Finite Group Databasebeta

← #38767387

Subgroups — #38767387

Order 25472.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#27164554C211yesmaximal, characteristic
2#27164072C211yesmaximal, characteristic, complemented
3#27163882C211yesmaximal, characteristic, complemented
4#27163755C211yesmaximal, characteristic, complemented
5#27163753C211yesmaximal, characteristic
6#27163751C211yesmaximal, characteristic
7#328×C199C211yesmaximal, characteristic, complemented
8#20263950—21no—
9#20263946C411yescharacteristic, complemented
10#20263944C411yescharacteristic, complemented
11#20263939—21no—
12#20263929C2211yescharacteristic
13Q8×Dic199C411yescharacteristic, complemented
14#20263917C411yescharacteristic, complemented
15(C22⋊D199)⋊C4—21no—
16#20263913—21no—
17C2×Dic199⋊C4—21no—
18C2×C4×Dic199C2211yescharacteristic
19Dic199⋊C8—21no—
20C22⋊(C199⋊C8)C2211yescharacteristic
21#20263865—21no—
22C199⋊#95—21no—
23C199⋊#95C2211yescharacteristic
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C2×D796C2×C411yescharacteristic
28C2×C22⋊D199—21no—
29C22×Dic199—21no—
30C22×Dic199D411yescharacteristic
31C22×Dic199D411yescharacteristic
32C2×C4×D199—21no—
33C2×C199⋊C8—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199C2×C411yescharacteristic
37Dic199⋊C4—21no—
38Dic199⋊C4—42no—
39(C2×D199)⋊C4—21no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43C4×Dic199—21no—
44C4×Dic199C2×C411yescharacteristic
45C4×Dic199C2×C411yescharacteristic
46C4×Dic199D411yescharacteristic
47C4×Dic199D411yescharacteristic
48C2×Q8×C199C2×C411yescharacteristic
49C22×C4×C199—21no—
50C22×C4×C199C2311yescharacteristic
51C2×C8×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199C2×C411yescharacteristic
54C22⋊C4×C199—21no—
55C22×D199—21no—
56D796—41no—
57C22⋊D199—41no—
58C22⋊D199—41no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—41no—
63C2×Dic199—41no—
64C2×Dic199—41no—
65C2×Dic199—41no—
66C2×Dic199C22⋊C411yescharacteristic
67C2×Dic199C22⋊C411yescharacteristic
68C2×Dic199C22⋊C411yescharacteristic
69C2×Dic199C22⋊C411yescharacteristic
70C4×D199—42no—
71C199⋊C8—41no—
72C23×C199C2×D411yescharacteristic
73Q8×C199—41no—
74C2×C4×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199—41no—
77C2×C4×C199—41no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199C22×C411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C8×C199—41no—
83C22×C199—21no—
84C22×C199—21no—
85C22×C199C2×C22⋊C411yescharacteristic
86C22×C199AGL(1,ℤ/8)11yescharacteristic
87C22×C199#13711yescharacteristic
88C2×D199—41no—
89C2×D199—41no—
90C2×D199—41no—
91C4×C199—21no—
92C4×C199—21no—
93C4×C199—41no—
94C4×C199—41no—
95C4×C199—81no—
96Dic199—21no—
97Dic199—21no—
98Dic199—21no—
99Dic199—21no—
100Dic199—41no—
101Dic199—81no—
102C2×C199—21no—
103C2×C199—21no—
104C2×C199#41011yescharacteristic
105C2×C199#41711yescharacteristic
106C2×C199#42111yescharacteristic
107D199—81no—
108C199#151211yescharacteristic, complemented, Sylow
109#1512—1991nomaximal, Sylow
110#529—1991no—
111#432—1991no—
112#382—1991no—
113#328—1991no—
114#328D19911yescharacteristic, complemented
115#327—1991no—
116#327—1991no—
117#126—3981no—
118#125—1991no—
119#124—1991no—
120#123—3981no—
121C22⋊Q8C2×D19911yescharacteristic
122#121—1991no—
123C4×Q8—1991no—
124C4×D4—3981no—
125C4×D4—1991no—
126#117—3981no—
127C2×C4⋊C4—3981no—
128C2×C42—1991no—
129C4⋊C8—3981no—
130C22⋊C8—1991no—
131C22⋊C8C2×D19911yescharacteristic
132#97—3981no—
133#95—3981no—
134#95—1991no—
135#95C2×D19911yescharacteristic
136C2×Q8C4×D19911yescharacteristic
137C2×D4—3981no—
138C2×D4—1991no—
139C22×C4—21no—
140C22×C4—3981no—
141C22×C4—3981no—
142C22×C4—1991no—
143C22×C4—1991no—
144C22×C4C22×D19911yescharacteristic
145C2×C8—21no—
146C2×C8—3981no—
147C4⋊C4—21no—
148C4⋊C4—7962no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C4⋊C4—3981no—
152C4⋊C4—1991no—
153C4⋊C4C4×D19911yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—3981no—
156C22⋊C4—3981no—
157C22⋊C4—3981no—
158C22⋊C4—3981no—
159C42—3981no—
160C42—1991no—
161C42—1991no—
162C42—1991no—
163C42—1991no—
164C23—3981no—
165C23D4×D19911yescharacteristic
166Q8—41no—
167D4—7961no—
168D4—7961no—
169D4—7961no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—7962no—
177C2×C4—7961no—
178C2×C4—7961no—
179C2×C4—7961no—
180C2×C4—7961no—
181C2×C4—3981no—
182C2×C4—3981no—
183C2×C4—3981no—
184C2×C4—1991no—
185C2×C4—1991no—
186C2×C4—1991no—
187C2×C4—1991no—
188C2×C4C2×C4×D19911yescharacteristic
189C2×C4D4×D19911yescharacteristic
190C8—41no—
191C8—7961no—
192C22—21no—
193C22—21no—
194C22—7961no—
195C22—7961no—
196C22—7961no—
197C22C22⋊C4×D19911yescharacteristic
198C22#2026398211yescharacteristic
199C22#2026398711yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—81no—
205C4—7961no—
206C4—15921no—
207C4—3981no—
208C4—3981no—
209C4—3981no—
210C4—3981no—
211C2—21no—
212C2—21no—
213C2—15921no—
214C2#2716399011yescharacteristic
215C2#2716401811yescharacteristic
216C2#2716403311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263929C22
cosocleC22×C4×C199C23
hypercenter#328D199
hypocenterC199#1512
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27164072C2semidirect
3. #27163882C2semidirect
4. #27163755C2semidirect
7. #328×C199C2semidirect
9. #20263946C4semidirect
10. #20263944C4semidirect
13. Q8×Dic199C4semidirect
14. #20263917C4semidirect
108. C199#1512semidirect
114. #328D199semidirect

Maximal subgroups

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

subgroupindex
#271645542
#271640722
#271638822
#271637552
#271637532
#271637512
#328×C1992
#1512199