Finite Group Databasebeta

← #44548951

Subgroups — #44548951

Order 32128.

234 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 73 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#30632221C211yesmaximal, characteristic
2C4×#16860794C211yesmaximal, characteristic, complemented
3#30631428C212yesmaximal, complemented
4#30631426C212yesmaximal, complemented
5C2×#97×C251C211yesmaximal, characteristic, complemented
6C2×#16860794—21no—
7#22548709—21no—
8#22548707C2211yescharacteristic
9#22548703C2211yescharacteristic
10#22548698—21no—
11#22548694C411yescharacteristic
12#22548689C411yescharacteristic
13C4×Dic502C411yescharacteristic
14C4×D1004C411yescharacteristic
15C4×C22⋊D251—21no—
16#22548672—21no—
17#22548626C2212yes—
18C4×C251⋊C8—21no—
19C22×C8×C251—21no—
20C2×C42×C251C2211yescharacteristic
21#97×C251—21no—
22#97×C251C2212yes—
23C2×Dic502C2×C411yescharacteristic
24C2×D1004C2×C411yescharacteristic
25C2×C22⋊D251—21no—
26C2×C4×D251—21no—
27#16860794—24no—
28C2×C251⋊C8—22no—
29C4⋊Dic251C2×C411yescharacteristic
30C4⋊Dic251C2×C411yescharacteristic
31Dic251⋊C4—21no—
32Dic251⋊C4—21no—
33Dic251⋊C4—21no—
34Dic251⋊C4—21no—
35C22⋊Dic251—21no—
36(C2×D251)⋊C4—21no—
37(C2×D251)⋊C4—21no—
38(C2×D251)⋊C4—21no—
39(C2×D251)⋊C4—21no—
40C4×Dic251—21no—
41C22×C4×C251C2×C411yescharacteristic
42C22×C4×C251D411yescharacteristic
43C22×C4×C251D411yescharacteristic
44C2×C8×C251—24no—
45C2×C8×C251—22no—
46C42×C251C2×C411yescharacteristic
47C42×C251D411yescharacteristic
48C42×C251D411yescharacteristic
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—41no—
52D1004—41no—
53C22⋊D251—42no—
54C2×Dic251—21no—
55C2×Dic251—21no—
56C2×Dic251—21no—
57C4×D251—42no—
58C251⋊C8—24no—
59C23×C251C22⋊C411yescharacteristic
60C2×C4×C251—22no—
61C2×C4×C251—22no—
62C2×C4×C251—41no—
63C2×C4×C251C22⋊C411yescharacteristic
64C2×C4×C251C22⋊C411yescharacteristic
65C2×C4×C251C22⋊C411yescharacteristic
66C2×C4×C251C22×C411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—24no—
70C22×C251—21no—
71C22×C251—21no—
72C22×C251#10412yes—
73C22×C251C2×C22⋊C411yescharacteristic
74C2×D251—41no—
75C2×D251—41no—
76C2×D251—41no—
77C4×C251—21no—
78C4×C251—21no—
79C4×C251—21no—
80C4×C251—21no—
81C4×C251—21no—
82C4×C251#10412yes—
83Dic251—41no—
84Dic251—41no—
85Dic251—41no—
86C2×C251—22no—
87C2×C251#41311yescharacteristic
88C2×C251C2×#10411yescharacteristic
89C2×C251C2×#10411yescharacteristic
90D251—81no—
91C251#152411yescharacteristic, complemented, Sylow
92#1524—2511nomaximal, Sylow
93#529—2511no—
94C4×M16—2511no—
95C2×#97D25111yescharacteristic, complemented
96#330—2512no—
97#329—2512no—
98C2×M16—5021no—
99C22×C8—21no—
100#126—5021no—
101#125—2511no—
102#124—2511no—
103#123—5021no—
104C22⋊Q8—2511no—
105#121—2511no—
106C4×Q8—2511no—
107C4×D4—5021no—
108C4×D4—2511no—
109#117—5021no—
110C2×C42C2×D25111yescharacteristic
111#97—21no—
112#97—2512no—
113#97C2×D25112yes—
114C4×C8—5021no—
115C2×Q8—2511no—
116C2×D4—5021no—
117C2×D4—2511no—
118C22×C4—5021no—
119C22×C4C4×D25111yescharacteristic
120C22×C4C22⋊D25111yescharacteristic
121C22×C4D100411yescharacteristic
122M16—5024no—
123C2×C8—24no—
124C2×C8—22no—
125C2×C8—5022no—
126C4⋊C4—5021no—
127C4⋊C4—5021no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—2511no—
131C4⋊C4—2511no—
132C22⋊C4—5021no—
133C22⋊C4—5021no—
134C22⋊C4—5021no—
135C22⋊C4—5021no—
136C22⋊C4—5021no—
137C42—5021no—
138C42C4×D25111yescharacteristic
139C42C22⋊D25111yescharacteristic
140C42D100411yescharacteristic
141C42C22×D25111yescharacteristic
142C23—5021no—
143C23(C2×D251)⋊C411yescharacteristic
144Q8—10041no—
145D4—10041no—
146D4—10042no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—10042no—
151C2×C4—5021no—
152C2×C4—5021no—
153C2×C4—5021no—
154C2×C4(C2×D251)⋊C411yescharacteristic
155C2×C4(C2×D251)⋊C411yescharacteristic
156C2×C4(C2×D251)⋊C411yescharacteristic
157C2×C4C2×C4×D25111yescharacteristic
158C2×C4C2×C22⋊D25111yescharacteristic
159C2×C4C2×D100411yescharacteristic
160C8—24no—
161C8—5024no—
162C22—21no—
163C22—21no—
164C22—10041no—
165C22—10041no—
166C22—10041no—
167C22#2254864112yes—
168C22C2×(C2×D251)⋊C411yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—10041no—
175C4—10041no—
176C4—10041no—
177C4#2254864312yes—
178C2—22no—
179C2—20081no—
180C2#3063166911yescharacteristic
181C2C2×#2254864111yescharacteristic
182C2C2×#2254864311yescharacteristic
subgroupG/H
centerC2×C4(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
FittingC2×#97×C251C2
socleC22×C251C2×C22⋊C4
prime radical#22548689C4
cosocleC42×C251C23
hypercenterC2×#97D251
hypocenterC251#1524
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
2. C4×#16860794C2semidirect
3. #30631428C2semidirect
4. #30631426C2semidirect
5. C2×#97×C251C2semidirect
91. C251#1524semidirect
95. C2×#97D251semidirect

Maximal subgroups

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

subgroupindex
#306322212
C4×#168607942
#306314282
#306314262
C2×#97×C2512
#1524251