Finite Group Databasebeta

← #37574204

Subgroups — #37574204

Order 24448.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#27133214C211yesmaximal, characteristic
2#27132646C212yesmaximal, complemented
3#27132462C211yesmaximal, characteristic, complemented
4#27132459C212yesmaximal, complemented
5#356×C191C211yesmaximal, characteristic, complemented
6C2×#15690288C411yescharacteristic
7C2×#15690288C2211yescharacteristic
8M16×D191—22no—
9#20259319C411yescharacteristic
10#20259311C411yescharacteristic
11#20259311C2211yescharacteristic
12#20259297—22no—
13#20259284—22no—
14C42×D191C411yescharacteristic
15#20259241C2212yes—
16#20259238—22no—
17C4⋊Q8×C191C2211yescharacteristic
18#101×C191C2212yes—
19#15690288—41no—
20#15690288—41no—
21#15690288—41no—
22#15690288—41no—
23C2×D764—21no—
24C2×D764—22no—
25C2×D764C2×C411yescharacteristic
26C2×D764C2×C411yescharacteristic
27C2×D764D412yes—
28C2×C4×D191—21no—
29C2×C4×D191—21no—
30C2×C4×D191C2×C411yescharacteristic
31C2×C4×D191D412yes—
32#15690271—22no—
33#15690270—42no—
34C8×D191—42no—
35(C2×D191)⋊C4—22no—
36(C2×D191)⋊C4—22no—
37C4×Dic191—21no—
38C4×Dic191C2×C411yescharacteristic
39C2×Q8×C191C2×C411yescharacteristic
40C2×Q8×C191C2311yescharacteristic
41M16×C191—22no—
42C4⋊C4×C191—22no—
43C42×C191C2×C411yescharacteristic
44C22×D191—21no—
45C22×D191C22⋊C412yes—
46C22×D191C22⋊C411yescharacteristic
47D764—41no—
48D764—42no—
49D764—41no—
50D764—44no—
51D764—42no—
52C2×Dic191—21no—
53C2×Dic191C22⋊C411yescharacteristic
54C4×D191—22no—
55C4×D191—22no—
56C4×D191—21no—
57C4×D191—21no—
58C4×D191—42no—
59C4×D191—41no—
60C4×D191—41no—
61C191⋊C8—42no—
62Q8×C191—41no—
63Q8×C191—41no—
64C2×C4×C191—21no—
65C2×C4×C191—21no—
66C2×C4×C191C22×C411yescharacteristic
67C2×C4×C191C2×D412yes—
68C8×C191—42no—
69C22×C191C2×C22⋊C411yescharacteristic
70C2×D191—21no—
71C2×D191—22no—
72C2×D191—41no—
73C2×D191—42no—
74C2×D191—42no—
75C2×D191—41no—
76C4×C191—22no—
77C4×C191—22no—
78C4×C191—21no—
79C4×C191—41no—
80Dic191—41no—
81Dic191#9911yescharacteristic
82Dic191#9911yescharacteristic
83C2×C191—21no—
84C2×C191C2×#9911yescharacteristic
85D191—41no—
86D191—42no—
87D191—81no—
88C191#213211yescharacteristic, complemented, Sylow
89#2132—1911nomaximal, Sylow
90#532—1911no—
91#413—1912no—
92#356—1911no—
93#356D19111yescharacteristic, complemented
94#355—1912no—
95C2×#41—1911no—
96C2×#41—1911no—
97C2×M16—3822no—
98C4⋊Q8C2×D19111yescharacteristic
99#127—1911no—
100#124—1911no—
101#124—1911no—
102#121—3822no—
103C4×D4—3822no—
104C2×C42—1911no—
105#101—1912no—
106#101C2×D19112yes—
107#100—3822no—
108#41—7641no—
109#41—7641no—
110#41—7641no—
111#41—7641no—
112C2×Q8C4×D19111yescharacteristic
113C2×Q8C22×D19111yescharacteristic
114C2×D4—3822no—
115C2×D4—3821no—
116C2×D4—1911no—
117C2×D4—1911no—
118C2×D4—1912no—
119C22×C4—3821no—
120C22×C4—3821no—
121C22×C4—1911no—
122C22×C4—1912no—
123M16—22no—
124M16—3822no—
125M16—7642no—
126C2×C8—7642no—
127C4⋊C4—22no—
128C22⋊C4—3822no—
129C22⋊C4—3822no—
130C42—3821no—
131C42—1911no—
132C42C4×D19111yescharacteristic
133C23—3821no—
134C23—1912no—
135C23—1911no—
136Q8—41no—
137Q8—41no—
138D4—7642no—
139D4—7641no—
140D4—7641no—
141D4—7644no—
142D4—7642no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—3822no—
146C2×C4—3822no—
147C2×C4—3821no—
148C2×C4—3821no—
149C2×C4—3821no—
150C2×C4—1911no—
151C2×C4—7642no—
152C2×C4—7641no—
153C2×C4—7641no—
154C2×C4C2×C4×D19111yescharacteristic
155C2×C4D4×D19112yes—
156C8—42no—
157C8—7642no—
158C22—3821no—
159C22—3822no—
160C22—7642no—
161C22—7641no—
162C22—7642no—
163C22—7641no—
164C22C22⋊C4×D19111yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—1911no—
170C4—1911no—
171C4—7641no—
172C2—21no—
173C2—15281no—
174C2—7641no—
175C2—7642no—
176C2#99×D19111yescharacteristic
subgroupG/H
centerC2#99×D191
commutatorC2×C4×C191C22×C4
FrattiniC2×Q8C22×D191
Fitting#356×C191C2
socleC2×C191C2×#99
prime radical#27133214C2
cosocleC2×Q8×C191C23
hypercenter#356D191
hypocenterC191#2132
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #27132646C2semidirect
3. #27132462C2semidirect
4. #27132459C2semidirect
5. #356×C191C2semidirect
88. C191#2132semidirect
93. #356D191semidirect

Maximal subgroups

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

subgroupindex
#271332142
#271326462
#271324622
#271324592
#356×C1912
#2132191