Finite Group Databasebeta

← C137×#23565 #46872951

Subgroups — C137×#23565 #46872951

Order 35072.

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 366 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
1C2×#543×C137C211yesmaximal, characteristic
2#2315×C137C213yesmaximal
3#2062×C137C218yesmaximal, complemented
4C2×#342×C137C213yesmaximal
5#543×C137—26no—
6#543×C137C2214yes—
7C2×C22⋊Q8×C137—26no—
8C22×C22⋊C4×C137C2211yescharacteristic
9C22×C22⋊C4×C137C2213yes—
10C22×C22⋊C4×C137C2213yes—
11#385×C137—26no—
12#385×C137C22112yes—
13#380×C137—26no—
14#342×C137C22112yes—
15C25×C137C2311yescharacteristic
16C22×Q8×C137—22no—
17C23×C4×C137—23no—
18C23×C4×C137D416yes—
19C23×C4×C137C2312yes—
20C22⋊Q8×C137—224no—
21C22⋊Q8×C137—224no—
22C2×C4⋊C4×C137—26no—
23C2×C4⋊C4×C137—212no—
24C2×C22⋊C4×C137—23no—
25C2×C22⋊C4×C137—26no—
26C2×C22⋊C4×C137—26no—
27C2×C22⋊C4×C137—26no—
28C2×C22⋊C4×C137—23no—
29C2×C22⋊C4×C137—23no—
30C2×C22⋊C4×C137D416yes—
31C2×C22⋊C4×C137Q8112yescomplemented
32C2×C22⋊C4×C137C2316yes—
33C2×C22⋊C4×C137C2316yes—
34#95×C137—212no—
35C24×C137—82no—
36C24×C137C2×D413yes—
37C24×C137C2×D413yes—
38C24×C137C2×Q816yes—
39C24×C137#4112yes—
40C24×C137C2411yescharacteristic
41C2×Q8×C137—28no—
42C2×Q8×C137—84no—
43C22×C4×C137—26no—
44C22×C4×C137—26no—
45C22×C4×C137—26no—
46C22×C4×C137—212no—
47C22×C4×C137—26no—
48C22×C4×C137—82no—
49C22×C4×C137—86no—
50C22×C4×C137—86no—
51C22×C4×C137C2×D4112yes—
52C4⋊C4×C137—224no—
53C4⋊C4×C137—812no—
54C22⋊C4×C137—212no—
55C22⋊C4×C137—212no—
56C22⋊C4×C137—212no—
57C22⋊C4×C137—26no—
58C22⋊C4×C137—43no—
59C22⋊C4×C137—41no—
60C22⋊C4×C137C2×Q8112yescomplemented
61C23×C137—46no—
62C23×C137—46no—
63C23×C137—42no—
64C23×C137—86no—
65C23×C137—82no—
66C23×C137#12013yes—
67C23×C137#12011yescharacteristic
68C23×C137C22⋊Q816yes—
69C23×C137C22⋊Q816yes—
70C23×C137C4⋊Q816yes—
71C23×C137C4⋊Q816yes—
72C23×C137C22×D413yes—
73C23×C137C22×Q813yes—
74C23×C137C2×#4111yescharacteristic
75Q8×C137—88no—
76C2×C4×C137—212no—
77C2×C4×C137—212no—
78C2×C4×C137—412no—
79C2×C4×C137—412no—
80C2×C4×C137—412no—
81C2×C4×C137—412no—
82C2×C4×C137—42no—
83C2×C4×C137—86no—
84C2×C4×C137—812no—
85C22×C137—22no—
86C22×C137—26no—
87C22×C137—26no—
88C22×C137—46no—
89C22×C137—46no—
90C22×C137—46no—
91C22×C137—46no—
92C22×C137—43no—
93C22×C137—41no—
94C22×C137#39516yes—
95C22×C137#39512yes—
96C22×C137C2×#12011yescharacteristic
97C22×C137C2×C22⋊Q813yes—
98C22×C137C2×C4⋊Q813yes—
99C4×C137—412no—
100C4×C137—412no—
101C4×C137—82no—
102C2×C137—23no—
103C2×C137—23no—
104C2×C137—22no—
105C2×C137—26no—
106C2×C137C2×#39511yescharacteristic
107C2×C137#302612yes—
108#23565C13711yesmaximal, characteristic, direct product factor, Sylow
