Finite Group Databasebeta

← #37582051

Subgroups — #37582051

Order 24448.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20259355C211yesmaximal, characteristic
2C22×Dic764C211yesmaximal, characteristic, complemented
3#27133239C211yesmaximal, characteristic
4#27133140C211yesmaximal, characteristic, complemented
5#27132886C214yesmaximal, complemented
6#27132874C214yesmaximal, complemented
7#27132673C211yesmaximal, characteristic, complemented
8#27132660C211yesmaximal, characteristic
9#424×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic
11C22×Dic382C2211yescharacteristic
12C22×Dic382C2211yescharacteristic
13#20259355—24no—
14C2×Dic764—24no—
15C2×Dic764C2214yes—
16C2×Dic764C2212yes—
17C2×#15690276C2212yes—
18#20259322C2212yes—
19#20259310C2212yes—
20#20259306—22no—
21#20259302—22no—
22C4×Dic382C2214yes—
23#20259287—22no—
24#20259281C2211yescharacteristic
25#20259278—22no—
26#20259247C2214yes—
27#20259247C2212yes—
28#20259244C2212yes—
29C2×M16×C191C2211yescharacteristic
30C22×C8×C191C2211yescharacteristic
31#117×C191C2211yescharacteristic
32C4⋊C8×C191C2214yes—
33#15690287—22no—
34#15690287—22no—
35C2×Dic382—21no—
36C2×Dic382—21no—
37C2×Dic382—22no—
38C2×Dic382—22no—
39C2×Dic382—22no—
40C2×Dic382D414yes—
41C2×Dic382C2312yes—
42C2×Dic382C2311yescharacteristic
43C2×Dic382C2312yes—
44C2×D764C2311yescharacteristic
45C2×C22⋊D191—21no—
46C22×Dic191—21no—
47C22×Dic191—21no—
48C2×C4×D191—21no—
49Dic764—28no—
50Dic764—24no—
51#15690276—24no—
52C4⋊Dic191—24no—
53C4⋊Dic191C2312yes—
54Dic191⋊C4—24no—
55C22⋊Dic191—21no—
56(C2×D191)⋊C4—24no—
57C4×Dic191—24no—
58C4×Dic191—21no—
59C22×C4×C191C2311yescharacteristic
60M16×C191—22no—
61C2×C8×C191—22no—
62C2×C8×C191C2312yes—
63C2×C8×C191C2312yes—
64C4⋊C4×C191D412yes—
65C22⋊C4×C191D412yes—
66C42×C191C2312yes—
67C22×D191—21no—
68Dic382—24no—
69Dic382—22no—
70Dic382—22no—
71Dic382—21no—
72Dic382—22no—
73Dic382—21no—
74Dic382—42no—
75Dic382C2×D414yes—
76D764—21no—
77D764—21no—
78C22⋊D191—42no—
79C2×Dic191—24no—
80C2×Dic191—22no—
81C2×Dic191—22no—
82C2×Dic191—22no—
83C2×Dic191—21no—
84C2×Dic191—42no—
85C4×D191—42no—
86C23×C191C2×D411yescharacteristic
87C2×C4×C191C2×D412yes—
88C2×C4×C191C2×D414yes—
89C2×C4×C191C2×D411yescharacteristic
90C2×C4×C191#4112yes—
91C2×C4×C191C2411yescharacteristic
92C8×C191—22no—
93C8×C191—22no—
94C22×C191—22no—
95C22×C191#12112yes—
96C22×C191C22×D411yescharacteristic
97C2×D191—41no—
98C2×D191—42no—
99C4×C191—24no—
100C4×C191#12112yes—
101C4×C191C22×D411yescharacteristic
102C4×C191C2×#4111yescharacteristic
103Dic191—24no—
104Dic191—41no—
105Dic191—42no—
106Dic191—42no—
107C2×C191—22no—
108C2×C191C2×#12111yescharacteristic
109C2×C191#57812yes—
110D191—81no—
