Finite Group Databasebeta

← #37574542

Subgroups — #37574542

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#27132848C211yesmaximal, characteristic, complemented
3#27132845C211yesmaximal, characteristic, complemented
4#27132838C211yesmaximal, characteristic
5#27132461C211yesmaximal, characteristic, complemented
6#27132459C211yesmaximal, characteristic, complemented
7#356×C191C211yesmaximal, characteristic, complemented
8#20259370C2211yescharacteristic
9#20259355—21no—
10#20259353—21no—
11#20259350—21no—
12#20259320C2211yescharacteristic
13#20259317—21no—
14#20259316—21no—
15#20259312—21no—
16#20259311C2211yescharacteristic
17#20259307—21no—
18#20259300—21no—
19#20259298—21no—
20#20259250—21no—
21#20259248—21no—
22#20259248—21no—
23#20259241C2211yescharacteristic
24C4⋊Q8×C191C2211yescharacteristic
25#101×C191C2211yescharacteristic
26#101×C191C2211yescharacteristic
27#15690288—41no—
28#15690287—21no—
29#15690287—21no—
30#15690287—21no—
31C2×Dic382D411yescharacteristic
32Q8×D191—41no—
33C2×D764D411yescharacteristic
34C2×C4×D191—21no—
35Dic764—41no—
36#15690277—41no—
37#15690276—41no—
38#15690276—41no—
39#15690274—41no—
40D1528—41no—
41#15690271—21no—
42C4⋊Dic191—21no—
43Dic191⋊C4—21no—
44Dic191⋊C4—21no—
45Dic191⋊C4—21no—
46Dic191⋊C4—21no—
47Dic191⋊C4—21no—
48(C2×D191)⋊C4—21no—
49(C2×D191)⋊C4—21no—
50(C2×D191)⋊C4—21no—
51(C2×D191)⋊C4—21no—
52(C2×D191)⋊C4—21no—
53C4×Dic191D411yescharacteristic
54C4×Dic191D411yescharacteristic
55C2×Q8×C191D411yescharacteristic
56C2×Q8×C191C2311yescharacteristic
57M16×C191—21no—
58M16×C191—21no—
59C4⋊C4×C191—21no—
60C4⋊C4×C191—21no—
61C42×C191D411yescharacteristic
62C22×D191—21no—
63Dic382—21no—
64Dic382—21no—
65Dic382—21no—
66Dic382—41no—
67D764—21no—
68D764—21no—
69D764—21no—
70D764—41no—
71C22⋊D191—41no—
72C2×Dic191—21no—
73C2×Dic191—21no—
74C2×Dic191—21no—
75C4×D191—41no—
76C4×D191—41no—
77C4×D191—41no—
78C4×D191—81no—
79C191⋊C8—41no—
80Q8×C191—41no—
81Q8×C191—41no—
82C2×C4×C191—21no—
83C2×C4×C191—21no—
84C2×C4×C191C2×D411yescharacteristic
85C2×C4×C191C2×D411yescharacteristic
86C2×C4×C191C2×D411yescharacteristic
87C8×C191—41no—
88C8×C191—41no—
89C22×C191#12011yescharacteristic
90C2×D191—42no—
91C2×D191—41no—
92C2×D191—41no—
93C4×C191—21no—
94C4×C191—21no—
95C4×C191—21no—
96C4×C191—41no—
97C4×C191—41no—
98Dic191—41no—
99Dic191—41no—
100Dic191—41no—
101Dic191—41no—
102C2×C191—21no—
103C2×C191B(4,𝔽2)11yescharacteristic
104D191—81no—
105D191—81no—
106C191#220311yescharacteristic, complemented, Sylow
107#2203—1911nomaximal, Sylow
108#563—1911no—
109#456—1911no—
110#456—1911no—
111#454—1911no—
112#356—1911no—
113#356D19111yescharacteristic, complemented
114#355—1911no—
1152−1+4—1911no—
116#137—3821no—
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—3821no—
129#104—3821no—
130#104—3821no—
131#101—1911no—
132#101C2×D19111yescharacteristic
133#101C2×D19111yescharacteristic
134#41—3821no—
135#41—3821no—
136#41—3821no—
137#41—7641no—
138C2×Q8—1911no—
139C2×Q8—7641no—
140C2×Q8D76411yescharacteristic
141C2×Q8C22×D19111yescharacteristic
142C2×D4—1911no—
143C22×C4—3821no—
144Q16—7641no—
145Q16—7641no—
146SD16—7641no—
147SD16—7641no—
148SD16—7641no—
149D8—7641no—
150M16—21no—
151M16—21no—
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—
168C42D76411yescharacteristic
169C23—3821no—
170Q8—41no—
171Q8—41no—
172Q8—3821no—
173Q8—3821no—
174Q8—3821no—
175Q8—7641no—
176D4—3821no—
177D4—3821no—
178D4—3821no—
179D4—7641no—
180D4—7641no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—3821no—
184C2×C4—3821no—
185C2×C4—3821no—
186C2×C4—15281no—
187C2×C4—7641no—
188C2×C4—7641no—
189C2×C4—7641no—
190C2×C4D4×D19111yescharacteristic
191C2×C4D4×D19111yescharacteristic
192C2×C4C2×D76411yescharacteristic
193C8—41no—
194C8—41no—
195C8—7641no—
196C22—7642no—
197C22—7641no—
198C22—7641no—
199C22#2025929211yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—7641no—
206C4—7641no—
207C4—7641no—
208C4—7641no—
209C2—21no—
210C2—15281no—
211C2—15281no—
212C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#356×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132838C2
hypercenter#356D191
hypocenterC191#2203
LCSG ▷ C2×Q8×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
1. #27133412C2semidirect
2. #27132848C2semidirect
3. #27132845C2semidirect
5. #27132461C2semidirect
6. #27132459C2semidirect
7. #356×C191C2semidirect
106. C191#2203semidirect
113. #356D191semidirect

Maximal subgroups

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

subgroupindex
#271334122
#271328482
#271328452
#271328382
#271324612
#271324592
#356×C1912
#2203191