Finite Group Databasebeta

← C137×#10584 #46859970

Subgroups — C137×#10584 #46859970

Order 35072.

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

normal subgroups only

This group has 374 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#2844×C137C211yesmaximal, characteristic
2#1847×C137C211yesmaximal, characteristic
3#1836×C137C212yesmaximal
4#1295×C137C212yesmaximal
5#1295×C137C211yesmaximal, characteristic
6C2×#107×C137—21no—
7C2×#106×C137—21no—
8C2×#97×C137C2212yes—
9C2×#97×C137C2211yescharacteristic
10#400×C137—21no—
11#400×C137—22no—
12#395×C137—21no—
13#395×C137—22no—
14#384×C137C2211yescharacteristic
15#382×C137C2212yes—
16#382×C137C2211yescharacteristic
17#340×C137—22no—
18#340×C137—22no—
19C22×Q8×C137—21no—
20C22×C8×C137—21no—
21C22×C8×C137—22no—
22C2×C4⋊C4×C137—21no—
23C2×C4⋊C4×C137—22no—
24C2×C4⋊C4×C137—21no—
25C2×C4⋊C4×C137—22no—
26C2×C4⋊C4×C137D411yescharacteristic
27C2×C4⋊C4×C137D412yes—
28C2×C4⋊C4×C137D411yescharacteristic
29C2×C4⋊C4×C137D412yes—
30C2×C42×C137C2311yescharacteristic
31#107×C137—42no—
32#106×C137—42no—
33#97×C137—21no—
34#97×C137—22no—
35#97×C137Q812yescomplemented
36#97×C137Q814yescomplemented
37#95×C137—22no—
38#95×C137—21no—
39#95×C137—21no—
40#95×C137—22no—
41#95×C137—22no—
42#95×C137—22no—
43C2×Q8×C137—81no—
44C2×Q8×C137—82no—
45C22×C4×C137—21no—
46C22×C4×C137—22no—
47C22×C4×C137—21no—
48C22×C4×C137—22no—
49C22×C4×C137—41no—
50C22×C4×C137C2×D411yescharacteristic
51C22×C4×C137C2×D412yes—
52C2×C8×C137—22no—
53C2×C8×C137—22no—
54C2×C8×C137—22no—
55C2×C8×C137—24no—
56C2×C8×C137—42no—
57C2×C8×C137—42no—
58C4⋊C4×C137—41no—
59C4⋊C4×C137—41no—
60C4⋊C4×C137—42no—
61C4⋊C4×C137—42no—
62C4⋊C4×C137—41no—
63C4⋊C4×C137—41no—
64C4⋊C4×C137—44no—
65C4⋊C4×C137—81no—
66C4⋊C4×C137—82no—
67C42×C137C2×Q811yescharacteristic
68C42×C137C2×Q812yes—
69C42×C137#4111yescharacteristic
70C23×C137#12011yescharacteristic
71Q8×C137—84no—
72C2×C4×C137—21no—
73C2×C4×C137—22no—
74C2×C4×C137—22no—
75C2×C4×C137—21no—
76C2×C4×C137—41no—
77C2×C4×C137—41no—
78C2×C4×C137—42no—
79C2×C4×C137—42no—
80C2×C4×C137—42no—
81C2×C4×C137—42no—
82C2×C4×C137—42no—
83C2×C4×C137—42no—
84C2×C4×C137—42no—
85C2×C4×C137—42no—
86C2×C4×C137—81no—
87C2×C4×C137—82no—
88C2×C4×C137C22⋊Q811yescharacteristic
89C2×C4×C137C22⋊Q812yes—
90C2×C4×C137C4⋊Q811yescharacteristic
91C2×C4×C137C4⋊Q812yes—
92C8×C137—24no—
93C8×C137—44no—
94C22×C137#39511yescharacteristic
95C22×C137#45311yescharacteristic
96C22×C137#45412yes—
97C22×C137#45512yes—
98C22×C137#45611yescharacteristic
99C4×C137—21no—
100C4×C137—21no—
101C4×C137—22no—
102C4×C137—22no—
103C4×C137—81no—
104C4×C137—81no—
105C4×C137—82no—
106C4×C137—82no—
107C4×C137—161no—
108C2×C137#165711yescharacteristic
109C2×C137#165911yescharacteristic
110C2×C137#166412yes—
111C2×C137#206011yescharacteristic
112C2×C137#206112yes—
113#10584C13711yesmaximal, characteristic, direct product factor, Sylow
114C137#1058411yescharacteristic, direct product factor, Sylow
115#2844C2×C13711yescharacteristic
116#1847C2×C13711yescharacteristic
117#1836C2×C13712yes—
118#1295C2×C13712yes—
119#1295C2×C13711yescharacteristic
120C2×#107—21no—
121C2×#106—21no—
122C2×#97C22×C13712yes—
123C2×#97C22×C13711yescharacteristic
124#400—22no—
125#400—21no—
126#395—22no—
127#395—21no—
128#384C22×C13711yescharacteristic
129#382C22×C13712yes—
130#382C22×C13711yescharacteristic
131#340—22no—
132#340—22no—
133C22×Q8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4D4×C13712yes—
141C2×C4⋊C4D4×C13712yes—
142C2×C4⋊C4D4×C13711yescharacteristic
143C2×C4⋊C4D4×C13711yescharacteristic
144C2×C42C23×C13711yescharacteristic
145#107—42no—
146#106—42no—
147#97—21no—
148#97—22no—
149#97Q8×C13712yescomplemented
150#97Q8×C13714yescomplemented
151#95—22no—
152#95—21no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4C2×D4×C13711yescharacteristic
165C22×C4C2×D4×C13712yes—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C42C2×Q8×C13711yescharacteristic
182C42C2×Q8×C13712yes—
183C42#41×C13711yescharacteristic
184C23#120×C13711yescharacteristic
185Q8—84no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—82no—
202C2×C4C22⋊Q8×C13711yescharacteristic
203C2×C4C22⋊Q8×C13712yes—
204C2×C4C4⋊Q8×C13711yescharacteristic
205C2×C4C4⋊Q8×C13712yes—
206C8—24no—
207C8—44no—
208C22#395×C13711yescharacteristic
209C22#453×C13711yescharacteristic
210C22#454×C13712yes—
211C22#455×C13712yes—
212C22#456×C13711yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—161no—
222C2#1657×C13711yescharacteristic
223C2#1659×C13711yescharacteristic
224C2#1664×C13712yes—
225C2#2060×C13711yescharacteristic
226C2#2061×C13712yes—
subgroupG/H
center, socle, prime radicalC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
35. #97×C137Q8semidirect
36. #97×C137Q8semidirect
113. #10584C137direct (N × H)
114. C137#10584direct (N × H)
149. #97Q8×C137semidirect
150. #97Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#2844×C1372
#1847×C1372
#1836×C1372
#1295×C1372
#10584137