Finite Group Databasebeta

← #354×D251 #44552024

Subgroups — #354×D251 #44552024

Order 32128.

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

normal subgroups only

This group has 284 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#124×D251C211yesmaximal, characteristic
2#99×D251C212yesmaximal
3#30631482C211yesmaximal, characteristic, complemented
4#30631481C212yesmaximal, complemented
5#354×C251C211yesmaximal, characteristic, complemented
6C2×Q8×D251C411yescharacteristic, complemented
7C2×D4×D251C2211yescharacteristic
8#22548704C411yescharacteristic, complemented
9#22548703C411yescharacteristic, complemented
10#22548702C2211yescharacteristic
11#22548701—22no—
12C22⋊C4×D251—22no—
13C22⋊C4×D251—22no—
14C42×D251C411yescharacteristic, complemented
15#22548630C2212yes—
16#22548629—22no—
17#124×C251C2211yescharacteristic
18#99×C251C2212yes—
19C23×D251D412yes—
20C2×Dic502—21no—
21C2×Dic502C2×C411yescharacteristic, complemented
22Q8×D251—41no—
23Q8×D251—41no—
24C2×D1004C2×C411yescharacteristic
25C2×C22⋊D251D412yes—
26D4×D251—41no—
27D4×D251—41no—
28C2×C4×D251—21no—
29C2×C4×D251—21no—
30C2×C4×D251—42no—
31C2×C4×D251C2×C411yescharacteristic
32C22⋊Dic251—22no—
33C22⋊Dic251—22no—
34(C2×D251)⋊C4—22no—
35(C2×D251)⋊C4—22no—
36(C2×D251)⋊C4—42no—
37C4×Dic251—21no—
38C4×Dic251C2×C411yescharacteristic, complemented
39C2×Q8×C251C2×C411yescharacteristic, complemented
40C2×D4×C251C2311yescharacteristic
41C22⋊C4×C251—22no—
42C22⋊C4×C251—22no—
43C42×C251C2×C411yescharacteristic, complemented
44C22×D251—22no—
45C22×D251—22no—
46C22×D251—44no—
47C22×D251C22⋊C411yescharacteristic
48C22×D251C22⋊C412yes—
49Dic502—42no—
50Dic502—41no—
51D1004—41no—
52C22⋊D251—42no—
53C2×Dic251—21no—
54C2×Dic251—21no—
55C2×Dic251—42no—
56C2×Dic251C22⋊C411yescharacteristic
57C4×D251—21no—
58C4×D251—21no—
59C4×D251—42no—
60C4×D251—41no—
61C4×D251—41no—
62C4×D251—84no—
63C23×C251C2×D412yes—
64Q8×C251—41no—
65D4×C251—41no—
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—42no—
69C2×C4×C251C22×C411yescharacteristic
70C22×C251—22no—
71C22×C251—42no—
72C22×C251C2×C22⋊C411yescharacteristic
73C2×D251—22no—
74C2×D251—22no—
75C2×D251—44no—
76C2×D251—42no—
77C2×D251—82no—
78C2×D251#9911yescharacteristic
79C2×D251#9911yescharacteristic
80C4×C251—21no—
81C4×C251—41no—
82C4×C251—41no—
83C4×C251—82no—
84Dic251—21no—
85Dic251—41no—
86Dic251—41no—
87Dic251—82no—
88C2×C251—21no—
89C2×C251—42no—
90C2×C251C2×#9911yescharacteristic
91D251—21no—
92D251—42no—
93D251#35412yesdirect product factor
94C251C2×#35411yescharacteristic, complemented, Sylow
95C2×#354—2511nomaximal, Sylow
96C2×#124—2511no—
97C2×#99—2512no—
98#354—2511no—
99#354—2512no—
100#354D25111yescharacteristic, direct product factor
101C22×Q8—2511no—
102C22×D4—2511no—
103#124—5022no—
104#124—2511no—
105#124—2511no—
106#124—2511no—
107#124C2×D25111yescharacteristic
108C2×C22⋊C4—5022no—
109C2×C22⋊C4—5022no—
110C2×C42—2511no—
111#99—5022no—
112#99—2512no—
113#99C2×D25112yes—
114C24—2512no—
115C2×Q8—10041no—
116C2×Q8—10041no—
117C2×Q8—5021no—
118C2×Q8—2511no—
119C2×Q8C4×D25111yescharacteristic, complemented
120C2×D4—10041no—
121C2×D4—10041no—
122C2×D4—2512no—
123C2×D4—2511no—
124C2×D4C22×D25111yescharacteristic
125C22×C4—10042no—
126C22×C4—5021no—
127C22×C4—5021no—
128C22×C4—2511no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—10042no—
132C22⋊C4—5022no—
133C22⋊C4—5022no—
134C22⋊C4—5022no—
135C22⋊C4—5022no—
136C42—5021no—
137C42—2511no—
138C42C4×D25111yescharacteristic, complemented
139C23—10044no—
140C23—5022no—
141C23—5022no—
142C23—2512no—
143C23—2511no—
144C23D4×D25112yes—
145Q8—41no—
146Q8—10041no—
147Q8—10042no—
148D4—41no—
149D4—10041no—
150D4—10042no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—20084no—
155C2×C4—10042no—
156C2×C4—10041no—
157C2×C4—10041no—
158C2×C4—10042no—
159C2×C4—5021no—
160C2×C4—5021no—
161C2×C4—5021no—
162C2×C4—5021no—
163C2×C4—2511no—
164C2×C4C2×C4×D25111yescharacteristic
165C22—22no—
166C22—42no—
167C22—20082no—
168C22—10042no—
169C22—10044no—
170C22—5022no—
171C22—5022no—
172C22—2511no—
173C22—2511no—
174C22C22⋊C4×D25111yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—20082no—
180C4—10041no—
181C4—10041no—
182C4—5021no—
183C2—21no—
184C2—42no—
185C2—10042no—
186C2—5021no—
187C2—2512no—
188C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC2×C4×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#354×C251C2
socleC2×C251C2×#99
prime radicalC2×D4×D251C22
cosocleC2×D4×C251C23
hypercenter#354D251
hypocenterC251C2×#354
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #30631482C2semidirect
4. #30631481C2semidirect
5. #354×C251C2semidirect
6. C2×Q8×D251C4semidirect
8. #22548704C4semidirect
9. #22548703C4semidirect
14. C42×D251C4semidirect
21. C2×Dic502C2×C4semidirect
38. C4×Dic251C2×C4semidirect
39. C2×Q8×C251C2×C4semidirect
43. C42×C251C2×C4semidirect
93. D251#354direct (N × H)
94. C251C2×#354semidirect
100. #354D251direct (N × H)
119. C2×Q8C4×D251semidirect
138. C42C4×D251semidirect

Maximal subgroups

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

subgroupindex
#124×D2512
#99×D2512
#306314822
#306314812
#354×C2512
C2×#354251