Finite Group Databasebeta

← #37576163

Subgroups — #37576163

Order 24448.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×Q8×D191C211yesmaximal, characteristic
2C2×#20259322C211yesmaximal, characteristic, complemented
3C2×#20259320C211yesmaximal, characteristic, complemented
4C2×#20259302C211yesmaximal, characteristic
5#27132584C214yesmaximal, complemented
6#27132568C211yesmaximal, characteristic, complemented
7#27132537C214yesmaximal
8C4×(C2×D191)⋊C4C211yesmaximal, characteristic
9#391×C191C211yesmaximal, characteristic, complemented
10C22×Dic382C2211yescharacteristic
11C22×Dic382C2212yes—
12C22×Dic382C2211yescharacteristic
13C2×Q8×D191—24no—
14C2×Q8×D191—24no—
15C2×Q8×D191—24no—
16C22×C4×D191C2212yes—
17C22×C4×D191C2211yescharacteristic
18#20259322—24no—
19#20259320—24no—
20#20259302—24no—
21C2×C4⋊Dic191C2214yes—
22C2×C4⋊Dic191C2212yes—
23C2×Dic191⋊C4C2214yes—
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2212yes—
26C2×C4×Dic191C2211yescharacteristic
27C191⋊#95C2212yes—
28C191⋊#95C2214yes—
29C22×Q8×C191C2211yescharacteristic
30C2×C4⋊C4×C191C2211yescharacteristic
31C2×C42×C191C2211yescharacteristic
32#95×C191C2214yes—
33C23×D191C2311yescharacteristic
34C2×Dic382—24no—
35C2×Dic382—24no—
36C2×Dic382—22no—
37C2×Dic382—24no—
38C2×Dic382—22no—
39C2×Dic382—22no—
40C2×Dic382—22no—
41C2×Dic382—22no—
42C2×Dic382D414yes—
43Q8×D191—232no—
44C22×Dic191C2312yes—
45C22×Dic191C2312yes—
46C22×Dic191C2311yescharacteristic
47C22×Dic191C2312yes—
48C2×C4×D191—24no—
49C2×C4×D191—24no—
50C2×C4×D191—22no—
51C2×C4×D191—24no—
52C2×C4×D191—22no—
53C2×C4×D191D414yes—
54C4⋊Dic191—24no—
55C4⋊Dic191—28no—
56Dic191⋊C4—28no—
57(C2×D191)⋊C4—24no—
58(C2×D191)⋊C4Q818yescomplemented
59C4×Dic191—22no—
60C2×Q8×C191—22no—
61C2×Q8×C191—22no—
62C2×Q8×C191D414yes—
63C22×C4×C191C2312yes—
64C22×C4×C191C2312yes—
65C22×C4×C191C2312yes—
66C22×C4×C191C2311yescharacteristic
67C4⋊C4×C191—22no—
68C42×C191—22no—
69C22×D191—21no—
70C22×D191—21no—
71C22×D191—22no—
72C22×D191C2×D412yes—
73C22×D191C2×Q814yes—
74Dic382—216no—
75Dic382—28no—
76Dic382—44no—
77C2×Dic191—22no—
78C2×Dic191—22no—
79C2×Dic191—21no—
80C2×Dic191—22no—
81C2×Dic191—21no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191C2×D414yes—
87C2×Dic191C2×D412yes—
88C2×Dic191C2×Q814yescomplemented
89C4×D191—216no—
90C4×D191—28no—
91C23×C191C2411yescharacteristic
92Q8×C191—28no—
93C2×C4×C191—22no—
94C2×C4×C191—22no—
95C2×C4×C191—22no—
96C2×C4×C191—22no—
97C2×C4×C191—22no—
98C2×C4×C191—24no—
99C2×C4×C191C2×D412yes—
100C2×C4×C191C2×D412yes—
101C2×C4×C191C2×D414yes—
102C2×C4×C191C2×D412yes—
103C2×C4×C191C2×Q814yescomplemented
104C22×C191C22×D411yescharacteristic
105C22×C191C22×D411yescharacteristic
106C22×C191C22×D411yescharacteristic
107C22×C191C22×Q812yes—
108C22×C1912−1+412yes—
109C2×D191—22no—
110C2×D191—22no—
111C2×D191—24no—
112C2×D191—24no—
113C2×D191C4⋊Q814yescomplemented
114C4×C191—24no—
115C4×C191—24no—
116C4×C191—42no—
117C4×C191#12014yes—
118Dic191—24no—
119Dic191—22no—
120Dic191—24no—
121Dic191—42no—
122C2×C191C2×#12011yescharacteristic
123C2×C191C2×C4⋊Q811yescharacteristic
124C2×C191#53611yescharacteristic
125C2×C191D4×Q812yes—
126C2×C191D4×Q812yes—
127D191—24no—
