Finite Group Databasebeta

← #353×D251 #44552021

Subgroups — #353×D251 #44552021

Order 32128.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D251C211yesmaximal, characteristic
2#99×D251C212yesmaximal, complemented
3#30631480C211yesmaximal, characteristic, complemented
4#30631479C212yesmaximal, complemented
5#353×C251C211yesmaximal, characteristic, complemented
6C2×D4×D251—22no—
7C2×D4×D251C411yescharacteristic, complemented
8C2×D4×D251C2211yescharacteristic
9#22548712C411yescharacteristic, complemented
10#22548711—22no—
11#22548711C411yescharacteristic, complemented
12#22548711C2211yescharacteristic
13C22⋊C4×D251—22no—
14C42×D251C411yescharacteristic, complemented
15#22548630C2212yescomplemented
16#22548629—22no—
17#127×C251C2211yescharacteristic
18#99×C251C2212yescomplemented
19C23×D251—21no—
20C23×D251D412yes—
21C2×D1004—22no—
22C2×D1004C2×C411yescharacteristic
23C2×D1004C2×C411yescharacteristic, complemented
24C2×C22⋊D251—22no—
25C2×C22⋊D251—21no—
26C2×C22⋊D251—22no—
27C2×C22⋊D251D412yes—
28D4×D251—41no—
29D4×D251—41no—
30D4×D251—41no—
31D4×D251—41no—
32D4×D251—48no—
33C2×C4×D251—21no—
34C2×C4×D251—42no—
35C2×C4×D251C2×C411yescharacteristic
36C22⋊Dic251—22no—
37(C2×D251)⋊C4—42no—
38C4×Dic251—21no—
39C4×Dic251C2×C411yescharacteristic, complemented
40C2×D4×C251—22no—
41C2×D4×C251C2×C411yescharacteristic, complemented
42C2×D4×C251C2311yescharacteristic
43C22⋊C4×C251—22no—
44C42×C251C2×C411yescharacteristic, complemented
45C22×D251—22no—
46C22×D251—22no—
47C22×D251—21no—
48C22×D251—44no—
49C22×D251—44no—
50C22×D251—41no—
51C22×D251—41no—
52C22×D251C22⋊C411yescharacteristic
53C22×D251C22⋊C412yes—
54D1004—41no—
55D1004—41no—
56D1004—44no—
57C22⋊D251—42no—
58C22⋊D251—44no—
59C22⋊D251—44no—
60C22⋊D251—42no—
61C2×Dic251—21no—
62C2×Dic251—42no—
63C2×Dic251C22⋊C411yescharacteristic
64C4×D251—24no—
65C4×D251—21no—
66C4×D251—21no—
67C4×D251—84no—
68C23×C251—21no—
69C23×C251C2×D412yes—
70D4×C251—41no—
71D4×C251—41no—
72D4×C251—44no—
73C2×C4×C251—21no—
74C2×C4×C251—42no—
75C2×C4×C251C22×C411yescharacteristic
76C22×C251—22no—
77C22×C251—42no—
78C22×C251—42no—
79C22×C251—41no—
80C22×C251C2×C22⋊C411yescharacteristic
81C2×D251—22no—
82C2×D251—22no—
83C2×D251—44no—
84C2×D251—42no—
85C2×D251—42no—
86C2×D251—41no—
87C2×D251—82no—
88C2×D251—82no—
89C2×D251—82no—
90C2×D251#9911yescharacteristic
91C2×D251#9911yescharacteristic
92C4×C251—22no—
93C4×C251—21no—
94C4×C251—82no—
95Dic251—22no—
96Dic251—21no—
97Dic251—82no—
98C2×C251—21no—
99C2×C251—42no—
100C2×C251—81no—
101C2×C251C2×#9911yescharacteristic
102D251—21no—
103D251—42no—
104D251—81no—
105D251#35312yesdirect product factor
106C251C2×#35311yescharacteristic, complemented, Sylow
107C2×#353—2511nomaximal, Sylow
108C2×#127—2511no—
109C2×#99—2512no—
110#353—2511no—
111#353—2512no—
112#353D25111yescharacteristic, direct product factor
113C22×D4—5022no—
114C22×D4—2511no—
115C22×D4—2511no—
116#127—5022no—
117#127—2511no—
118#127—2511no—
119#127—2511no—
120#127C2×D25111yescharacteristic
121C2×C22⋊C4—5022no—
122C2×C42—2511no—
123#99—5022no—
124#99—2512no—
125#99C2×D25112yescomplemented
126C24—5021no—
127C24—2512no—
128C2×D4—22no—
129C2×D4—10041no—
130C2×D4—10041no—
131C2×D4—10048no—
132C2×D4—10041no—
133C2×D4—10041no—
134C2×D4—5022no—
135C2×D4—5022no—
136C2×D4—5022no—
137C2×D4—5021no—
138C2×D4—2512no—
139C2×D4—2511no—
140C2×D4—2511no—
141C2×D4C4×D25111yescharacteristic, complemented
142C2×D4C22×D25111yescharacteristic
143C22×C4—10042no—
144C22×C4—5021no—
145C22×C4—2511no—
146C22⋊C4—22no—
147C22⋊C4—10042no—
148C22⋊C4—5022no—
149C42—5021no—
150C42—2511no—
151C42C4×D25111yescharacteristic, complemented
152C23—21no—
153C23—10044no—
154C23—10041no—
155C23—10041no—
156C23—10044no—
157C23—5022no—
158C23—5022no—
159C23—5021no—
160C23—2512no—
161C23—2511no—
162C23D4×D25112yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—10041no—
167D4—10044no—
168D4—10041no—
169D4—10042no—
170D4—10044no—
171D4—10044no—
172D4—10042no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—20084no—
176C2×C4—10042no—
177C2×C4—5024no—
178C2×C4—5021no—
179C2×C4—5021no—
180C2×C4—5021no—
181C2×C4—2511no—
182C2×C4C2×C4×D25111yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—20082no—
188C22—20082no—
189C22—20082no—
190C22—10042no—
191C22—10042no—
192C22—10044no—
193C22—10041no—
194C22—5022no—
195C22—5022no—
196C22—2511no—
197C22—2511no—
198C22C22⋊C4×D25111yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—20082no—
203C4—5022no—
204C4—5021no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—20081no—
209C2—10042no—
210C2—5021no—
211C2—2512no—
212C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC2×C4×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#353×C251C2
socleC2×C251C2×#99
prime radical#127×D251C2
cosocleC2×D4×C251C23
hypercenter#353D251
hypocenterC251C2×#353
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D251C2semidirect
3. #30631480C2semidirect
4. #30631479C2semidirect
5. #353×C251C2semidirect
7. C2×D4×D251C4semidirect
9. #22548712C4semidirect
11. #22548711C4semidirect
14. C42×D251C4semidirect
15. #22548630C22semidirect
18. #99×C251C22semidirect
23. C2×D1004C2×C4semidirect
39. C4×Dic251C2×C4semidirect
41. C2×D4×C251C2×C4semidirect
44. C42×C251C2×C4semidirect
105. D251#353direct (N × H)
106. C251C2×#353semidirect
112. #353D251direct (N × H)
125. #99C2×D251semidirect
141. C2×D4C4×D251semidirect
151. C42C4×D251semidirect

Maximal subgroups

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

subgroupindex
#127×D2512
#99×D2512
#306314802
#306314792
#353×C2512
C2×#353251