Finite Group Databasebeta

← #31782442

Subgroups — #31782442

Order 17536.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#22577974C211yesmaximal, characteristic
2#22577387C211yesmaximal, characteristic
3#22577192C214yesmaximal, complemented
4#410×C137C211yesmaximal, characteristic
5C22×Dic274C411yescharacteristic, complemented
6C2×#14555494—21no—
7#17984086—21no—
8#17984075—21no—
9#17984069C412yescomplemented
10#17984069C2212yes—
11#17984067—21no—
12#17984067—21no—
13#17984047C411yescharacteristic, complemented
14#17984007—21no—
15#17984005C2212yes—
16C2×#41×C137C2211yescharacteristic
17#117×C137—21no—
18#99×C137—21no—
19#99×C137C2212yes—
20C2×Dic274—21no—
21C2×Dic274—21no—
22C2×Dic274—41no—
23C2×Dic274—41no—
24C22×Dic137C2×C412yes—
25#14555494—22no—
26#14555494—41no—
27C2×#11764040—41no—
28C4⋊Dic137—21no—
29#14555489—22no—
30#14555489—22no—
31C22⋊Dic137—22no—
32C22⋊Dic137—22no—
33C22⋊Dic137C2×C412yes—
34C4×Dic137—21no—
35#41×C137—41no—
36#41×C137—41no—
37C2×Q8×C137D411yescharacteristic
38C2×D4×C137C2×C411yescharacteristic
39C2×D4×C137D411yescharacteristic
40C2×D4×C137C2311yescharacteristic
41C22×C4×C137C2×C411yescharacteristic
42C22×C4×C137D411yescharacteristic
43C22×C4×C137D411yescharacteristic
44C4⋊C4×C137—41no—
45C22⋊C4×C137—22no—
46C42×C137—41no—
47Dic274—42no—
48Dic274—41no—
49Dic274—41no—
50C2×Dic137—22no—
51C2×Dic137—22no—
52C2×Dic137—42no—
53#11764040—42no—
54C23×C137C22⋊C411yescharacteristic
55C23×C137C22×C411yescharacteristic
56C23×C137C2×D411yescharacteristic
57Q8×C137—41no—
58D4×C137—41no—
59D4×C137—41no—
60D4×C137—41no—
61C2×C4×C137—21no—
62C2×C4×C137—21no—
63C2×C4×C137—21no—
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—42no—
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22⋊C411yescharacteristic
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C22×C137—21no—
73C22×C137—21no—
74C22×C137—21no—
75C22×C137—41no—
76C22×C137—41no—
77C22×C137—41no—
78C22×C137C2×C22⋊C411yescharacteristic
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—82no—
83C4×C137#9911yescharacteristic
84C4×C137#9911yescharacteristic
85Dic137—42no—
86Dic137—42no—
87C2×C137—21no—
88C2×C137—41no—
89C2×C137—41no—
90C2×C137—41no—
91C2×C137C2×#9911yescharacteristic
92C137#212311yescharacteristic, complemented, Sylow
93#2123—1371nomaximal, Sylow
94#536—1371no—
95#413—1371no—
96#410D13711yescharacteristic
97#352—1374no—
98C2×#41C2×D13711yescharacteristic
99C22×Q8—1371no—
100C2×M16—2741no—
101C4⋊Q8—2741no—
102#124—2741no—
103#123—1372no—
104#123—1372no—
105C22⋊Q8—2741no—
106C22⋊Q8—2741no—
107#117—21no—
108#117—1371no—
109#101—2741no—
110#100—1372no—
111#99—21no—
112#99C2×D13712yes—
113#41—41no—
114#41—41no—
115C2×Q8—2741no—
116C2×Q8—2741no—
117C2×Q8—5481no—
118C2×Q8—5481no—
119C2×Q8D54811yescharacteristic
120C2×D4C4×D13711yescharacteristic
121C2×D4C22⋊D13711yescharacteristic
122C2×D4C22×D13711yescharacteristic
123C22×C4—1372no—
124C22×C4C4×D13711yescharacteristic
125C22×C4C22⋊D13711yescharacteristic
126C22×C4D54811yescharacteristic
127M16—2742no—
128M16—5481no—
129C2×C8—5481no—
130C4⋊C4—41no—
131C4⋊C4—2742no—
132C4⋊C4—2741no—
133C4⋊C4—2742no—
134C22⋊C4—22no—
135C22⋊C4—2742no—
136C22⋊C4—2742no—
137C22⋊C4—1372no—
138C42—41no—
139C42—2741no—
140C23#1455548711yescharacteristic
141C23C2×C4×D13711yescharacteristic
142C23C2×C22⋊D13711yescharacteristic
143Q8—41no—
144Q8—5482no—
145Q8—5481no—
146Q8—5481no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—2742no—
158C2×C4—2742no—
159C2×C4—5482no—
160C2×C4#1455548711yescharacteristic
161C2×C4#1455548711yescharacteristic
162C2×C4#1455548711yescharacteristic
163C2×C4C2×D54811yescharacteristic
164C8—5482no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#1455548711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—5482no—
177C4—5482no—
178C4#1798400211yescharacteristic
179C4#1798400211yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1798400211yescharacteristic
subgroupG/H
centerC2C2×#17984002
commutatorC23×C137C22×C4
FrattiniC2×D4C22×D137
Fitting#410×C137C2
socleC2×C137C2×#99
prime radicalC2×#41×C137C22
cosocleC2×D4×C137C23
hypercenter#410D137
hypocenterC137#2123
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #22577192C2semidirect
5. C22×Dic274C4semidirect
9. #17984069C4semidirect
13. #17984047C4semidirect
92. C137#2123semidirect

Maximal subgroups

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

subgroupindex
#225779742
#225773872
#225771922
#410×C1372
#2123137