Finite Group Databasebeta

← #37574539

Subgroups — #37574539

Order 24448.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133412C211yesmaximal, characteristic, complemented
2#27132846C211yesmaximal, characteristic, complemented
3#27132836C211yesmaximal, characteristic
4#27132462C211yesmaximal, characteristic, complemented
5#27132461C211yesmaximal, characteristic, complemented
6#27132459C211yesmaximal, characteristic, complemented
7#456×C191C211yesmaximal, characteristic, complemented
8#20259352—21no—
9#20259347—21no—
10#20259320C2211yescharacteristic
11#20259317—21no—
12#20259316—21no—
13#20259312—21no—
14#20259311C2211yescharacteristic
15#20259307—21no—
16#20259300—21no—
17#20259298—21no—
18#20259249—21no—
19#20259249—21no—
20#20259241C2211yescharacteristic
21#20259241C2211yescharacteristic
222−1+4×C191C2211yescharacteristic
23#137×C191—21no—
24C4⋊Q8×C191C2211yescharacteristic
25#104×C191—21no—
26#101×C191C2211yescharacteristic
27C2×Dic382D411yescharacteristic
28C2×D764D411yescharacteristic
29C2×C4×D191—21no—
30#15690277—41no—
31#15690275—41no—
32#15690274—41no—
33#15690272—41no—
34#15690271—21no—
35#15690271—21no—
36C4⋊Dic191—21no—
37Dic191⋊C4—21no—
38Dic191⋊C4—21no—
39Dic191⋊C4—21no—
40Dic191⋊C4—21no—
41Dic191⋊C4—21no—
42(C2×D191)⋊C4—21no—
43(C2×D191)⋊C4—21no—
44(C2×D191)⋊C4—21no—
45(C2×D191)⋊C4—21no—
46(C2×D191)⋊C4—21no—
47C4×Dic191D411yescharacteristic
48C4×Dic191D411yescharacteristic
49#41×C191—21no—
50#41×C191—21no—
51#41×C191—21no—
52#41×C191—41no—
53C2×Q8×C191—41no—
54C2×Q8×C191D411yescharacteristic
55C2×Q8×C191C2311yescharacteristic
56Q16×C191—41no—
57SD16×C191—41no—
58M16×C191—21no—
59C4⋊C4×C191—21no—
60C4⋊C4×C191—21no—
61C42×C191D411yescharacteristic
62C22×D191—21no—
63Dic382—41no—
64D764—41no—
65C2×Dic191—21no—
66C2×Dic191—21no—
67C2×Dic191—21no—
68C4×D191—81no—
69C191⋊C8—41no—
70C191⋊C8—41no—
71Q8×C191—21no—
72Q8×C191—21no—
73Q8×C191—21no—
74Q8×C191—41no—
75Q8×C191—41no—
76D4×C191—21no—
77D4×C191—21no—
78D4×C191—21no—
79D4×C191—41no—
80C2×C4×C191—21no—
81C2×C4×C191—21no—
82C2×C4×C191—41no—
83C2×C4×C191—41no—
84C2×C4×C191—41no—
85C2×C4×C191C2×D411yescharacteristic
86C2×C4×C191C2×D411yescharacteristic
87C2×C4×C191C2×D411yescharacteristic
88C8×C191—41no—
89C22×C191—41no—
90C22×C191#12011yescharacteristic
91C2×D191—42no—
92C2×D191—41no—
93C4×C191—21no—
94C4×C191—21no—
95C4×C191—21no—
96C4×C191—41no—
97C4×C191—41no—
98C4×C191—41no—
99Dic191—41no—
100Dic191—41no—
101Dic191—41no—
102C2×C191—21no—
103C2×C191—81no—
104C2×C191B(4,𝔽2)11yescharacteristic
105D191—81no—
106C191#220311yescharacteristic, complemented, Sylow
107#2203—1911nomaximal, Sylow
108#563—1911no—
109#456—1911no—
110#456D19111yescharacteristic, complemented
111#454—1911no—
112#356—1911no—
113#356—1911no—
114#355—1911no—
1152−1+4C2×D19111yescharacteristic
116#137—21no—
117#137—3821no—
118AGL(1,ℤ/8)—3821no—
119C4⋊Q8—1911no—
120C4⋊Q8C2×D19111yescharacteristic
121#126—3821no—
122#126—3821no—
123#125—3821no—
124#124—1911no—
125#123—3821no—
126C22⋊Q8—3821no—
127C22⋊Q8—3821no—
128#104—21no—
129#104—3821no—
130#104—3821no—
131#101—1911no—
132#101—1911no—
133#101C2×D19111yescharacteristic
134#41—21no—
135#41—21no—
136#41—21no—
137#41—41no—
138C2×Q8—41no—
139C2×Q8—1911no—
140C2×Q8C22⋊D19111yescharacteristic
141C2×Q8C22×D19111yescharacteristic
142C2×D4—1911no—
143C22×C4—3821no—
144Q16—41no—
145Q16—7641no—
146SD16—41no—
147SD16—7641no—
148SD16—7641no—
149D8—7641no—
150M16—21no—
151M16—3821no—
152M16—3821no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—3821no—
156C4⋊C4—3821no—
157C4⋊C4—3821no—
158C4⋊C4—3821no—
159C4⋊C4—3821no—
160C4⋊C4—3821no—
161C22⋊C4—3821no—
162C22⋊C4—3821no—
163C22⋊C4—3821no—
164C22⋊C4—3821no—
165C22⋊C4—3821no—
166C42—1911no—
167C42—1911no—
168C42C22⋊D19111yescharacteristic
169C23—3821no—
170Q8—21no—
171Q8—21no—
172Q8—21no—
173Q8—41no—
174Q8—41no—
175Q8—7641no—
176D4—21no—
177D4—21no—
178D4—21no—
179D4—41no—
180D4—7641no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3821no—
187C2×C4—3821no—
188C2×C4—3821no—
189C2×C4—15281no—
190C2×C4D4×D19111yescharacteristic
191C2×C4D4×D19111yescharacteristic
192C2×C4C2×C22⋊D19111yescharacteristic
193C8—41no—
194C8—7641no—
195C8—7641no—
196C22—41no—
197C22—7642no—
198C22—7641no—
199C22#2025929011yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—41no—
206C4—7641no—
207C4—7641no—
208C4—7641no—
209C2—21no—
210C2—81no—
211C2—15281no—
212C2#2713284911yescharacteristic
subgroupG/H
centerC2#27132849
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#456×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132836C2
hypercenter#456D191
hypocenterC191#2203
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #27133412C2semidirect
2. #27132846C2semidirect
4. #27132462C2semidirect
5. #27132461C2semidirect
6. #27132459C2semidirect
7. #456×C191C2semidirect
106. C191#2203semidirect
110. #456D191semidirect

Maximal subgroups

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

subgroupindex
#271334122
#271328462
#271328362
#271324622
#271324612
#271324592
#456×C1912
#2203191