#486—21no—
195#480—22no—
196#474—22no—
197#431C22×C10911yescharacteristic
198C2×#107—22no—
199#423C22×C10911yescharacteristic
200C2×#101—22no—
201C4×M16C22×C10911yescharacteristic
202C2×C4×C8C22×C10911yescharacteristic
203#332—22no—
204D4⋊C8—22no—
205C2×#41—21no—
206C22×Q8—21no—
207C22×Q8—21no—
208#137—44no—
209#137—44no—
210C2×Q16—24no—
211C2×SD16—24no—
212C2×M16—21no—
213C2×M16D4×C10912yes—
214C22×C8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8D4×C10912yes—
223C4⋊Q8D4×C10911yescharacteristic
224#127D4×C10911yescharacteristic
225#124—21no—
226#121—21no—
227#121—21no—
228C4×D4—21no—
229C4×D4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4D4×C10912yes—
234C2×C42C23×C10911yescharacteristic
235#107—28no—
236#107—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8D4×C10912yes—
240#102—24no—
241#101—44no—
242C22⋊C8—21no—
243#97—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8D4×C10912yes—
248#41—42no—
249#41—42no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×D4×C10911yescharacteristic
267C22×C4C2×D4×C10911yescharacteristic
268C22×C4#41×C10911yescharacteristic
269Q16—48no—
270SD16—48no—
271M16—22no—
272M16—42no—
273M16D8×C10914yes—
274C2×C8—24no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42C2×D4×C10911yescharacteristic
292C42C2×D4×C10911yescharacteristic
293C42C2×D4×C10911yescharacteristic
294C42C2×D4×C10911yescharacteristic
295C23—41no—
296C23#121×C10911yescharacteristic
297Q8—44no—
298Q8—42no—
299Q8—82no—
300Q8—81no—
301Q8—81no—
302Q8—81no—
303Q8—81no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#120×C10911yescharacteristic
318C2×C4#120×C10911yescharacteristic
319C2×C4#120×C10911yescharacteristic
320C2×C4#121×C10911yescharacteristic
321C2×C4#121×C10911yescharacteristic
322C2×C4#127×C10911yescharacteristic
323C2×C4C2×D8×C10912yes—
324C2×C4AGL(1,ℤ/8)×C10912yes—
325C8—24no—
326C8—24no—
327C8—42no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#392×C10911yescharacteristic
335C22#459×C10912yes—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—81no—
340C4—82no—
341C4#447×C10912yes—
342C4#456×C10912yes—
343C4#473×C10912yes—
344C4#493×C10911yescharacteristic
345C4#496×C10911yescharacteristic
346C2—22no—
347C2—161no—
348C2C109×#200511yescharacteristic
349C2C109×#201111yescharacteristic
350C2C109×#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
2. C2×#500×C109C2semidirect
4. C109×#1966C2semidirect
5. C109×#1546C2semidirect
7. C109×#1278C2semidirect
170. #10174C109direct (N × H)
177. C2×#500C2×C109semidirect
179. #1966C2×C109semidirect
180. #1546C2×C109semidirect
182. #1278C2×C109semidirect
183. C109#10174direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32262
C2×#500×C1092
C109×#31522
C109×#19662
C109×#15462
C109×#14922
C109×#12782
#10174109