Finite Group Databasebeta

← C137×#10368 #46859754

Subgroups — C137×#10368 #46859754

Order 35072.

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

normal subgroups only

This group has 402 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
1#2864×C137C211yesmaximal, characteristic, complemented
2#1962×C137C211yesmaximal, characteristic
3#1517×C137C212yesmaximal
4#1286×C137C212yesmaximal, complemented
5#1276×C137C211yesmaximal, characteristic, complemented
6#532×C137C2211yescharacteristic
7#431×C137C2212yes—
8#423×C137C2211yescharacteristic
9C2×#104×C137—22no—
10#389×C137—22no—
11#387×C137—24no—
12#382×C137C2212yes—
13C43×C137C2211yescharacteristic
14#327×C137—24no—
15C2×#41×C137—21no—
16C2×M16×C137—21no—
17C4⋊Q8×C137Q811yescharacteristic
18#127×C137Q811yescharacteristic
19#124×C137—21no—
20#121×C137—22no—
21C4×D4×C137—22no—
22C2×C4⋊C4×C137—24no—
23C2×C4⋊C4×C137—22no—
24C2×C42×C137—22no—
25C2×C42×C137D412yes—
26C2×C42×C137C2311yescharacteristic
27C4⋊C8×C137—22no—
28C4⋊C8×C137—22no—
29#104×C137—44no—
30C22⋊C8×C137—22no—
31C4×C8×C137—22no—
32#95×C137—24no—
33#95×C137—22no—
34#95×C137D414yes—
35#41×C137—42no—
36#41×C137—42no—
37C2×Q8×C137—21no—
38C2×D4×C137—21no—
39C2×D4×C137—22no—
40C2×D4×C137—41no—
41C22×C4×C137—22no—
42C22×C4×C137—24no—
43C22×C4×C137—41no—
44C22×C4×C137C2×D412yes—
45C22×C4×C137C2×D411yescharacteristic
46M16×C137—42no—
47C2×C8×C137—42no—
48C2×C8×C137—41no—
49C4⋊C4×C137—22no—
50C4⋊C4×C137—48no—
51C4⋊C4×C137—42no—
52C4⋊C4×C137—42no—
53C22⋊C4×C137—42no—
54C42×C137—22no—
55C42×C137—22no—
56C42×C137—44no—
57C42×C137C2×Q811yescharacteristic
58C42×C137#4111yescharacteristic
59C42×C137#4112yes—
60C23×C137—41no—
61C23×C137#12011yescharacteristic
62Q8×C137—42no—
63D4×C137—42no—
64D4×C137—82no—
65D4×C137—82no—
66C2×C4×C137—24no—
67C2×C4×C137—42no—
68C2×C4×C137—42no—
69C2×C4×C137—42no—
70C2×C4×C137—48no—
71C2×C4×C137—41no—
72C2×C4×C137—44no—
73C2×C4×C137—81no—
74C2×C4×C137—81no—
75C2×C4×C137C22⋊Q812yes—
76C2×C4×C137C22⋊Q811yescharacteristic
77C2×C4×C137#12412yes—
78C2×C4×C137#12411yescharacteristic
79C2×C4×C137#13512yes—
80C2×C4×C137AGL(1,ℤ/8)12yes—
81C8×C137—82no—
82C8×C137—81no—
83C22×C137—22no—
84C22×C137—81no—
85C22×C137—82no—
86C22×C137#39311yescharacteristic
87C22×C137#44912yes—
88C4×C137—22no—
89C4×C137—22no—
90C4×C137—44no—
91C4×C137—84no—
92C4×C137—81no—
93C4×C137#47812yes—
94C4×C137#48811yescharacteristic
95C4×C137#49011yescharacteristic
96C2×C137—22no—
97C2×C137—161no—
98C2×C137#202511yescharacteristic
99C2×C137#203112yes—
100#10368C13711yesmaximal, characteristic, direct product factor, Sylow
101C137#1036811yescharacteristic, direct product factor, Sylow
102#2864C2×C13711yescharacteristic, complemented
103#1962C2×C13711yescharacteristic
104#1517C2×C13712yes—
105#1286C2×C13712yescomplemented
106#1276C2×C13711yescharacteristic, complemented
107#532C22×C13711yescharacteristic
108#431C22×C13712yes—
109#423C22×C13711yescharacteristic
110C2×#104—22no—
111#389—22no—
112#387—24no—
113#382C22×C13712yes—
114C43C22×C13711yescharacteristic
115#327—24no—
116C2×#41—21no—
117C2×M16—21no—
118C4⋊Q8Q8×C13711yescharacteristic
119#127Q8×C13711yescharacteristic
120#124—21no—
121#121—22no—
122C4×D4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42D4×C13712yes—
127C2×C42C23×C13711yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—22no—
130#104—44no—
131C22⋊C8—22no—
132C4×C8—22no—
133#95—24no—
134#95—22no—
135#95D4×C13714yes—
136#41—42no—
137#41—42no—
138C2×Q8—21no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—41no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—41no—
145C22×C4C2×D4×C13712yes—
146C22×C4C2×D4×C13711yescharacteristic
147M16—42no—
148C2×C8—42no—
149C2×C8—41no—
150C4⋊C4—22no—
151C4⋊C4—48no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—44no—
158C42C2×Q8×C13711yescharacteristic
159C42#41×C13711yescharacteristic
160C42#41×C13712yes—
161C23—41no—
162C23#120×C13711yescharacteristic
163Q8—42no—
164D4—42no—
165D4—82no—
166D4—82no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—48no—
172C2×C4—41no—
173C2×C4—44no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4C22⋊Q8×C13712yes—
177C2×C4C22⋊Q8×C13711yescharacteristic
178C2×C4#124×C13712yes—
179C2×C4#124×C13711yescharacteristic
180C2×C4#135×C13712yes—
181C2×C4AGL(1,ℤ/8)×C13712yes—
182C8—82no—
183C8—81no—
184C22—22no—
185C22—81no—
186C22—82no—
187C22#393×C13711yescharacteristic
188C22#449×C13712yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—81no—
194C4#478×C13712yes—
195C4#488×C13711yescharacteristic
196C4#490×C13711yescharacteristic
197C2—22no—
198C2—161no—
199C2#2025×C13711yescharacteristic
200C2#2031×C13712yes—
subgroupG/H
center, socleC22×C137#393
commutator, Frattini, cosocleC2×C42C23×C137
prime radical#532×C137C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2864×C137C2semidirect
4. #1286×C137C2semidirect
5. #1276×C137C2semidirect
100. #10368C137direct (N × H)
101. C137#10368direct (N × H)
102. #2864C2×C137semidirect
105. #1286C2×C137semidirect
106. #1276C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2864×C1372
#1962×C1372
#1517×C1372
#1286×C1372
#1276×C1372
#10368137