Finite Group Databasebeta

← #37572469

Subgroups — #37572469

Order 24448.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#27133499C211yesmaximal, characteristic
2#27132645C212yesmaximal, complemented
3#27132631C212yesmaximal, complemented
4#27132630C211yesmaximal, characteristic, complemented
5#413×C191C211yesmaximal, characteristic, complemented
6#20259370—21no—
7#20259369—22no—
8#20259368—22no—
9#20259367—21no—
10#20259367C414yes—
11C2×#15690287—22no—
12C2×#15690287C2212yes—
13#41×D191—24no—
14C2×#15690271C2211yescharacteristic, complemented
15C22×C191⋊C8C2211yescharacteristic, complemented
16#20259240—22no—
17#20259238—22no—
18C22⋊(C191⋊C8)—22no—
19(C2×D191)⋊C8—24no—
20C2×#41×C191C2211yescharacteristic
21C2×M16×C191C2212yescomplemented
22#101×C191—21no—
23#100×C191—21no—
24#15690288—24no—
25#15690287—24no—
26#15690287—24no—
27#15690287—24no—
28#15690287—22no—
29#15690287—22no—
30#15690287—22no—
31#15690287—22no—
32#15690286—24no—
33#15690286—28no—
34C2×Dic382—22no—
35C2×Dic382D412yes—
36Q8×D191—24no—
37C2×D764—22no—
38C2×D764D412yes—
39C2×C22⋊D191—24no—
40C2×C22⋊D191C2×C414yes—
41D4×D191—24no—
42D4×D191—28no—
43C2×C4×D191—24no—
44C2×C4×D191D414yes—
45#15690271—22no—
46C2×C191⋊C8—22no—
47C2×C191⋊C8—21no—
48C2×C191⋊C8—21no—
49#41×C191—22no—
50#41×C191—22no—
51C2×Q8×C191D411yescharacteristic
52C2×D4×C191C2×C412yes—
53C2×D4×C191D411yescharacteristic
54C22×C4×C191D412yes—
55C22×C4×C191C2311yescharacteristic
56M16×C191—22no—
57M16×C191—22no—
58C2×C8×C191—22no—
59C22×D191—24no—
60C22×D191C22⋊C414yes—
61Dic382—24no—
62Dic382—22no—
63Dic382—22no—
64D764—24no—
65D764—22no—
66D764—22no—
67C22⋊D191—28no—
68C22⋊D191—24no—
69C22⋊D191—24no—
70C2×Dic191—24no—
71C2×Dic191C22⋊C414yes—
72C4×D191—28no—
73C4×D191—24no—
74C4×D191—24no—
75C191⋊C8—41no—
76C191⋊C8—41no—
77C23×C191C22⋊C412yes—
78C23×C191C22×C411yescharacteristic
79Q8×C191—22no—
80D4×C191—24no—
81D4×C191—22no—
82C2×C4×C191—22no—
83C2×C4×C191—22no—
84C2×C4×C191C22⋊C412yes—
85C2×C4×C191C2×D412yes—
86C2×C4×C191C2×D412yes—
87C2×C4×C191C2×D411yescharacteristic
88C2×C4×C191C2×D411yescharacteristic
89C8×C191—42no—
90C22×C191—22no—
91C22×C191—41no—
92C22×C191—42no—
93C22×C191C2×C22⋊C412yes—
94C22×C191C2×C22⋊C411yescharacteristic
95C2×D191—24no—
96C2×D191—44no—
97C2×D191—84no—
98C4×C191—22no—
99C4×C191#12012yes—
100C4×C191#12011yescharacteristic
101C4×C191#12011yescharacteristic
102Dic191—24no—
103C2×C191—22no—
104C2×C191—21no—
105C2×C191—42no—
106C2×C191#37911yescharacteristic
107D191—44no—
108C191#179111yescharacteristic, complemented, Sylow
109#1791—1911nomaximal, Sylow
110#585—1911no—
111#413—1912no—
112#413D19111yescharacteristic, complemented
113#408—1911no—
114#408—1912no—
1152−1+4—3822no—
1162−1+4—3821no—
117H(5,𝔽2)—3822no—
118H(5,𝔽2)—3821no—
119H(5,𝔽2)—1914no—
120C2×#41—3824no—
121C2×#41—3822no—
122C2×#41—1912no—
123C2×#41C2×D19111yescharacteristic
124C2×M16—1911no—
125C2×M16C2×D19112yescomplemented
126C22×C8—1911no—
127#101—21no—
128#101—3822no—
129#100—21no—
130#100—3822no—
131C22⋊C8—3824no—
132C22⋊C8—3822no—
133#41—22no—
134#41—22no—
135#41—3824no—
136#41—3824no—
137#41—3824no—
138#41—3824no—
139#41—3828no—
140#41—3824no—
141#41—3822no—
142#41—3822no—
143#41—3822no—
144#41—3822no—
145C2×Q8—3822no—
146C2×Q8—3824no—
147C2×Q8—1912no—
148C2×Q8C22⋊D19111yescharacteristic
149C2×D4—3822no—
150C2×D4—3824no—
151C2×D4—3828no—
152C2×D4—3824no—
153C2×D4—1914no—
154C2×D4—1912no—
155C2×D4C4×D19112yes—
156C2×D4C22⋊D19111yescharacteristic
157C22×C4—3824no—
158C22×C4—1914no—
159C22×C4D76412yes—
160C22×C4C22×D19111yescharacteristic
161M16—22no—
162M16—22no—
163M16—3822no—
164C2×C8—22no—
165C2×C8—3822no—
166C2×C8—3821no—
167C2×C8—3821no—
168C23—3824no—
169C23—1914no—
170C23(C2×D191)⋊C412yes—
171C23C2×C4×D19111yescharacteristic
172Q8—22no—
173Q8—3824no—
174Q8—3822no—
175Q8—3822no—
176D4—24no—
177D4—22no—
178D4—3824no—
179D4—3828no—
180D4—3824no—
181D4—3824no—
182D4—3822no—
183D4—3822no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3828no—
187C2×C4—3824no—
188C2×C4—3824no—
189C2×C4—3824no—
190C2×C4—1914no—
191C2×C4(C2×D191)⋊C412yes—
192C2×C4D4×D19112yes—
193C2×C4D4×D19112yes—
194C2×C4C2×C22⋊D19111yescharacteristic
195C2×C4C2×D76411yescharacteristic
196C8—42no—
197C8—7641no—
198C8—7641no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—3824no—
203C22—15284no—
204C22—7644no—
205C22C22⋊C4×D19112yes—
206C22C2×(C2×D191)⋊C411yescharacteristic
207C4—22no—
208C4—3824no—
209C4#2025929012yes—
210C4#2025929211yescharacteristic
211C4#2025929211yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—7644no—
216C2#2713251811yescharacteristic
subgroupG/H
centerC4#20259292
commutatorC23×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#413×C191C2
socleC2×C191#379
prime radical#27133499C2
cosocleC22×C4×C191C23
hypercenter#413D191
hypocenterC191#1791
LCSG ▷ C23×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27132645C2semidirect
3. #27132631C2semidirect
4. #27132630C2semidirect
5. #413×C191C2semidirect
14. C2×#15690271C22semidirect
15. C22×C191⋊C8C22semidirect
21. C2×M16×C191C22semidirect
108. C191#1791semidirect
112. #413D191semidirect
125. C2×M16C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334992
#271326452
#271326312
#271326302
#413×C1912
#1791191