Finite Group Databasebeta

← C109×#10336 #42128330

Subgroups — C109×#10336 #42128330

Order 27904.

1566 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2403C213yesmaximal, complemented
2C109×#1781C213yesmaximal, complemented
3C109×#1304C211yesmaximal, characteristic, complemented
4C23×D4×C109C2213yes—
5C2×#120×C109—26no—
6C2×#99×C109—26no—
7#392×C109—26no—
8#386×C109—26no—
9#379×C109C2211yescharacteristic
10#379×C109C2213yes—
11C25×C109D413yescomplemented
12C25×C109C2311yescharacteristic
13C22×D4×C109—26no—
14C22×D4×C109—23no—
15C22×D4×C109—26no—
16C22×D4×C109—23no—
17C22×D4×C109—26no—
18C22×D4×C109—412no—
19C23×C4×C109D413yescomplemented
20#120×C109—424no—
21C2×C22⋊C4×C109—212no—
22C2×C22⋊C4×C109—26no—
23C2×C22⋊C4×C109—23no—
24C2×C22⋊C4×C109D416yescomplemented
25#99×C109—412no—
26#95×C109—26no—
27C24×C109—26no—
28C24×C109—23no—
29C24×C109—46no—
30C24×C109—46no—
31C24×C109—412no—
32C24×C109C2×D413yes—
33C24×C109C2×D413yes—
34C24×C109#4111yescharacteristic
35C2×D4×C109—212no—
36C2×D4×C109—224no—
37C2×D4×C109—412no—
38C2×D4×C109—46no—
39C2×D4×C109—412no—
40C2×D4×C109—46no—
41C2×D4×C109—412no—
42C2×D4×C109—46no—
43C2×D4×C109—43no—
44C2×D4×C109—46no—
45C2×D4×C109—43no—
46C2×D4×C109—424no—
47C2×D4×C109—86no—
48C22×C4×C109—26no—
49C22×C4×C109—23no—
50C22×C4×C109—26no—
51C22×C4×C109—46no—
52C22×C4×C109—41no—
53C22⋊C4×C109—424no—
54C22⋊C4×C109—412no—
55C22⋊C4×C109—46no—
56C22⋊C4×C109—46no—
57C23×C109—212no—
58C23×C109—26no—
59C23×C109—212no—
60C23×C109—26no—
61C23×C109—43no—
62C23×C109—46no—
63C23×C109—43no—
64C23×C109—412no—
65C23×C109—46no—
66C23×C109—43no—
67C23×C109—43no—
68C23×C109—412no—
69C23×C109—86no—
70C23×C109—88no—
71C23×C109—824no—
72C23×C109#12013yes—
73C23×C109#12113yes—
74C23×C109#12711yescharacteristic
75D4×C109—412no—
76D4×C109—424no—
77D4×C109—412no—
78D4×C109—824no—
79C2×C4×C109—212no—
80C2×C4×C109—26no—
81C2×C4×C109—46no—
82C2×C4×C109—43no—
83C2×C4×C109—43no—
84C2×C4×C109—412no—
85C2×C4×C109—46no—
86C2×C4×C109—83no—
87C22×C109—26no—
88C22×C109—23no—
89C22×C109—23no—
90C22×C109—26no—
91C22×C109—412no—
92C22×C109—424no—
93C22×C109—46no—
94C22×C109—86no—
95C22×C109—812no—
96C22×C109—82no—
97C22×C109—86no—
98C22×C109—86no—
99C22×C109—86no—
100C22×C109#39211yescharacteristic
101C22×C109B(4,𝔽2)16yes—
102C4×C109—46no—
103C4×C109—86no—
104C4×C109—161no—
105#10336C10911yesmaximal, characteristic, direct product factor, Sylow
106C2×C109—26no—
107C2×C109—41no—
108C2×C109—43no—
109C2×C109—86no—
110C2×C109#202113yes—
111#2403C2×C10913yescomplemented
112#1781C2×C10913yescomplemented
113#1304C2×C10911yescharacteristic, complemented
114C109#1033611yescharacteristic, direct product factor, Sylow
115C23×D4C22×C10913yes—
116C2×#120—26no—
117C2×#99—26no—
118#392—26no—
119#386—26no—
120#379C22×C10911yescharacteristic
121#379C22×C10913yes—
122C25D4×C10913yescomplemented
123C25C23×C10911yescharacteristic
124C22×D4—26no—
125C22×D4—23no—
126C22×D4—26no—
127C22×D4—23no—
128C22×D4—26no—
129C22×D4—412no—
130C23×C4D4×C10913yescomplemented
131#120—424no—
132C2×C22⋊C4—212no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—23no—
135C2×C22⋊C4D4×C10916yescomplemented
136#99—412no—
137#95—26no—
138C24—26no—
139C24—23no—
140C24—46no—
141C24—46no—
142C24—412no—
143C24C2×D4×C10913yes—
144C24C2×D4×C10913yes—
145C24#41×C10911yescharacteristic
146C2×D4—212no—
147C2×D4—224no—
148C2×D4—412no—
149C2×D4—46no—
150C2×D4—412no—
151C2×D4—46no—
152C2×D4—412no—
153C2×D4—46no—
154C2×D4—43no—
155C2×D4—46no—
156C2×D4—43no—
157C2×D4—424no—
158C2×D4—86no—
159C22×C4—26no—
160C22×C4—23no—
161C22×C4—26no—
162C22×C4—46no—
163C22×C4—41no—
164C22⋊C4—424no—
165C22⋊C4—412no—
166C22⋊C4—46no—
167C22⋊C4—46no—
168C23—212no—
169C23—26no—
170C23—212no—
171C23—26no—
172C23—43no—
173C23—46no—
174C23—43no—
175C23—412no—
176C23—46no—
177C23—43no—
178C23—43no—
179C23—412no—
180C23—86no—
181C23—88no—
182C23—824no—
183C23#120×C10913yes—
184C23#121×C10913yes—
185C23#127×C10911yescharacteristic
186D4—412no—
187D4—424no—
188D4—412no—
189D4—824no—
190C2×C4—212no—
191C2×C4—26no—
192C2×C4—46no—
193C2×C4—43no—
194C2×C4—43no—
195C2×C4—412no—
196C2×C4—46no—
197C2×C4—83no—
198C22—26no—
199C22—23no—
200C22—23no—
201C22—26no—
202C22—412no—
203C22—424no—
204C22—46no—
205C22—86no—
206C22—812no—
207C22—82no—
208C22—86no—
209C22—86no—
210C22—86no—
211C22#392×C10911yescharacteristic
212C22B(4,𝔽2)×C10916yes—
213C4—46no—
214C4—86no—
215C4—161no—
216C2—26no—
217C2—41no—
218C2—43no—
219C2—86no—
220C2C109×#202113yes—
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC25C23×C109
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C25×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#2403C2semidirect
2. C109×#1781C2semidirect
3. C109×#1304C2semidirect
11. C25×C109D4semidirect
19. C23×C4×C109D4semidirect
24. C2×C22⋊C4×C109D4semidirect
105. #10336C109direct (N × H)
111. #2403C2×C109semidirect
112. #1781C2×C109semidirect
113. #1304C2×C109semidirect
114. C109#10336direct (N × H)
122. C25D4×C109semidirect
130. C23×C4D4×C109semidirect
135. C2×C22⋊C4D4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#24032
C109×#17812
C109×#13042
#10336109