Finite Group Databasebeta

← #2194×C137 #31777205

Subgroups — #2194×C137 #31777205

Order 17536.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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#576×C137C211yesmaximal, characteristic
2#510×C137C211yesmaximal, characteristic, complemented
3C2×SD32×C137C211yesmaximal, characteristic, complemented
4#473×C137C211yesmaximal, characteristic, complemented
5#361×C137C211yesmaximal, characteristic, complemented
6#359×C137C211yesmaximal, characteristic, complemented
7#350×C137C211yesmaximal, characteristic, complemented
8H(5,𝔽2)×C137—21no—
9#137×C137—21no—
10AGL(1,ℤ/8)×C137—22no—
11AGL(1,ℤ/8)×C137—21no—
12#135×C137—21no—
13#135×C137C2211yescharacteristic
14C2×Q16×C137C2211yescharacteristic, complemented
15C2×D8×C137—21no—
16C2×D8×C137C2211yescharacteristic
17#131×C137C2211yescharacteristic
18Q32×C137—21no—
19SD32×C137—22no—
20SD32×C137—21no—
21M32×C137C2211yescharacteristic, complemented
22C2×C16×C137C2211yescharacteristic, complemented
23#108×C137C2211yescharacteristic, complemented
24#101×C137—21no—
25#41×C137—22no—
26#41×C137—21no—
27#41×C137—41no—
28#41×C137D411yescharacteristic
29C2×Q8×C137—21no—
30C2×D4×C137—22no—
31C2×D4×C137—21no—
32C2×D4×C137—42no—
33C2×D4×C137—41no—
34Q16×C137—21no—
35Q16×C137—21no—
36Q16×C137—41no—
37Q16×C137D411yescharacteristic
38SD16×C137—22no—
39SD16×C137—21no—
40SD16×C137—41no—
41D8×C137—22no—
42D8×C137—21no—
43D8×C137D411yescharacteristic
44D8×C137D412yes—
45M16×C137—21no—
46M16×C137—21no—
47M16×C137D411yescharacteristic
48C2×C8×C137—21no—
49C2×C8×C137C2311yescharacteristic
50C16×C137—21no—
51C16×C137—21no—
52C23×C137—42no—
53C23×C137—41no—
54Q8×C137—21no—
55Q8×C137—41no—
56Q8×C137—41no—
57Q8×C137D811yescharacteristic
58D4×C137—21no—
59D4×C137—21no—
60D4×C137—22no—
61D4×C137—42no—
62D4×C137—42no—
63D4×C137—41no—
64D4×C137—41no—
65D4×C137D811yescharacteristic
66C2×C4×C137—21no—
67C2×C4×C137—42no—
68C2×C4×C137—41no—
69C2×C4×C137—41no—
70C2×C4×C137C2×D411yescharacteristic
71C8×C137—21no—
72C8×C137—41no—
73C8×C137C2×D411yescharacteristic
74C8×C137C2×D411yescharacteristic
75C22×C137—21no—
76C22×C137—42no—
77C22×C137—41no—
78C22×C137—161no—
79C22×C137—162no—
80C22×C137AGL(1,ℤ/8)11yescharacteristic
81C4×C137—21no—
82C4×C137—41no—
83C4×C137—81no—
84C4×C137#12011yescharacteristic
85C4×C137C2×D811yescharacteristic
86C2×C137—21no—
87C2×C137—41no—
88C2×C137—82no—
89C2×C137—81no—
90C2×C137#44711yescharacteristic
91C137#219411yescharacteristic, direct product factor, Sylow
92#2194C13711yesmaximal, characteristic, direct product factor, Sylow
93#576C2×C13711yescharacteristic
94#510C2×C13711yescharacteristic, complemented
95C2×SD32C2×C13711yescharacteristic, complemented
96#473C2×C13711yescharacteristic, complemented
97#361C2×C13711yescharacteristic, complemented
98#359C2×C13711yescharacteristic, complemented
99#350C2×C13711yescharacteristic, complemented
100H(5,𝔽2)—21no—
101#137—21no—
102AGL(1,ℤ/8)—22no—
103AGL(1,ℤ/8)—21no—
104#135—21no—
105#135C22×C13711yescharacteristic
106C2×Q16C22×C13711yescharacteristic, complemented
107C2×D8—21no—
108C2×D8C22×C13711yescharacteristic
109#131C22×C13711yescharacteristic
110Q32—21no—
111SD32—22no—
112SD32—21no—
113M32C22×C13711yescharacteristic, complemented
114C2×C16C22×C13711yescharacteristic, complemented
115#108C22×C13711yescharacteristic, complemented
116#101—21no—
117#41—22no—
118#41—21no—
119#41—41no—
120#41D4×C13711yescharacteristic
121C2×Q8—21no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—41no—
126Q16—21no—
127Q16—21no—
128Q16—41no—
129Q16D4×C13711yescharacteristic
130SD16—22no—
131SD16—21no—
132SD16—41no—
133D8—22no—
134D8—21no—
135D8D4×C13711yescharacteristic
136D8D4×C13712yes—
137M16—21no—
138M16—21no—
139M16D4×C13711yescharacteristic
140C2×C8—21no—
141C2×C8C23×C13711yescharacteristic
142C16—21no—
143C16—21no—
144C23—42no—
145C23—41no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8D8×C13711yescharacteristic
150D4—21no—
151D4—21no—
152D4—22no—
153D4—42no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4D8×C13711yescharacteristic
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4C2×D4×C13711yescharacteristic
163C8—21no—
164C8—41no—
165C8C2×D4×C13711yescharacteristic
166C8C2×D4×C13711yescharacteristic
167C22—21no—
168C22—42no—
169C22—41no—
170C22—161no—
171C22—162no—
172C22AGL(1,ℤ/8)×C13711yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4#120×C13711yescharacteristic
177C4C2×D8×C13711yescharacteristic
178C2—21no—
179C2—41no—
180C2—82no—
181C2—81no—
182C2#447×C13711yescharacteristic
subgroupG/H
center, socleC2×C137#447
commutator, Frattini, cosocleC2×C8C23×C137
prime radical#576×C137C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C2×C4×C137 ◁ #131×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #510×C137C2semidirect
3. C2×SD32×C137C2semidirect
4. #473×C137C2semidirect
5. #361×C137C2semidirect
6. #359×C137C2semidirect
7. #350×C137C2semidirect
14. C2×Q16×C137C22semidirect
21. M32×C137C22semidirect
22. C2×C16×C137C22semidirect
23. #108×C137C22semidirect
91. C137#2194direct (N × H)
92. #2194C137direct (N × H)
94. #510C2×C137semidirect
95. C2×SD32C2×C137semidirect
96. #473C2×C137semidirect
97. #361C2×C137semidirect
98. #359C2×C137semidirect
99. #350C2×C137semidirect
106. C2×Q16C22×C137semidirect
113. M32C22×C137semidirect
114. C2×C16C22×C137semidirect
115. #108C22×C137semidirect

Maximal subgroups

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

subgroupindex
#576×C1372
#510×C1372
C2×SD32×C1372
#473×C1372
#361×C1372
#359×C1372
#350×C1372
#2194137