Finite Group Databasebeta

← #37574207

Subgroups — #37574207

Order 24448.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#27133237C211yesmaximal, characteristic
2C2×#20259241C211yesmaximal, characteristic, complemented
3#27132461C212yesmaximal, complemented
4#27132459C212yesmaximal, complemented
5C2×#101×C191C211yesmaximal, characteristic, complemented
6C2×#15690287C411yescharacteristic
7C2×#15690271—21no—
8#20259320C411yescharacteristic
9#20259320C2211yescharacteristic
10#20259311C411yescharacteristic
11#20259311C2211yescharacteristic
12#20259305—21no—
13#20259305—21no—
14#20259298—21no—
15#20259298—21no—
16#20259281C411yescharacteristic
17#20259241—21no—
18#20259241C2212yes—
19C22×Q8×C191C2211yescharacteristic
20C2×M16×C191—21no—
21#101×C191—21no—
22#101×C191C2212yes—
23#15690287—41no—
24#15690287—41no—
25C2×Dic382C2×C411yescharacteristic
26C2×D764C2×C411yescharacteristic
27C2×C22⋊D191—21no—
28C2×C4×D191—21no—
29#15690271—22no—
30#15690271—41no—
31C2×C191⋊C8—41no—
32C4⋊Dic191—21no—
33Dic191⋊C4—21no—
34Dic191⋊C4—21no—
35Dic191⋊C4—21no—
36Dic191⋊C4—21no—
37C22⋊Dic191—21no—
38(C2×D191)⋊C4—21no—
39(C2×D191)⋊C4—21no—
40(C2×D191)⋊C4—21no—
41(C2×D191)⋊C4—21no—
42C4×Dic191C2×C411yescharacteristic
43C4×Dic191C2×C411yescharacteristic
44C2×Q8×C191—41no—
45C2×Q8×C191—41no—
46C2×Q8×C191C2×C411yescharacteristic
47C2×Q8×C191D411yescharacteristic
48C2×Q8×C191D411yescharacteristic
49C2×Q8×C191C2311yescharacteristic
50C22×C4×C191C2×C411yescharacteristic
51C22×C4×C191D411yescharacteristic
52C22×C4×C191D411yescharacteristic
53M16×C191—22no—
54M16×C191—41no—
55C2×C8×C191—41no—
56C22×D191—21no—
57Dic382—41no—
58D764—41no—
59C22⋊D191—41no—
60C22⋊D191—41no—
61C2×Dic191—21no—
62C2×Dic191—21no—
63C2×Dic191—21no—
64C4×D191—42no—
65C191⋊C8—42no—
66C23×C191C22⋊C411yescharacteristic
67Q8×C191—41no—
68Q8×C191—41no—
69Q8×C191—41no—
70Q8×C191—41no—
71C2×C4×C191—21no—
72C2×C4×C191—21no—
73C2×C4×C191—21no—
74C2×C4×C191—21no—
75C2×C4×C191—21no—
76C2×C4×C191—21no—
77C2×C4×C191C22⋊C411yescharacteristic
78C2×C4×C191C22⋊C411yescharacteristic
79C2×C4×C191C22⋊C411yescharacteristic
80C2×C4×C191C22×C411yescharacteristic
81C2×C4×C191C2×D411yescharacteristic
82C2×C4×C191C2×D411yescharacteristic
83C8×C191—42no—
84C22×C191—41no—
85C22×C191#9911yescharacteristic
86C22×C191#9911yescharacteristic
87C22×C191C2×C22⋊C411yescharacteristic
88C2×D191—42no—
89C2×D191—41no—
90C4×C191—21no—
91C4×C191—21no—
92C4×C191—21no—
93C4×C191—21no—
94C4×C191—21no—
95C4×C191—21no—
96Dic191—41no—
97Dic191—41no—
98Dic191—41no—
99C2×C191—21no—
100C2×C191—21no—
101C2×C191—21no—
102C2×C191C2×#9911yescharacteristic
103D191—81no—
104C191#213311yescharacteristic, complemented, Sylow
105#2133—1911nomaximal, Sylow
106#536—1911no—
107C2×#101—1911no—
108C2×#101D19111yescharacteristic, complemented
109#356—1912no—
110#355—1912no—
111C2×#41—1911no—
112C22×Q8C2×D19111yescharacteristic
113C2×M16—21no—
114C2×M16—3821no—
115C4⋊Q8—1911no—
116C4⋊Q8—1911no—
117#124—1911no—
118#124—1911no—
119#123—3821no—
120#123—3821no—
121C22⋊Q8—3821no—
122C22⋊Q8—3821no—
123#117—1911no—
124#101—21no—
125#101—3821no—
126#101—1912no—
127#101C2×D19112yes—
128#41—7641no—
129#41—7641no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—1911no—
133C2×Q8C4×D19111yescharacteristic
134C2×Q8C22⋊D19111yescharacteristic
135C2×Q8D76411yescharacteristic
136C2×Q8C22×D19111yescharacteristic
137C2×D4—3821no—
138C2×D4—1911no—
139C22×C4—3821no—
140C22×C4C4×D19111yescharacteristic
141C22×C4C22⋊D19111yescharacteristic
142C22×C4D76411yescharacteristic
143M16—22no—
144M16—41no—
145M16—3822no—
146M16—7641no—
147C2×C8—41no—
148C2×C8—7641no—
149C4⋊C4—3821no—
150C4⋊C4—3821no—
151C4⋊C4—3821no—
152C4⋊C4—3821no—
153C4⋊C4—3821no—
154C22⋊C4—3821no—
155C22⋊C4—3821no—
156C22⋊C4—3821no—
157C22⋊C4—3821no—
158C22⋊C4—3821no—
159C42—1911no—
160C42—1911no—
161C23—3821no—
162C23(C2×D191)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—7641no—
168D4—7641no—
169D4—7641no—
170D4—7641no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—3821no—
178C2×C4—3821no—
179C2×C4—3821no—
180C2×C4—7642no—
181C2×C4(C2×D191)⋊C411yescharacteristic
182C2×C4(C2×D191)⋊C411yescharacteristic
183C2×C4(C2×D191)⋊C411yescharacteristic
184C2×C4C2×C4×D19111yescharacteristic
185C2×C4C2×C22⋊D19111yescharacteristic
186C2×C4C2×D76411yescharacteristic
187C8—42no—
188C8—7642no—
189C22—41no—
190C22—7642no—
191C22—7641no—
192C22#2025923611yescharacteristic
193C22#2025923611yescharacteristic
194C22C2×(C2×D191)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—7641no—
202C4—7641no—
203C4—7641no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—15281no—
208C2C2×#2025923611yescharacteristic
subgroupG/H
centerC2C2×#20259236
commutatorC2×C4×C191C22×C4
FrattiniC2×Q8C22×D191
FittingC2×#101×C191C2
socleC2×C191C2×#99
prime radicalC2×#15690287C4
cosocleC2×Q8×C191C23
hypercenterC2×#101D191
hypocenterC191#2133
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#20259241C2semidirect
3. #27132461C2semidirect
4. #27132459C2semidirect
5. C2×#101×C191C2semidirect
104. C191#2133semidirect
108. C2×#101D191semidirect

Maximal subgroups

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

subgroupindex
#271332372
C2×#202592412
#271324612
#271324592
C2×#101×C1912
#2133191