Finite Group Databasebeta

← #38769990

Subgroups — #38769990

Order 25472.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#20263954C211yesmaximal, characteristic
2C2×#20263880C211yesmaximal, characteristic
3C2×#20263878C211yesmaximal, characteristic
4#27163923C211yesmaximal, characteristic
5#27163780C211yesmaximal, characteristic
6#27163772C211yesmaximal, characteristic
7#340×C199C211yesmaximal, characteristic
8C22×Dic398—21no—
9C22×Dic398C2211yescharacteristic
10C22×C199⋊C8C2211yescharacteristic
11#20263954—22no—
12#20263954—22no—
13C2×C4⋊Dic199C2211yescharacteristic
14C2×Dic199⋊C4—21no—
15C2×C4×Dic199C2211yescharacteristic
16#20263880—22no—
17#20263878—22no—
18C199⋊#95—21no—
19C199⋊#95—21no—
20C22×C8×C199C2211yescharacteristic
21C2×C4⋊C4×C199C2211yescharacteristic
22C2×C4⋊C4×C199C2211yescharacteristic
23C2×Dic398—24no—
24C2×Dic398—21no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—42no—
30C2×Dic398—42no—
31C2×Dic398Q812yescomplemented
32C22×Dic199—21no—
33C22×Dic199—21no—
34C22×Dic199D411yescharacteristic
35C22×Dic199D411yescharacteristic
36C2×C199⋊C8—21no—
37C2×C199⋊C8—21no—
38C2×C199⋊C8—21no—
39C4⋊Dic199—21no—
40C4⋊Dic199D412yes—
41Dic199⋊C4—24no—
42C4×Dic199—21no—
43C4×Dic199—21no—
44C22×C4×C199—21no—
45C22×C4×C199—21no—
46C22×C4×C199C2311yescharacteristic
47C2×C8×C199—21no—
48C2×C8×C199—21no—
49C2×C8×C199—21no—
50C4⋊C4×C199—21no—
51C4⋊C4×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199D412yes—
54Dic398—22no—
55Dic398—22no—
56Dic398—48no—
57Dic398—41no—
58Dic398—41no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—22no—
64C2×Dic199—22no—
65C2×Dic199—41no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C2×Dic199SD1612yes—
70C2×Dic199Q1612yes—
71C199⋊C8—41no—
72C23×C199C2×D411yescharacteristic
73C2×C4×C199—22no—
74C2×C4×C199—41no—
75C2×C4×C199—41no—
76C2×C4×C199—41no—
77C2×C4×C199—41no—
78C2×C4×C199—41no—
79C2×C4×C199C2×D411yescharacteristic
80C2×C4×C199C2×D411yescharacteristic
81C2×C4×C199C2×Q811yescharacteristic
82C2×C4×C199#4111yescharacteristic
83C2×C4×C199#4111yescharacteristic
84C2×C4×C199#4111yescharacteristic
85C8×C199—41no—
86C22×C199#12011yescharacteristic
87C22×C199C22⋊Q811yescharacteristic
88C22×C199#12411yescharacteristic
89C22×C199C2×SD1611yescharacteristic
90C22×C199C2×Q1611yescharacteristic
91C22×C199AGL(1,ℤ/8)11yescharacteristic
92C22×C199#13711yescharacteristic
93C4×C199—42no—
94C4×C199—81no—
95C4×C199C22⋊Q811yescharacteristic
96C4×C199C22⋊Q811yescharacteristic
97C4×C199#12411yescharacteristic
98C4×C199#12411yescharacteristic
99Dic199—22no—
100Dic199—22no—
101Dic199—42no—
102Dic199—42no—
103C2×C199#39311yescharacteristic
104C2×C199#45011yescharacteristic
105C2×C199#45111yescharacteristic
106C2×C199#47511yescharacteristic
107C2×C199#47711yescharacteristic
108C2×C199#48711yescharacteristic
109C2×C199#48911yescharacteristic
110C199#202411yescharacteristic, complemented, Sylow
111#2024—1991nomaximal, Sylow
112C2×C4⋊Q8—1991no—
113C2×#103—1991no—
114C2×#103—1991no—
115#391—1991no—
116#340—1991no—
117#340D19911yescharacteristic
118#336—1991no—
119C22×Q8—3981no—
120C22×Q8—1991no—
121C22×C8—1991no—
122C22×C8C2×D19911yescharacteristic
123C4⋊Q8—3982no—
124C4⋊Q8—3982no—
125C2×C4⋊C4—3981no—
126C2×C4⋊C4—1991no—
127C2×C4⋊C4C2×D19911yescharacteristic
128C2×C4⋊C4C2×D19911yescharacteristic
129C2×C42—1991no—
130#103—3982no—
131#103—3982no—
132#95—3981no—
133#95—3981no—
134C2×Q8—7962no—
135C2×Q8—7962no—
136C2×Q8—3984no—
137C2×Q8—3981no—
138C2×Q8—3981no—
139C2×Q8—3981no—
140C2×Q8—3981no—
141C2×Q8—3981no—
142C2×Q8—1992no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3981no—
146C22×C4—3981no—
147C22×C4—1991no—
148C22×C4—1991no—
149C22×C4C22×D19911yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—3981no—
154C2×C8—3981no—
155C2×C8—3981no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—3984no—
160C4⋊C4—3981no—
161C4⋊C4—1992no—
162C4⋊C4D79612yes—
163C42—3981no—
164C42—3981no—
165C23D4×D19911yescharacteristic
166Q8—7968no—
167Q8—7961no—
168Q8—7961no—
169Q8—3982no—
170Q8—3982no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—7961no—
178C2×C4—7961no—
179C2×C4—7961no—
180C2×C4—7961no—
181C2×C4—3982no—
182C2×C4—3982no—
183C2×C4—3981no—
184C2×C4—3981no—
185C2×C4—3981no—
186C2×C4—3981no—
187C2×C4—1992no—
188C2×C4—1992no—
189C2×C4D4×D19911yescharacteristic
190C2×C4C2×D79611yescharacteristic
191C2×C4Q8×D19911yescharacteristic
192C2×C4#1569188511yescharacteristic
193C2×C4#1569188611yescharacteristic
194C2×C4#1569188611yescharacteristic
195C8—41no—
196C8—7961no—
197C22#2026392511yescharacteristic
198C22#2026393311yescharacteristic
199C22#2026394111yescharacteristic
200C22SD16×D19911yescharacteristic
201C22Q16×D19911yescharacteristic
202C22#2026397911yescharacteristic
203C22#2026398411yescharacteristic
204C4—42no—
205C4—81no—
206C4—7962no—
207C4—7962no—
208C4—3982no—
209C4—3982no—
210C4#2026393311yescharacteristic
211C4#2026393511yescharacteristic
212C4#2026394111yescharacteristic
213C4#2026394311yescharacteristic
214C2#2716393111yescharacteristic
215C2#2716417011yescharacteristic
216C2#2716417511yescharacteristic
217C2#2716430211yescharacteristic
218C2#2716431211yescharacteristic
219C2#2716436711yescharacteristic
220C2#2716437611yescharacteristic
subgroupG/H
centerC23D4×D199
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#340×C199C2
socle, prime radicalC23×C199C2×D4
hypercenter#340D199
hypocenterC199#2024
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic398Q8semidirect
110. C199#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#202639542
C2×#202638802
C2×#202638782
#271639232
#271637802
#271637722
#340×C1992
#2024199