Finite Group Databasebeta

← #37571524

Subgroups — #37571524

Order 24448.

254 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 51 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#27133240C211yesmaximal, characteristic
2#27132977C211yesmaximal, characteristic, complemented
3#27132974C211yesmaximal, characteristic, complemented
4#27132595C211yesmaximal, characteristic, complemented
5#27132396C211yesmaximal, characteristic, complemented
6#27132388C211yesmaximal, characteristic, complemented
7#327×C191C211yesmaximal, characteristic, complemented
8C2×#15690286—21no—
9#20259300—21no—
10#20259299—21no—
11#20259296C2211yescharacteristic
12#20259295—21no—
13#20259294—21no—
14#20259294—21no—
15C2×C4⋊Dic191—21no—
16C2×C4⋊Dic191C2211yescharacteristic
17C2×Dic191⋊C4C2211yescharacteristic
18#20259256—21no—
19#20259255—21no—
20#20259244—21no—
21#20259243—21no—
22C22⋊(C191⋊C8)C2211yescharacteristic
23C191⋊#95—21no—
24#121×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic
26#95×C191C2211yescharacteristic
27#15690286—42no—
28#15690286—42no—
29C2×Dic382—21no—
30C2×D764D411yescharacteristic
31C2×C22⋊D191—21no—
32C2×C22⋊D191—21no—
33C2×C22⋊D191—21no—
34C22×Dic191—21no—
35C22×Dic191—21no—
36C22×Dic191D411yescharacteristic
37C22×Dic191D411yescharacteristic
38C2×C4×D191—21no—
39C2×C191⋊C8—21no—
40C4⋊Dic191—21no—
41C4⋊Dic191—21no—
42C4⋊Dic191—42no—
43C4⋊Dic191D411yescharacteristic
44Dic191⋊C4—21no—
45Dic191⋊C4—21no—
46Dic191⋊C4—21no—
47C22⋊Dic191—21no—
48(C2×D191)⋊C4—21no—
49(C2×D191)⋊C4—21no—
50C2×D4×C191—21no—
51C2×D4×C191D411yescharacteristic
52C22×C4×C191—21no—
53C22×C4×C191C2311yescharacteristic
54C2×C8×C191—21no—
55C4⋊C4×C191D411yescharacteristic
56C22⋊C4×C191—21no—
57C22×D191—21no—
58Dic382—42no—
59D764—41no—
60C22⋊D191—42no—
61C22⋊D191—41no—
62C22⋊D191—41no—
63C22⋊D191—81no—
64C2×Dic191—21no—
65C2×Dic191—21no—
66C2×Dic191—21no—
67C2×Dic191—21no—
68C2×Dic191—42no—
69C2×Dic191—41no—
70C2×Dic191—41no—
71C2×Dic191—41no—
72C2×Dic191—41no—
73C2×Dic191D812yes—
74C4×D191—42no—
75C191⋊C8—41no—
76C23×C191—21no—
77C23×C191C2×D411yescharacteristic
78D4×C191—41no—
79D4×C191—41no—
80D4×C191—41no—
81C2×C4×C191—22no—
82C2×C4×C191—21no—
83C2×C4×C191—41no—
84C2×C4×C191—41no—
85C2×C4×C191—41no—
86C2×C4×C191C2×D411yescharacteristic
87C2×C4×C191C2×D411yescharacteristic
88C8×C191—41no—
89C22×C191—21no—
90C22×C191—21no—
91C22×C191—41no—
92C22×C191—41no—
93C22×C191—41no—
94C22×C191#12011yescharacteristic
95C22×C191C2×D811yescharacteristic
96C22×C191AGL(1,ℤ/8)11yescharacteristic
97C2×D191—41no—
98C2×D191—41no—
99C2×D191—41no—
100C4×C191—21no—
101C4×C191—21no—
102C4×C191—42no—
103C4×C191—41no—
104Dic191—22no—
105Dic191—41no—
106Dic191—41no—
107Dic191—81no—
108C2×C191—21no—
109C2×C191—21no—
110C2×C191—81no—
111C2×C191#44711yescharacteristic
112C2×C191#45611yescharacteristic
113C2×C191#45811yescharacteristic
114D191—81no—
115C191#161011yescharacteristic, complemented, Sylow
116#1610—1911nomaximal, Sylow
117#537—1911no—
118#480—1911no—
119#480—1911no—
120#398—1911no—
121#327—1911no—
122#327D19111yescharacteristic, complemented
123#324—1911no—
124C2×#41—3821no—
125C22⋊Q8—3821no—
126C22⋊Q8—3821no—
127#121—3821no—
128#121—3821no—
129#121—3821no—
130#121—1911no—
131#121C2×D19111yescharacteristic
132C2×C4⋊C4—3821no—
133C2×C4⋊C4—1911no—
134C2×C4⋊C4—1911no—
135#107—3821no—
136#107—3821no—
137#102—3821no—
138#102—3821no—
139C22⋊C8—1911no—
140C22⋊C8C2×D19111yescharacteristic
141#95—3821no—
142#95C2×D19111yescharacteristic
143#41—7642no—
144#41—7642no—
145C2×Q8—3821no—
146C2×D4—21no—
147C2×D4—3821no—
148C2×D4—3821no—
149C2×D4—3821no—
150C2×D4—1911no—
151C2×D4D76411yescharacteristic
152C22×C4—21no—
153C22×C4—3821no—
154C22×C4—3821no—
155C22×C4—3821no—
156C22×C4—1911no—
157C22×C4—1911no—
158C22×C4C22×D19111yescharacteristic
159C2×C8—21no—
160C2×C8—3821no—
161C4⋊C4—3821no—
162C4⋊C4—3821no—
163C4⋊C4—3821no—
164C4⋊C4—3821no—
165C4⋊C4—3821no—
166C4⋊C4—1911no—
167C4⋊C4—7642no—
168C4⋊C4D76411yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—3821no—
171C22⋊C4—3821no—
172C22⋊C4—3821no—
173C23—21no—
174C23—3821no—
175C23D4×D19111yescharacteristic
176Q8—7642no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—15281no—
181D4—7641no—
182D4—7642no—
183D4—7641no—
184D4—7641no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3821no—
191C2×C4—3821no—
192C2×C4—3821no—
193C2×C4—3821no—
194C2×C4—1912no—
195C2×C4—7642no—
196C2×C4—7642no—
197C2×C4—7641no—
198C2×C4—7641no—
199C2×C4—7641no—
200C2×C4—7641no—
201C2×C4D4×D19111yescharacteristic
202C2×C4C2×D76411yescharacteristic
203C8—41no—
204C8—7641no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—7641no—
211C22—7641no—
212C22—7641no—
213C22#2025929211yescharacteristic
214C22D8×D19111yescharacteristic
215C22#2025934911yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—3822no—
221C4—15281no—
222C4—7641no—
223C4—7641no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—15281no—
228C2#2713279811yescharacteristic
229C2#2713284711yescharacteristic
230C2#2713285411yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#327×C191C2
socleC22×C191#120
prime radical#27133240C2
hypercenter#327D191
hypocenterC191#1610
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27132977C2semidirect
3. #27132974C2semidirect
4. #27132595C2semidirect
5. #27132396C2semidirect
6. #27132388C2semidirect
7. #327×C191C2semidirect
115. C191#1610semidirect
122. #327D191semidirect

Maximal subgroups

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

subgroupindex
#271332402
#271329772
#271329742
#271325952
#271323962
#271323882
#327×C1912
#1610191