Finite Group Databasebeta

← #31781173

Subgroups — #31781173

Order 17536.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#22578170C211yesmaximal, characteristic, complemented
2#22577420C212yesmaximal, complemented
3#22577377C211yesmaximal, characteristic, complemented
4#22577163C212yesmaximal, complemented
5#532×C137C211yesmaximal, characteristic
6#17984095—24no—
7C2×#14555494—22no—
8C2×#14555494C2212yes—
9#17984047C2212yes—
10#17984015—24no—
11#17984003—22no—
12C2×#41×C137C2211yescharacteristic
13C2×#41×C137C2211yescharacteristic
14C4⋊Q8×C137C411yescharacteristic, complemented
15#127×C137C411yescharacteristic, complemented
16#124×C137C411yescharacteristic, complemented
17#124×C137C411yescharacteristic, complemented
18#121×C137—22no—
19C4×D4×C137—22no—
20C2×C42×C137C2211yescharacteristic
21#14555494—24no—
22#14555494—22no—
23#14555494D414yescomplemented
24C2×#11764040—24no—
25C2×#11764040—22no—
26C4⋊Dic137—22no—
27C22⋊Dic137—22no—
28C4×Dic137—22no—
29#41×C137—22no—
30#41×C137—22no—
31#41×C137—22no—
32#41×C137—22no—
33C2×Q8×C137C2×C411yescharacteristic
34C2×Q8×C137C2×C411yescharacteristic
35C2×D4×C137—22no—
36C2×D4×C137—21no—
37C2×D4×C137C2×C411yescharacteristic
38C2×D4×C137C2×C411yescharacteristic
39C2×D4×C137D412yes—
40C22×C4×C137—21no—
41C22×C4×C137—21no—
42C22×C4×C137D412yes—
43C22×C4×C137C2311yescharacteristic
44C4⋊C4×C137—22no—
45C22⋊C4×C137—22no—
46C22⋊C4×C137—22no—
47C42×C137—21no—
48C42×C137C2×C411yescharacteristic
49C42×C137C2×C411yescharacteristic
50C2×Dic137—42no—
51#11764040—24no—
52C23×C137—21no—
53C23×C137C22⋊C412yes—
54C23×C137C2×D411yescharacteristic
55Q8×C137—22no—
56Q8×C137—22no—
57D4×C137—24no—
58D4×C137—22no—
59D4×C137—22no—
60D4×C137—44no—
61D4×C137—42no—
62C2×C4×C137—21no—
63C2×C4×C137—22no—
64C2×C4×C137—22no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—42no—
68C2×C4×C137—41no—
69C2×C4×C137—41no—
70C2×C4×C137C22⋊C412yes—
71C2×C4×C137C22×C411yescharacteristic
72C2×C4×C137C2×D412yes—
73C2×C4×C137C2×D411yescharacteristic
74C2×C4×C137#4112yes—
75C22×C137—22no—
76C22×C137—41no—
77C22×C137—42no—
78C22×C137—42no—
79C22×C137—41no—
80C22×C137C2×C22⋊C411yescharacteristic
81C22×C137#12112yes—
82C4×C137—22no—
83C4×C137—22no—
84C4×C137—41no—
85C4×C137—41no—
86C4×C137C4×D412yes—
87C4×C137#12011yescharacteristic
88C4×C137#12311yescharacteristic
89Dic137—82no—
90C2×C137—22no—
91C2×C137—21no—
92C2×C137—42no—
93C2×C137—81no—
94C2×C137#38611yescharacteristic
95C137#188411yescharacteristic, complemented, Sylow
96#1884—1371nomaximal, Sylow
97#568—1371no—
98#532D13711yescharacteristic
99#421—1372no—
100#410—1371no—
101#339—1372no—
102C2×#41C2×D13711yescharacteristic
103C2×#41C2×D13711yescharacteristic
104#131—2744no—
105C2×M16—2742no—
106C2×M16—1372no—
107C4⋊Q8Dic13711yescharacteristic, complemented
108#127Dic13711yescharacteristic, complemented
109#124Dic13711yescharacteristic, complemented
110#124Dic13711yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—1372no—
114C2×C42C2×D13711yescharacteristic
115#104—2744no—
116#99—2742no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic13711yescharacteristic
122C2×Q8C2×Dic13711yescharacteristic
123C2×D4—22no—
124C2×D4—21no—
125C2×D4C2×Dic13711yescharacteristic
126C2×D4C2×Dic13711yescharacteristic
127C2×D4C22⋊D13712yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D13712yes—
131C22×C4C22×D13711yescharacteristic
132M16—2744no—
133M16—2742no—
134M16—1374no—
135C2×C8—2744no—
136C2×C8—2742no—
137C4⋊C4—22no—
138C4⋊C4—2742no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—2742no—
142C42—21no—
143C42—2742no—
144C42C2×Dic13711yescharacteristic
145C42C2×Dic13711yescharacteristic
146C23—21no—
147C23C22⋊Dic13712yes—
148C23C2×C22⋊D13711yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—5482no—
165C2×C4C22⋊Dic13712yes—
166C2×C4C22×Dic13711yescharacteristic
167C2×C4D4×D13712yes—
168C2×C4C2×C22⋊D13711yescharacteristic
169C2×C4#1455550912yes—
170C8—2744no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic13711yescharacteristic
177C22#1798406012yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—10962no—
183C4D4×Dic13712yes—
184C4#1798405611yescharacteristic
185C4#1798406911yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#2257728311yescharacteristic
subgroupG/H
centerC4#17984056
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting, prime radical#532×C137C2
socleC2×C137#386
cosocleC22×C4×C137C23
hypercenter#532D137
hypocenterC137#1884
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22578170C2semidirect
2. #22577420C2semidirect
3. #22577377C2semidirect
4. #22577163C2semidirect
14. C4⋊Q8×C137C4semidirect
15. #127×C137C4semidirect
16. #124×C137C4semidirect
17. #124×C137C4semidirect
23. #14555494D4semidirect
95. C137#1884semidirect
107. C4⋊Q8Dic137semidirect
108. #127Dic137semidirect
109. #124Dic137semidirect
110. #124Dic137semidirect

Maximal subgroups

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

subgroupindex
#225781702
#225774202
#225773772
#225771632
#532×C1372
#1884137