111C191#304411yescharacteristic, complemented, Sylow
112#3044—1911nomaximal, Sylow
113C2×#137—1911no—
114C22×Q16—1911no—
115#536—1911no—
116#519—1911no—
117#464—1914no—
118#462—1914no—
119#424D19111yescharacteristic, complemented
120#419—1911no—
121#416—1911no—
122C2×#41—1911no—
123C22×Q8—1911no—
124C22×Q8—1911no—
125#137—3824no—
126C2×Q16—3824no—
127C2×Q16—1914no—
128C2×Q16—1912no—
129C2×SD16—1912no—
130C2×M16C2×D19111yescharacteristic
131C22×C8C2×D19111yescharacteristic
132C4⋊Q8—1912no—
133#124—1912no—
134#123—3822no—
135C22⋊Q8—3822no—
136C4×Q8—3822no—
137C4×Q8—1914no—
138#117—3822no—
139#117—1911no—
140#117C2×D19111yescharacteristic
141C4⋊C8C2×D19114yes—
142#103—1912no—
143#103—1914no—
144#102—1912no—
145#41—3822no—
146#41—3822no—
147C2×Q8—3821no—
148C2×Q8—3821no—
149C2×Q8—3822no—
150C2×Q8—3822no—
151C2×Q8—3822no—
152C2×Q8—1914no—
153C2×Q8—1912no—
154C2×Q8—1911no—
155C2×Q8—1912no—
156C2×D4—3821no—
157C2×D4—1911no—
158C22×C4—3821no—
159C22×C4—3821no—
160C22×C4—3821no—
161C22×C4C22×D19111yescharacteristic
162Q16—3828no—
163Q16—3824no—
164SD16—3824no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D19112yes—
168C2×C8C22×D19112yes—
169C4⋊C4—3824no—
170C4⋊C4—3824no—
171C4⋊C4—1912no—
172C4⋊C4D76412yes—
173C22⋊C4—3821no—
174C22⋊C4—3824no—
175C22⋊C4D76412yes—
176C42—3824no—
177C42—3821no—
178C42C22×D19112yes—
179C23—3821no—
180C23C2×D76411yescharacteristic
181Q8—3822no—
182Q8—3824no—
183Q8—3822no—
184Q8—3821no—
185Q8—3822no—
186Q8—3821no—
187Q8—1914no—
188Q8—7642no—
189D4—3821no—
190D4—3821no—
191D4—7642no—
192C2×C4—3822no—
193C2×C4—3824no—
194C2×C4—3822no—
195C2×C4—3822no—
196C2×C4—3821no—
197C2×C4—7642no—
198C2×C4—7642no—
199C2×C4D4×D19112yes—
200C2×C4C2×D76414yes—
201C2×C4C2×D76411yescharacteristic
202C2×C4#1569028812yes—
203C2×C4C23×D19111yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—7642no—
208C22—7641no—
209C22#2025929712yes—
210C22C22×D76411yescharacteristic
211C4—24no—
212C4—3824no—
213C4—7642no—
214C4—7641no—
215C4—7642no—
216C4#2025929712yes—
217C4C2×D4×D19111yescharacteristic
218C4C2×#1569028811yescharacteristic
219C2—22no—
220C2—15281no—
221C2C2×#2025929711yescharacteristic
222C2#2713348012yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#424×C191C2
socleC22×C191C22×D4
prime radicalC2×#15690287C22
hypercenter#424D191
hypocenterC191#3044
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic764C2semidirect
4. #27133140C2semidirect
5. #27132886C2semidirect
6. #27132874C2semidirect
7. #27132673C2semidirect
9. #424×C191C2semidirect
111. C191#3044semidirect
119. #424D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C22×Dic7642
#271332392
#271331402
#271328862
#271328742
#271326732
#271326602
#424×C1912
#3044191