109C137#2356511yescharacteristic, direct product factor, Sylow
110C2×#543C2×C13711yescharacteristic
111#2315C2×C13713yes—
112#2062C2×C13718yescomplemented
113C2×#342C2×C13713yes—
114#543—26no—
115#543C22×C13714yes—
116C2×C22⋊Q8—26no—
117C22×C22⋊C4C22×C13711yescharacteristic
118C22×C22⋊C4C22×C13713yes—
119C22×C22⋊C4C22×C13713yes—
120#385—26no—
121#385C22×C137112yes—
122#380—26no—
123#342C22×C137112yes—
124C25C23×C13711yescharacteristic
125C22×Q8—22no—
126C23×C4—23no—
127C23×C4D4×C13716yes—
128C23×C4C23×C13712yes—
129C22⋊Q8—224no—
130C22⋊Q8—224no—
131C2×C4⋊C4—26no—
132C2×C4⋊C4—212no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4—23no—
136C2×C22⋊C4—26no—
137C2×C22⋊C4—23no—
138C2×C22⋊C4—23no—
139C2×C22⋊C4D4×C13716yes—
140C2×C22⋊C4Q8×C137112yescomplemented
141C2×C22⋊C4C23×C13716yes—
142C2×C22⋊C4C23×C13716yes—
143#95—212no—
144C24—82no—
145C24C2×D4×C13713yes—
146C24C2×D4×C13713yes—
147C24C2×Q8×C13716yes—
148C24#41×C13712yes—
149C24C24×C13711yescharacteristic
150C2×Q8—28no—
151C2×Q8—84no—
152C22×C4—26no—
153C22×C4—26no—
154C22×C4—26no—
155C22×C4—212no—
156C22×C4—26no—
157C22×C4—82no—
158C22×C4—86no—
159C22×C4—86no—
160C22×C4C2×D4×C137112yes—
161C4⋊C4—224no—
162C4⋊C4—812no—
163C22⋊C4—212no—
164C22⋊C4—212no—
165C22⋊C4—212no—
166C22⋊C4—26no—
167C22⋊C4—43no—
168C22⋊C4—41no—
169C22⋊C4C2×Q8×C137112yescomplemented
170C23—46no—
171C23—46no—
172C23—42no—
173C23—86no—
174C23—82no—
175C23#120×C13713yes—
176C23#120×C13711yescharacteristic
177C23C22⋊Q8×C13716yes—
178C23C22⋊Q8×C13716yes—
179C23C4⋊Q8×C13716yes—
180C23C4⋊Q8×C13716yes—
181C23C22×D4×C13713yes—
182C23C22×Q8×C13713yes—
183C23C2×#41×C13711yescharacteristic
184Q8—88no—
185C2×C4—212no—
186C2×C4—212no—
187C2×C4—412no—
188C2×C4—412no—
189C2×C4—412no—
190C2×C4—412no—
191C2×C4—42no—
192C2×C4—86no—
193C2×C4—812no—
194C22—22no—
195C22—26no—
196C22—26no—
197C22—46no—
198C22—46no—
199C22—46no—
200C22—46no—
201C22—43no—
202C22—41no—
203C22#395×C13716yes—
204C22#395×C13712yes—
205C22C2×#120×C13711yescharacteristic
206C22C2×C22⋊Q8×C13713yes—
207C22C2×C4⋊Q8×C13713yes—
208C4—412no—
209C4—412no—
210C4—82no—
211C2—23no—
212C2—22no—
213C2—26no—
214C2—23no—
215C2C2×#395×C13711yescharacteristic
216C2#3026×C13712yes—
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radicalC25×C137C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C25×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #2062×C137C2semidirect
31. C2×C22⋊C4×C137Q8semidirect
60. C22⋊C4×C137C2×Q8semidirect
108. #23565C137direct (N × H)
109. C137#23565direct (N × H)
112. #2062C2×C137semidirect
140. C2×C22⋊C4Q8×C137semidirect
169. C22⋊C4C2×Q8×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#543×C1372
#2315×C1372
#2062×C1372
C2×#342×C1372
#23565137