Finite Group Databasebeta

← C137×#10363 #46859749

Subgroups — C137×#10363 #46859749

Order 35072.

416 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 114 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#2862×C137C211yesmaximal, characteristic
2#1994×C137C211yesmaximal, characteristic
3#1518×C137C212yesmaximal
4#1286×C137C212yesmaximal
5#1276×C137C211yesmaximal, characteristic
6#533×C137C2211yescharacteristic
7C2×C4⋊Q8×C137—21no—
8#431×C137C2212yes—
9#423×C137C2211yescharacteristic
10#391×C137—24no—
11#382×C137—21no—
12#382×C137C2212yes—
13C43×C137C2211yescharacteristic
14#339×C137—21no—
15#339×C137—21no—
16#328×C137—24no—
17C22×Q8×C137—22no—
18C2×M16×C137—21no—
19C4⋊Q8×C137—42no—
20C4⋊Q8×C137—42no—
21C4⋊Q8×C137Q812yescomplemented
22#125×C137—21no—
23C22⋊Q8×C137—22no—
24C4×Q8×C137—22no—
25#117×C137—21no—
26C2×C4⋊C4×C137—22no—
27C2×C4⋊C4×C137—22no—
28C2×C42×C137—22no—
29C2×C42×C137D411yescharacteristic
30C2×C42×C137D411yescharacteristic
31C2×C42×C137C2311yescharacteristic
32C4⋊C8×C137—22no—
33C4⋊C8×C137—22no—
34C22⋊C8×C137—22no—
35C4×C8×C137—22no—
36#95×C137—24no—
37#95×C137—22no—
38#95×C137D414yes—
39C2×Q8×C137—22no—
40C2×Q8×C137—44no—
41C2×Q8×C137—48no—
42C22×C4×C137—21no—
43C22×C4×C137—21no—
44C22×C4×C137—24no—
45C22×C4×C137C2×D412yes—
46C22×C4×C137C2×D411yescharacteristic
47M16×C137—41no—
48M16×C137—41no—
49C2×C8×C137—42no—
50C2×C8×C137—41no—
51C4⋊C4×C137—22no—
52C4⋊C4×C137—22no—
53C4⋊C4×C137—44no—
54C4⋊C4×C137—42no—
55C4⋊C4×C137—42no—
56C4⋊C4×C137—42no—
57C22⋊C4×C137—41no—
58C42×C137—21no—
59C42×C137—21no—
60C42×C137—44no—
61C42×C137—41no—
62C42×C137SD1612yes—
63C42×C137Q1612yes—
64C42×C137C2×Q811yescharacteristic
65C42×C137#4111yescharacteristic
66C42×C137#4112yes—
67C23×C137#12011yescharacteristic
68Q8×C137—82no—
69Q8×C137—88no—
70C2×C4×C137—24no—
71C2×C4×C137—22no—
72C2×C4×C137—22no—
73C2×C4×C137—42no—
74C2×C4×C137—42no—
75C2×C4×C137—48no—
76C2×C4×C137—42no—
77C2×C4×C137—44no—
78C2×C4×C137C22⋊Q812yes—
79C2×C4×C137C22⋊Q811yescharacteristic
80C2×C4×C137#12412yes—
81C2×C4×C137#12411yescharacteristic
82C2×C4×C137C2×SD1611yescharacteristic
83C2×C4×C137C2×Q1611yescharacteristic
84C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
85C2×C4×C137#13711yescharacteristic
86C8×C137—82no—
87C8×C137—81no—
88C22×C137—22no—
89C22×C137#39311yescharacteristic
90C22×C137#45011yescharacteristic
91C22×C137#45111yescharacteristic
92C4×C137—24no—
93C4×C137—22no—
94C4×C137—22no—
95C4×C137—42no—
96C4×C137—84no—
97C4×C137—82no—
98C4×C137#47511yescharacteristic
99C4×C137#47711yescharacteristic
100C4×C137#48711yescharacteristic
101C4×C137#48911yescharacteristic
102C2×C137—21no—
103C2×C137—21no—
104C2×C137#202411yescharacteristic
105C2×C137#203112yes—
106#10363C13711yesmaximal, characteristic, direct product factor, Sylow
107C137#1036311yescharacteristic, direct product factor, Sylow
108#2862C2×C13711yescharacteristic
109#1994C2×C13711yescharacteristic
110#1518C2×C13712yes—
111#1286C2×C13712yes—
112#1276C2×C13711yescharacteristic
113#533C22×C13711yescharacteristic
114C2×C4⋊Q8—21no—
115#431C22×C13712yes—
116#423C22×C13711yescharacteristic
117#391—24no—
118#382—21no—
119#382C22×C13712yes—
120C43C22×C13711yescharacteristic
121#339—21no—
122#339—21no—
123#328—24no—
124C22×Q8—22no—
125C2×M16—21no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8Q8×C13712yescomplemented
129#125—21no—
130C22⋊Q8—22no—
131C4×Q8—22no—
132#117—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C42—22no—
136C2×C42D4×C13711yescharacteristic
137C2×C42D4×C13711yescharacteristic
138C2×C42C23×C13711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—22no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—24no—
144#95—22no—
145#95D4×C13714yes—
146C2×Q8—22no—
147C2×Q8—44no—
148C2×Q8—48no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4C2×D4×C13712yes—
153C22×C4C2×D4×C13711yescharacteristic
154M16—41no—
155M16—41no—
156C2×C8—42no—
157C2×C8—41no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—44no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C22⋊C4—41no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—41no—
169C42SD16×C13712yes—
170C42Q16×C13712yes—
171C42C2×Q8×C13711yescharacteristic
172C42#41×C13711yescharacteristic
173C42#41×C13712yes—
174C23#120×C13711yescharacteristic
175Q8—82no—
176Q8—88no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—48no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4C22⋊Q8×C13712yes—
186C2×C4C22⋊Q8×C13711yescharacteristic
187C2×C4#124×C13712yes—
188C2×C4#124×C13711yescharacteristic
189C2×C4C2×SD16×C13711yescharacteristic
190C2×C4C2×Q16×C13711yescharacteristic
191C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
192C2×C4#137×C13711yescharacteristic
193C8—82no—
194C8—81no—
195C22—22no—
196C22#393×C13711yescharacteristic
197C22#450×C13711yescharacteristic
198C22#451×C13711yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—84no—
204C4—82no—
205C4#475×C13711yescharacteristic
206C4#477×C13711yescharacteristic
207C4#487×C13711yescharacteristic
208C4#489×C13711yescharacteristic
209C2—21no—
210C2—21no—
211C2#2024×C13711yescharacteristic
212C2#2031×C13712yes—
subgroupG/H
center, socleC22×C137#393
commutator, Frattini, cosocleC2×C42C23×C137
prime radicalC23×C137#120
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
21. C4⋊Q8×C137Q8semidirect
106. #10363C137direct (N × H)
107. C137#10363direct (N × H)
128. C4⋊Q8Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#2862×C1372
#1994×C1372
#1518×C1372
#1286×C1372
#1276×C1372
#10363137