Finite Group Databasebeta

← #37573569

Subgroups — #37573569

Order 24448.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20259355C212yesmaximal, complemented
2#27133244C211yesmaximal, characteristic
3C2×#20259248C212yesmaximal, complemented
4#27132568C211yesmaximal, characteristic, complemented
5#343×C191C211yesmaximal, characteristic, complemented
6C2×#15690287C2212yes—
7C22×Dic382C2211yescharacteristic
8#20259355—24no—
9#20259355—24no—
10C2×Dic764—22no—
11C2×#15690276—22no—
12#20259302—22no—
13#20259297—22no—
14#20259296—22no—
15C2×C4×Dic191C2211yescharacteristic
16#20259248—24no—
17C191⋊#95—22no—
18C2×M16×C191C2212yes—
19C2×C4⋊C4×C191C2211yescharacteristic
20#15690287—22no—
21#15690287—22no—
22#15690287—22no—
23#15690287—22no—
24C2×Dic382—22no—
25C2×Dic382—22no—
26C2×Dic382—21no—
27C2×Dic382D412yes—
28C2×Dic382D412yescomplemented
29C2×D764—22no—
30C2×D764D412yes—
31C2×C22⋊D191—22no—
32C22×Dic191—21no—
33C22×Dic191—21no—
34C2×C4×D191—22no—
35Dic764—28no—
36#15690276—28no—
37C4⋊Dic191—22no—
38(C2×D191)⋊C4—24no—
39C4×Dic191—22no—
40C22×C4×C191D412yes—
41C22×C4×C191C2311yescharacteristic
42M16×C191—22no—
43M16×C191D414yescomplemented
44C2×C8×C191—22no—
45C4⋊C4×C191—22no—
46C22×D191—22no—
47Dic382—24no—
48Dic382—22no—
49Dic382—22no—
50Dic382—42no—
51D764—22no—
52D764—22no—
53D764—82no—
54C22⋊D191—42no—
55C22⋊D191—42no—
56C2×Dic191—22no—
57C2×Dic191—22no—
58C2×Dic191—22no—
59C2×Dic191—41no—
60C2×Dic191—41no—
61C2×Dic191—41no—
62C2×Dic191—41no—
63C4×D191—42no—
64C4×D191—42no—
65C23×C191C2×D411yescharacteristic
66C2×C4×C191—22no—
67C2×C4×C191—42no—
68C2×C4×C191C2×D411yescharacteristic
69C2×C4×C191C2×D412yes—
70C2×C4×C191C2×D412yes—
71C2×C4×C191#4111yescharacteristic
72C8×C191—24no—
73C22×C191—21no—
74C22×C191—21no—
75C22×C191#12011yescharacteristic
76C22×C191#12111yescharacteristic
77C22×C191#12711yescharacteristic
78C2×D191—42no—
79C2×D191—42no—
80C2×D191—42no—
81C4×C191—42no—
82C4×C191#12012yes—
83C4×C191#12112yes—
84Dic191—42no—
85Dic191—42no—
86Dic191—42no—
87C2×C191—21no—
88C2×C191—21no—
89C2×C191#39211yescharacteristic
90C2×C191#45511yescharacteristic
91C2×C191#45611yescharacteristic
92D191—82no—
93C191#200711yescharacteristic, complemented, Sylow
94#2007—1911nomaximal, Sylow
95C2×#137—1912no—
96#537—1911no—
97C2×#104—1912no—
98#391—1911no—
99#343D19111yescharacteristic, complemented
100C2×#41—1912no—
101C22×Q8—1911no—
102#137—3824no—
103#137—3824no—
104C2×Q16—3822no—
105C2×SD16—3822no—
106C2×M16C2×D19112yes—
107C22⋊Q8—3822no—
108#121—3822no—
109#121—3822no—
110C2×C4⋊C4C2×D19111yescharacteristic
111C2×C42—1911no—
112#104—3824no—
113#95—3822no—
114#41—3822no—
115#41—3822no—
116#41—3822no—
117#41—3822no—
118C2×Q8—3821no—
119C2×Q8—3822no—
120C2×Q8—3822no—
121C2×Q8—1912no—
122C2×Q8—1912no—
123C2×D4—3822no—
124C2×D4—3822no—
125C2×D4—1912no—
126C22×C4—3821no—
127C22×C4—3821no—
128C22×C4—3822no—
129C22×C4D76412yes—
130C22×C4C22×D19111yescharacteristic
131Q16—3828no—
132SD16—3828no—
133M16—22no—
134M16D76414yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—3822no—
138C22⋊C4—3824no—
139C42—3822no—
140C23—3822no—
141C23C2×D76411yescharacteristic
142Q8—3824no—
143Q8—3822no—
144Q8—3822no—
145Q8—7642no—
146D4—3822no—
147D4—3822no—
148D4—15282no—
149D4—7642no—
150D4—7642no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—3822no—
154C2×C4—3822no—
155C2×C4—3822no—
156C2×C4—7641no—
157C2×C4—7641no—
158C2×C4—7641no—
159C2×C4—7641no—
160C2×C4—7642no—
161C2×C4—7642no—
162C2×C4D4×D19111yescharacteristic
163C2×C4D4×D19112yes—
164C2×C4C2×D76412yes—
165C2×C4#1569028811yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—7642no—
170C22—7642no—
171C22—7642no—
172C22#2025929211yescharacteristic
173C22#2025929711yescharacteristic
174C22#2025931911yescharacteristic
175C4—42no—
176C4—7642no—
177C4—7642no—
178C4—7642no—
179C4#2025929212yes—
180C4#2025929712yes—
181C2—21no—
182C2—21no—
183C2—15282no—
184C2#2713257311yescharacteristic
185C2#2713284311yescharacteristic
186C2#2713284811yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#343×C191C2
socleC22×C191#120
prime radical#27133244C2
hypercenter#343D191
hypocenterC191#2007
LCSG ▷ C22×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
3. C2×#20259248C2semidirect
4. #27132568C2semidirect
5. #343×C191C2semidirect
28. C2×Dic382D4semidirect
43. M16×C191D4semidirect
93. C191#2007semidirect
99. #343D191semidirect
134. M16D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
#271332442
C2×#202592482
#271325682
#343×C1912
#2007191