128C191#243411yescharacteristic, complemented, Sylow
129#2434—1911nomaximal, Sylow
130C23×Q8—1911no—
131C2×C4⋊Q8—1911no—
132C2×C4⋊Q8—1911no—
133C2×C22⋊Q8—1911no—
134#395—1914no—
135#391—1911no—
136#391D19111yescharacteristic, complemented
137#385—1914no—
138C4×C22⋊C4—1911no—
139C22×Q8—3824no—
140C22×Q8—3824no—
141C22×Q8—3824no—
142C22×Q8—1911no—
143C22×Q8—1912no—
144C22×Q8—1911no—
145C22×Q8C2×D19111yescharacteristic
146C23×C4—1912no—
147C23×C4—1911no—
148C4⋊Q8—3824no—
149C4⋊Q8—3824no—
150C22⋊Q8—3824no—
151C2×C4⋊C4—1912no—
152C2×C4⋊C4—1914no—
153C2×C4⋊C4—1914no—
154C2×C4⋊C4C2×D19111yescharacteristic
155C2×C22⋊C4—1912no—
156C2×C22⋊C4—1912no—
157C2×C42—1911no—
158C2×C42C2×D19111yescharacteristic
159#95—1912no—
160#95—1914no—
161#95C2×D19114yes—
162C24—1911no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—3824no—
166C2×Q8—3824no—
167C2×Q8—38232no—
168C2×Q8—3822no—
169C2×Q8—3822no—
170C2×Q8—3822no—
171C2×Q8—3824no—
172C2×Q8—3822no—
173C2×Q8—3822no—
174C2×Q8—1914no—
175C2×Q8D76414yes—
176C22×C4—3824no—
177C22×C4—3824no—
178C22×C4—3822no—
179C22×C4—3824no—
180C22×C4—3822no—
181C22×C4—1912no—
182C22×C4—1912no—
183C22×C4—1912no—
184C22×C4—1911no—
185C22×C4—1914no—
186C22×C4C22×D19112yes—
187C22×C4C22×D19112yes—
188C22×C4C22×D19112yes—
189C22×C4C22×D19111yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—3824no—
192C4⋊C4—3828no—
193C4⋊C4—3828no—
194C22⋊C4—3824no—
195C22⋊C4—1918no—
196C42—22no—
197C42—3822no—
198C23—3821no—
199C23—3821no—
200C23—3822no—
201C23—1912no—
202C23—1914no—
203C23C23×D19111yescharacteristic
204Q8—28no—
205Q8—38216no—
206Q8—3828no—
207Q8—7644no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—38216no—
215C2×C4—3822no—
216C2×C4—3821no—
217C2×C4—3822no—
218C2×C4—3822no—
219C2×C4—3821no—
220C2×C4—3822no—
221C2×C4—3828no—
222C2×C4—3822no—
223C2×C4—3822no—
224C2×C4—3822no—
225C2×C4—1914no—
226C2×C4—1912no—
227C2×C4—1914no—
228C2×C4D4×D19112yes—
229C2×C4D4×D19112yes—
230C2×C4C2×D76414yes—
231C2×C4C2×D76412yes—
232C2×C4Q8×D19114yescomplemented
233C22—3822no—
234C22—3824no—
235C22—3824no—
236C22—3822no—
237C22—1914no—
238C22C2×D4×D19111yescharacteristic
239C22C2×D4×D19111yescharacteristic
240C22C22×D76411yescharacteristic
241C22C2×Q8×D19112yes—
242C22#2025936912yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—3824no—
247C4—3824no—
248C4—3822no—
249C4—7642no—
250C4#2025929214yes—
251C2—3824no—
252C2C2×#2025929211yescharacteristic
253C2C4⋊Q8×D19111yescharacteristic
254C2#2713323311yescharacteristic
255C2#2713332612yes—
256C2Q8×D76412yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#391×C191C2
prime radicalC23×D191C23
hypercenter#391D191
hypocenterC191#2434
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#20259322C2semidirect
3. C2×#20259320C2semidirect
5. #27132584C2semidirect
6. #27132568C2semidirect
9. #391×C191C2semidirect
58. (C2×D191)⋊C4Q8semidirect
88. C2×Dic191C2×Q8semidirect
103. C2×C4×C191C2×Q8semidirect
113. C2×D191C4⋊Q8semidirect
128. C191#2434semidirect
136. #391D191semidirect
232. C2×C4Q8×D191semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1912
C2×#202593222
C2×#202593202
C2×#202593022
#271325842
#271325682
#271325372
C4×(C2×D191)⋊C42
#391×C1912
#2434191