Finite Group Databasebeta

← C109×#12315 #42130309

Subgroups — C109×#12315 #42130309

Order 27904.

362 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3291C211yesmaximal, characteristic
2C109×#2214C212yesmaximal, complemented
3C109×#1722C211yesmaximal, characteristic, complemented
4C109×#1359C211yesmaximal, characteristic, complemented
5C109×#1336C212yesmaximal, complemented
6#577×C109—22no—
7#510×C109—22no—
8#500×C109—21no—
9#494×C109C2211yescharacteristic, complemented
10#492×C109C2211yescharacteristic
11#486×C109—21no—
12#471×C109—22no—
13#456×C109—21no—
14#453×C109—21no—
15#443×C109C2212yes—
16#364×C109C2212yescomplemented
17#362×C109—22no—
18#322×C109C2211yescharacteristic, complemented
192−1+4×C109—21no—
20H(5,𝔽2)×C109—21no—
21#137×C109—22no—
22#137×C109—42no—
23AGL(1,ℤ/8)×C109—22no—
24AGL(1,ℤ/8)×C109—42no—
25#135×C109—42no—
26#135×C109D412yes—
27C2×Q16×C109—22no—
28C2×SD16×C109—22no—
29C2×SD16×C109—42no—
30#131×C109—22no—
31C4⋊Q8×C109—21no—
32C4⋊Q8×C109D411yescharacteristic
33#127×C109D411yescharacteristic
34Q32×C109—42no—
35SD32×C109—42no—
36M32×C109—22no—
37#108×C109D412yes—
38#107×C109—42no—
39#104×C109—22no—
40#102×C109—42no—
41#101×C109—21no—
42#100×C109—21no—
43C4×C8×C109—21no—
44C4×C8×C109C2311yescharacteristic
45#41×C109—22no—
46#41×C109—42no—
47#41×C109—42no—
48#41×C109—81no—
49C2×Q8×C109—21no—
50C2×Q8×C109—42no—
51C2×Q8×C109—41no—
52C2×D4×C109—21no—
53C2×D4×C109—41no—
54C2×D4×C109—42no—
55C2×D4×C109—81no—
56Q16×C109—42no—
57Q16×C109—42no—
58Q16×C109—42no—
59Q16×C109D812yes—
60SD16×C109—22no—
61SD16×C109—42no—
62SD16×C109—42no—
63SD16×C109—42no—
64SD16×C109—42no—
65D8×C109—42no—
66D8×C109D812yes—
67M16×C109—22no—
68M16×C109—42no—
69C2×C8×C109—22no—
70C2×C8×C109—42no—
71C2×C8×C109C2×D412yes—
72C4⋊C4×C109—41no—
73C4⋊C4×C109—41no—
74C4⋊C4×C109—41no—
75C42×C109C2×D411yescharacteristic
76C16×C109—42no—
77C23×C109—41no—
78C23×C109—81no—
79Q8×C109—22no—
80Q8×C109—42no—
81Q8×C109—81no—
82Q8×C109—82no—
83D4×C109—22no—
84D4×C109—42no—
85D4×C109—42no—
86D4×C109—82no—
87D4×C109—82no—
88D4×C109—81no—
89C2×C4×C109—21no—
90C2×C4×C109—42no—
91C2×C4×C109—41no—
92C2×C4×C109—81no—
93C2×C4×C109—82no—
94C2×C4×C109—81no—
95C2×C4×C109#12011yescharacteristic
96C8×C109—42no—
97C8×C109—42no—
98C8×C109C2×D812yes—
99C8×C109AGL(1,ℤ/8)12yes—
100C22×C109—42no—
101C22×C109—81no—
102C22×C109—161no—
103C22×C109—321no—
104C22×C109#45311yescharacteristic
105C4×C109—22no—
106C4×C109—42no—
107C4×C109—81no—
108C4×C109—161no—
109C4×C109#44712yes—
110#12315C10911yesmaximal, characteristic, direct product factor, Sylow
111C2×C109—21no—
112C2×C109—82no—
113C2×C109—161no—
114C2×C109#161911yescharacteristic
115#3291C2×C10911yescharacteristic
116#2214C2×C10912yescomplemented
117#1722C2×C10911yescharacteristic, complemented
118#1359C2×C10911yescharacteristic, complemented
119#1336C2×C10912yescomplemented
120C109#1231511yescharacteristic, direct product factor, Sylow
121#577—22no—
122#510—22no—
123#500—21no—
124#494C22×C10911yescharacteristic, complemented
125#492C22×C10911yescharacteristic
126#486—21no—
127#471—22no—
128#456—21no—
129#453—21no—
130#443C22×C10912yes—
131#364C22×C10912yescomplemented
132#362—22no—
133#322C22×C10911yescharacteristic, complemented
1342−1+4—21no—
135H(5,𝔽2)—21no—
136#137—22no—
137#137—42no—
138AGL(1,ℤ/8)—22no—
139AGL(1,ℤ/8)—42no—
140#135—42no—
141#135D4×C10912yes—
142C2×Q16—22no—
143C2×SD16—22no—
144C2×SD16—42no—
145#131—22no—
146C4⋊Q8—21no—
147C4⋊Q8D4×C10911yescharacteristic
148#127D4×C10911yescharacteristic
149Q32—42no—
150SD32—42no—
151M32—22no—
152#108D4×C10912yes—
153#107—42no—
154#104—22no—
155#102—42no—
156#101—21no—
157#100—21no—
158C4×C8—21no—
159C4×C8C23×C10911yescharacteristic
160#41—22no—
161#41—42no—
162#41—42no—
163#41—81no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—81no—
171Q16—42no—
172Q16—42no—
173Q16—42no—
174Q16D8×C10912yes—
175SD16—22no—
176SD16—42no—
177SD16—42no—
178SD16—42no—
179SD16—42no—
180D8—42no—
181D8D8×C10912yes—
182M16—22no—
183M16—42no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8C2×D4×C10912yes—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C42C2×D4×C10911yescharacteristic
191C16—42no—
192C23—41no—
193C23—81no—
194Q8—22no—
195Q8—42no—
196Q8—81no—
197Q8—82no—
198D4—22no—
199D4—42no—
200D4—42no—
201D4—82no—
202D4—82no—
203D4—81no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4#120×C10911yescharacteristic
211C8—42no—
212C8—42no—
213C8C2×D8×C10912yes—
214C8AGL(1,ℤ/8)×C10912yes—
215C22—42no—
216C22—81no—
217C22—161no—
218C22—321no—
219C22#453×C10911yescharacteristic
220C4—22no—
221C4—42no—
222C4—81no—
223C4—161no—
224C4#447×C10912yes—
225C2—21no—
226C2—82no—
227C2—161no—
228C2C109×#161911yescharacteristic
subgroupG/H
center, socleC2×C109#1619
commutator, Frattini, cosocleC4×C8C23×C109
prime radicalC109×#3291C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C4×C8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2214C2semidirect
3. C109×#1722C2semidirect
4. C109×#1359C2semidirect
5. C109×#1336C2semidirect
9. #494×C109C22semidirect
16. #364×C109C22semidirect
18. #322×C109C22semidirect
110. #12315C109direct (N × H)
116. #2214C2×C109semidirect
117. #1722C2×C109semidirect
118. #1359C2×C109semidirect
119. #1336C2×C109semidirect
120. C109#12315direct (N × H)
124. #494C22×C109semidirect
131. #364C22×C109semidirect
133. #322C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32912
C109×#22142
C109×#17222
C109×#13592
C109×#13362
#12315109