Finite Group Databasebeta

← #37582050

Subgroups — #37582050

Order 24448.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259353C211yesmaximal, characteristic
2C22×#15690277C211yesmaximal, characteristic, complemented
3#27133239C211yesmaximal, characteristic
4#27132885C214yesmaximal, complemented
5#27132873C214yesmaximal, complemented
6#27132690C211yesmaximal, characteristic, complemented
7#27132671C211yesmaximal, characteristic, complemented
8#27132657C211yesmaximal, characteristic
9#519×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic
11C22×Dic382C2211yescharacteristic
12#20259353—24no—
13C2×#15690277—24no—
14C2×#15690277C2212yes—
15C2×#15690277C2214yes—
16C2×#15690274C2212yes—
17C2×#15690271C2211yescharacteristic
18C22×C191⋊C8C2211yescharacteristic
19#20259322C2212yes—
20#20259310C2212yes—
21#20259306—22no—
22#20259302—22no—
23C4⋊(C191⋊C8)C2214yes—
24#20259245C2212yes—
25#20259245C2214yes—
26#20259242C2212yes—
27C22×Q8×C191C2211yescharacteristic
28C4×Q8×C191—22no—
29C4×Q8×C191C2214yes—
30#117×C191—22no—
31#117×C191C2211yescharacteristic
32#117×C191C2211yescharacteristic
33#15690287—22no—
34#15690287—22no—
35C2×Dic382—21no—
36C2×Dic382—22no—
37C2×Dic382—22no—
38C2×Dic382C2312yes—
39C2×Dic382C2311yescharacteristic
40C2×D764C2311yescharacteristic
41C2×C22⋊D191—21no—
42C22×Dic191—21no—
43C2×C4×D191—21no—
44#15690277—28no—
45#15690277—24no—
46#15690274—24no—
47#15690271—22no—
48C2×C191⋊C8—22no—
49C2×C191⋊C8C2312yes—
50C2×C191⋊C8C2312yes—
51C4⋊Dic191—24no—
52(C2×D191)⋊C4—24no—
53C2×Q8×C191—21no—
54C2×Q8×C191—22no—
55C2×Q8×C191D414yes—
56C2×Q8×C191C2312yes—
57C22×C4×C191—21no—
58C22×C4×C191C2311yescharacteristic
59C4⋊C4×C191—24no—
60C4⋊C4×C191D412yes—
61C4⋊C4×C191C2312yes—
62C22⋊C4×C191—21no—
63C22⋊C4×C191D412yes—
64C42×C191—24no—
65C42×C191—21no—
66C42×C191C2312yes—
67C22×D191—21no—
68Dic382—22no—
69Dic382—21no—
70Dic382—22no—
71Dic382—21no—
72Dic382—42no—
73D764—21no—
74D764—21no—
75C22⋊D191—42no—
76C2×Dic191—22no—
77C2×Dic191—21no—
78C2×Dic191—42no—
79C4×D191—42no—
80C191⋊C8—22no—
81C191⋊C8—22no—
82C23×C191C2×D411yescharacteristic
83Q8×C191—24no—
84Q8×C191—22no—
85Q8×C191C2×D414yes—
86C2×C4×C191—24no—
87C2×C4×C191—22no—
88C2×C4×C191—22no—
89C2×C4×C191C2×D414yes—
90C2×C4×C191C2×D411yescharacteristic
91C2×C4×C191C2×D412yes—
92C2×C4×C191#4112yes—
93C2×C4×C191C2411yescharacteristic
94C22×C191—22no—
95C22×C191#12112yes—
96C22×C191C22×D411yescharacteristic
97C2×D191—41no—
98C2×D191—42no—
99C4×C191—24no—
100C4×C191—24no—
101C4×C191—42no—
102C4×C191#12112yes—
103C4×C191C22×D411yescharacteristic
104C4×C191C2×#4111yescharacteristic
105Dic191—41no—
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#519D19111yescharacteristic, complemented
117#464—1914no—
118#462—1914no—
119#424—1911no—
120#419—1911no—
121#416—1911no—
122C2×#41—1911no—
123C22×Q8—1911no—
124C22×Q8C2×D19111yescharacteristic
125#137—3824no—
126C2×Q16—3824no—
127C2×Q16—1914no—
128C2×Q16—1912no—
129C2×SD16—1912no—
130C2×M16—1911no—
131C22×C8—1911no—
132C4⋊Q8—1912no—
133#124—1912no—
134#123—3822no—
135C22⋊Q8—3822no—
136C4×Q8—22no—
137C4×Q8C2×D19114yes—
138#117—22no—
139#117C2×D19111yescharacteristic
140#117C2×D19111yescharacteristic
141C4⋊C8—1914no—
142#103—1912no—
143#103—1914no—
144#102—1912no—
145#41—3822no—
146#41—3822no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—3821no—
150C2×Q8—3822no—
151C2×Q8—3822no—
152C2×Q8—1912no—
153C2×Q8—1911no—
154C2×Q8D76414yes—
155C2×Q8C22×D19112yes—
156C2×D4—3821no—
157C2×D4—1911no—
158C22×C4—21no—
159C22×C4—3821no—
160C22×C4—3821no—
161C22×C4C22×D19111yescharacteristic
162Q16—3828no—
163Q16—3824no—
164SD16—3824no—
165M16—3822no—
166C2×C8—3822no—
167C2×C8—1912no—
168C2×C8—1912no—
169C4⋊C4—24no—
170C4⋊C4—3824no—
171C4⋊C4C22⋊D19112yes—
172C4⋊C4C22×D19112yes—
173C22⋊C4—21no—
174C22⋊C4—3824no—
175C22⋊C4C22⋊D19112yes—
176C42—24no—
177C42—21no—
178C42C22×D19112yes—
179C23—3821no—
180C23C2×C22⋊D19111yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—3822no—
184Q8—3821no—
185Q8—3822no—
186Q8—3821no—
187Q8—7642no—
188Q8C2×D76414yes—
189D4—3821no—
190D4—3821no—
191D4—7642no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—3822no—
196C2×C4—3821no—
197C2×C4—7642no—
198C2×C4—7642no—
199C2×C4C2×C22⋊D19114yes—
200C2×C4C2×C22⋊D19111yescharacteristic
201C2×C4C2×D76412yes—
202C2×C4#1569028712yes—
203C2×C4C23×D19111yescharacteristic
204C8—3822no—
205C8—3822no—
206C22—22no—
207C22—7642no—
208C22—7641no—
209C22#2025929612yes—
210C22C22×C22⋊D19111yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—7641no—
215C4—7642no—
216C4#2025929612yes—
217C4C22×D76411yescharacteristic
218C4C2×#1569028711yescharacteristic
219C2—22no—
220C2—15281no—
221C2C2×#2025929611yescharacteristic
222C2#2713347812yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#519×C191C2
socleC22×C191C22×D4
prime radicalC2×#15690287C22
hypercenter#519D191
hypocenterC191#3044
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#15690277C2semidirect
4. #27132885C2semidirect
5. #27132873C2semidirect
6. #27132690C2semidirect
7. #27132671C2semidirect
9. #519×C191C2semidirect
111. C191#3044semidirect
116. #519D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C22×#156902772
#271332392
#271328852
#271328732
#271326902
#271326712
#271326572
#519×C1912
#3044191