Finite Group Databasebeta

← #2011×C173 #36349796

Subgroups — #2011×C173 #36349796

Order 22144.

510 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C22×D8×C173C212yesmaximal, complemented
2C2×#127×C173C211yesmaximal, characteristic, complemented
3C2×#102×C173C212yesmaximal, complemented
4#390×C173C211yesmaximal, characteristic, complemented
5#336×C173C211yesmaximal, characteristic, complemented
6C22×D4×C173—22no—
7C22×D4×C173C2212yes—
8C22×D4×C173C2211yescharacteristic
9C2×D8×C173—24no—
10C2×D8×C173—24no—
11C2×D8×C173—24no—
12C22×C8×C173C2212yes—
13#127×C173—24no—
14C2×C4⋊C4×C173C2211yescharacteristic
15C2×C22⋊C4×C173—22no—
16C2×C42×C173C2211yescharacteristic
17#102×C173—24no—
18C24×C173—22no—
19C24×C173—21no—
20C2×D4×C173—28no—
21C2×D4×C173—22no—
22C2×D4×C173—22no—
23C2×D4×C173—22no—
24C2×D4×C173—22no—
25C2×D4×C173—22no—
26C2×D4×C173—22no—
27C2×D4×C173—22no—
28C2×D4×C173—21no—
29C2×D4×C173—44no—
30C2×D4×C173—44no—
31C2×D4×C173D414yes—
32C2×D4×C173D412yescomplemented
33C22×C4×C173—21no—
34C22×C4×C173D412yes—
35C22×C4×C173C2311yescharacteristic
36D8×C173—216no—
37C2×C8×C173—22no—
38C2×C8×C173—22no—
39C2×C8×C173D414yescomplemented
40C4⋊C4×C173—22no—
41C22⋊C4×C173—44no—
42C42×C173—22no—
43C23×C173—24no—
44C23×C173—22no—
45C23×C173—41no—
46C23×C173—42no—
47C23×C173—41no—
48C23×C173—42no—
49C23×C173—42no—
50C23×C173—44no—
51C23×C173—44no—
52C23×C173—42no—
53C23×C173C2×D411yescharacteristic
54D4×C173—24no—
55D4×C173—24no—
56D4×C173—24no—
57D4×C173—416no—
58D4×C173—42no—
59D4×C173—42no—
60D4×C173—42no—
61C2×C4×C173—22no—
62C2×C4×C173—22no—
63C2×C4×C173—41no—
64C2×C4×C173—42no—
65C2×C4×C173D814yescomplemented
66C2×C4×C173C2×D411yescharacteristic
67C2×C4×C173C2×D412yes—
68C2×C4×C173C2×D412yes—
69C2×C4×C173#4111yescharacteristic
70C8×C173—24no—
71C22×C173—44no—
72C22×C173—42no—
73C22×C173—48no—
74C22×C173—44no—
75C22×C173—82no—
76C22×C173—84no—
77C22×C173—81no—
78C22×C173—82no—
79C22×C173—82no—
80C22×C173—81no—
81C22×C173#12011yescharacteristic
82C22×C173#12111yescharacteristic
83C22×C173#12711yescharacteristic
84C22×C173C2×D812yes—
85C22×C173AGL(1,ℤ/8)12yes—
86C4×C173—24no—
87C4×C173—81no—
88C4×C173#12012yes—
89C4×C173#12112yes—
90C2×C173—84no—
91C2×C173—82no—
92C2×C173#39211yescharacteristic
93C2×C173#44712yes—
94C2×C173#45912yes—
95C2×C173#49311yescharacteristic
96C2×C173#49611yescharacteristic
97C173#201111yescharacteristic, direct product factor, Sylow
98#2011C17311yesmaximal, characteristic, direct product factor, Sylow
99C22×D8C2×C17312yescomplemented
100C2×#127C2×C17311yescharacteristic, complemented
101C2×#102C2×C17312yescomplemented
102#390C2×C17311yescharacteristic, complemented
103#336C2×C17311yescharacteristic, complemented
104C22×D4—22no—
105C22×D4C22×C17312yes—
106C22×D4C22×C17311yescharacteristic
107C2×D8—24no—
108C2×D8—24no—
109C2×D8—24no—
110C22×C8C22×C17312yes—
111#127—24no—
112C2×C4⋊C4C22×C17311yescharacteristic
113C2×C22⋊C4—22no—
114C2×C42C22×C17311yescharacteristic
115#102—24no—
116C24—21no—
117C24—22no—
118C2×D4—28no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4D4×C17314yes—
130C2×D4D4×C17312yescomplemented
131C22×C4—21no—
132C22×C4D4×C17312yes—
133C22×C4C23×C17311yescharacteristic
134D8—216no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8D4×C17314yescomplemented
138C4⋊C4—22no—
139C22⋊C4—44no—
140C42—22no—
141C23—24no—
142C23—22no—
143C23—42no—
144C23—44no—
145C23—44no—
146C23—42no—
147C23—41no—
148C23—42no—
149C23—42no—
150C23—41no—
151C23C2×D4×C17311yescharacteristic
152D4—24no—
153D4—24no—
154D4—24no—
155D4—416no—
156D4—42no—
157D4—42no—
158D4—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4D8×C17314yescomplemented
164C2×C4C2×D4×C17311yescharacteristic
165C2×C4C2×D4×C17312yes—
166C2×C4C2×D4×C17312yes—
167C2×C4#41×C17311yescharacteristic
168C8—24no—
169C22—44no—
170C22—42no—
171C22—48no—
172C22—44no—
173C22—82no—
174C22—84no—
175C22—81no—
176C22—82no—
177C22—82no—
178C22—81no—
179C22#120×C17311yescharacteristic
180C22#121×C17311yescharacteristic
181C22#127×C17311yescharacteristic
182C22C2×D8×C17312yes—
183C22AGL(1,ℤ/8)×C17312yes—
184C4—24no—
185C4—81no—
186C4#120×C17312yes—
187C4#121×C17312yes—
188C2—84no—
189C2—82no—
190C2#392×C17311yescharacteristic
191C2#447×C17312yes—
192C2#459×C17312yes—
193C2#493×C17311yescharacteristic
194C2#496×C17311yescharacteristic
subgroupG/H
center, socleC23×C173C2×D4
commutator, Frattini, cosocleC22×C4C23×C173
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C173 ◁ C2×C42×C173 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C173C2semidirect
2. C2×#127×C173C2semidirect
3. C2×#102×C173C2semidirect
4. #390×C173C2semidirect
5. #336×C173C2semidirect
32. C2×D4×C173D4semidirect
39. C2×C8×C173D4semidirect
65. C2×C4×C173D8semidirect
97. C173#2011direct (N × H)
98. #2011C173direct (N × H)
99. C22×D8C2×C173semidirect
100. C2×#127C2×C173semidirect
101. C2×#102C2×C173semidirect
102. #390C2×C173semidirect
103. #336C2×C173semidirect
130. C2×D4D4×C173semidirect
137. C2×C8D4×C173semidirect
163. C2×C4D8×C173semidirect

Maximal subgroups

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

subgroupindex
C22×D8×C1732
C2×#127×C1732
C2×#102×C1732
#390×C1732
#336×C1732
#2011173