Finite Group Databasebeta

← #37583967

Subgroups — #37583967

Order 24448.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#27133275C211yesmaximal, characteristic, complemented
2#27133268C211yesmaximal, characteristic
3#27132971C212yesmaximal, complemented
4#27132964C212yesmaximal, complemented
5#27132909C212yesmaximal, complemented
6#27132892C212yesmaximal, complemented
7#27132869C212yesmaximal, complemented
8#27132864C212yesmaximal, complemented
9#424×C191C211yesmaximal, characteristic, complemented
10C2×#15690276C2212yes—
11C2×#15690276C2212yes—
12#20259322C2211yescharacteristic
13#20259319C2211yescharacteristic
14#20259315—22no—
15#20259314C2211yescharacteristic
16#20259314C2211yescharacteristic
17#20259312—22no—
18#20259307—22no—
19#20259301C2212yes—
20#20259296C2212yes—
21#20259295—22no—
22C4×Dic382C2212yes—
23C4×D764C2212yes—
24#20259256C2212yes—
25#20259254C2212yes—
26#20259247C2212yes—
27#20259247C2212yes—
28#20259244C2212yes—
29#20259244C2212yes—
30C2×M16×C191C2211yescharacteristic
31C22×C8×C191C2211yescharacteristic
32#117×C191C2211yescharacteristic
33C4⋊C8×C191C2212yes—
34C4⋊C8×C191C2212yes—
35C2×Dic382C2312yes—
36C2×D764—22no—
37C2×D764C2312yes—
38C2×C22⋊D191—22no—
39C2×C4×D191—22no—
40#15690276—42no—
41#15690276—42no—
42C4⋊Dic191—22no—
43C4⋊Dic191C2312yes—
44C4⋊Dic191C2312yes—
45Dic191⋊C4—22no—
46Dic191⋊C4—22no—
47Dic191⋊C4—22no—
48Dic191⋊C4—22no—
49C22⋊Dic191—22no—
50(C2×D191)⋊C4—22no—
51(C2×D191)⋊C4—22no—
52C4×Dic191—22no—
53C22×C4×C191C2311yescharacteristic
54M16×C191—41no—
55C2×C8×C191—41no—
56C2×C8×C191C2312yes—
57C2×C8×C191C2312yes—
58C4⋊C4×C191D412yes—
59C22⋊C4×C191D412yes—
60C42×C191C2311yescharacteristic
61C42×C191C2311yescharacteristic
62C22×D191—22no—
63Dic382—22no—
64Dic382—22no—
65D764—22no—
66D764—22no—
67D764—82no—
68C22⋊D191—82no—
69C2×Dic191—22no—
70C2×Dic191—22no—
71C2×Dic191—22no—
72C4×D191—82no—
73C23×C191C2×D411yescharacteristic
74C2×C4×C191—21no—
75C2×C4×C191—21no—
76C2×C4×C191C2×D412yes—
77C2×C4×C191C2×D412yes—
78C2×C4×C191C2×D411yescharacteristic
79C2×C4×C191C2411yescharacteristic
80C8×C191—22no—
81C8×C191—22no—
82C22×C191—21no—
83C22×C191—22no—
84C22×C191C22×D411yescharacteristic
85C2×D191—42no—
86C2×D191—42no—
87C2×D191—42no—
88C4×C191—21no—
89C4×C191—22no—
90C4×C191—22no—
91C4×C191H(5,𝔽2)11yescharacteristic
92C4×C1912−1+411yescharacteristic
93Dic191—42no—
94Dic191—42no—
95Dic191—42no—
96C2×C191—41no—
97C2×C191#53711yescharacteristic
98C2×C191#57712yes—
99D191—82no—
100C191#326011yescharacteristic, complemented, Sylow
101#3260—1911nomaximal, Sylow
102#541—1911no—
103#540—1911no—
104#479—1912no—
105#478—1912no—
106#468—1912no—
107#465—1912no—
108#461—1912no—
109#460—1912no—
110#424D19111yescharacteristic, complemented
111C2×SD16—1912no—
112C2×SD16—1912no—
113C2×M16C2×D19111yescharacteristic
114C22×C8C2×D19111yescharacteristic
115C4⋊Q8—1911no—
116#127—1911no—
117#126—3822no—
118#125—3822no—
119#125—1911no—
120#125—1911no—
121#123—3822no—
122C22⋊Q8—1912no—
123#121—3822no—
124#121—1912no—
125C4×Q8—1912no—
126C4×D4—1912no—
127#117C2×D19111yescharacteristic
128#107—1912no—
129#106—1912no—
130C4⋊C8C2×D19112yes—
131C4⋊C8C2×D19112yes—
132#103—1912no—
133#103—1912no—
134#102—1912no—
135#102—1912no—
136C2×Q8—1912no—
137C2×D4—3822no—
138C2×D4—3822no—
139C2×D4—1912no—
140C22×C4—3822no—
141C22×C4C22×D19111yescharacteristic
142SD16—7642no—
143SD16—7642no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D19112yes—
147C2×C8C22×D19112yes—
148C4⋊C4—3822no—
149C4⋊C4—3822no—
150C4⋊C4—3822no—
151C4⋊C4—3822no—
152C4⋊C4—3822no—
153C4⋊C4—1912no—
154C4⋊C4—1912no—
155C4⋊C4D76412yes—
156C22⋊C4—3822no—
157C22⋊C4—3822no—
158C22⋊C4—3822no—
159C22⋊C4D76412yes—
160C42—3822no—
161C42C22×D19111yescharacteristic
162C42C22×D19111yescharacteristic
163C23—3822no—
164C23C2×D76411yescharacteristic
165Q8—3822no—
166Q8—3822no—
167D4—3822no—
168D4—3822no—
169D4—15282no—
170D4—15282no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—3822no—
174C2×C4—3822no—
175C2×C4—3822no—
176C2×C4—15282no—
177C2×C4C2×D76412yes—
178C2×C4C2×D76412yes—
179C2×C4C2×D76411yescharacteristic
180C2×C4C23×D19111yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—7642no—
186C22—7642no—
187C22—7642no—
188C22C22×D76411yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—7642no—
193C4—7642no—
194C4—7642no—
195C4#2025936711yescharacteristic
196C4#2025937011yescharacteristic
197C2—41no—
198C2—15282no—
199C2#2713324411yescharacteristic
200C2#2713347512yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#424×C191C2
socleC22×C191C22×D4
prime radical#27133268C2
hypercenter#424D191
hypocenterC191#3260
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
1. #27133275C2semidirect
3. #27132971C2semidirect
4. #27132964C2semidirect
5. #27132909C2semidirect
6. #27132892C2semidirect
7. #27132869C2semidirect
8. #27132864C2semidirect
9. #424×C191C2semidirect
100. C191#3260semidirect
110. #424D191semidirect

Maximal subgroups

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

subgroupindex
#271332752
#271332682
#271329712
#271329642
#271329092
#271328922
#271328692
#271328642
#424×C1912
#3260191