Finite Group Databasebeta

← #37546653

Subgroups — #37546653

Order 23168.

326 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic362C211yesmaximal, characteristic
2#26041689C212yesmaximal
3#26041675C211yesmaximal, characteristic
4#26041583C212yesmaximal
5#26041574C211yesmaximal, characteristic
6#26041564C212yesmaximal
7#26041561C211yesmaximal, characteristic
8#26041410C211yesmaximal, characteristic
9#26041384C212yesmaximal
10C4×#15668439C211yesmaximal, characteristic
11#557×C181C211yesmaximal, characteristic
12C2×#15668439C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153816C2212yes—
15#19153815—21no—
16#19153815—22no—
17Q8×Dic181—21no—
18C4×Dic362C2211yescharacteristic
19C4×Dic362C2212yes—
20#19153749C2211yescharacteristic
21#19153749C2212yes—
22#19153747C2212yes—
23#19153747C2212yes—
24#19153747C2212yes—
25C4⋊#13089678C2211yescharacteristic
26C4⋊#13089678C2212yes—
27#19153740C2211yescharacteristic
28#19153740C2212yes—
29#19153739C2211yescharacteristic
30#19153739C2212yes—
31#19153727C2212yes—
32C4×#13089678C2211yescharacteristic
33C4⋊Q8×C181—21no—
34C4⋊Q8×C181C2211yescharacteristic
35#125×C181—22no—
36#125×C181C2212yes—
37C4×Q8×C181—21no—
38C4×Q8×C181C2211yescharacteristic
39C4×Q8×C181C2212yes—
40C4×Q8×C181C2211yescharacteristic
41C2×Dic362—22no—
42C2×Dic362—21no—
43C2×Dic362C2311yescharacteristic
44#15668439Q814yescomplemented
45C2×#13089678C2311yescharacteristic
46C2×#13089678C2312yes—
47C2×#13089678C2311yescharacteristic
48C4⋊Dic181—22no—
49C4⋊Dic181—21no—
50C4⋊Dic181C2311yescharacteristic
51C4⋊Dic181C2312yes—
52#15668428—22no—
53#15668428—21no—
54C4×Dic181—22no—
55C4×Dic181—21no—
56C2×Q8×C181—21no—
57C2×Q8×C181D411yescharacteristic
58C2×Q8×C181C2311yescharacteristic
59C4⋊C4×C181—22no—
60C4⋊C4×C181—21no—
61C4⋊C4×C181—22no—
62C4⋊C4×C181—21no—
63C4⋊C4×C181—22no—
64C4⋊C4×C181D411yescharacteristic
65C4⋊C4×C181D412yes—
66C4⋊C4×C181C2311yescharacteristic
67C4⋊C4×C181C2312yes—
68C42×C181—22no—
69C42×C181—21no—
70C42×C181C2312yes—
71C42×C181C2311yescharacteristic
72Dic362—21no—
73Dic362—82no—
74Dic362—81no—
75Dic362C2×Q812yes—
76C2×Dic181—21no—
77C2×Dic181—22no—
78C2×Dic181—21no—
79#13089678—21no—
80#13089678—22no—
81#13089678C2×Q812yes—
82Q8×C181—22no—
83Q8×C181—21no—
84Q8×C181—42no—
85Q8×C181C2×Q812yes—
86C2×C4×C181—21no—
87C2×C4×C181—21no—
88C2×C4×C181—22no—
89C2×C4×C181C2×D412yes—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D412yes—
92C2×C4×C181C2×D411yescharacteristic
93C2×C4×C181C2411yescharacteristic
94C22×C181C22×D411yescharacteristic
95C4×C181—22no—
96C4×C181—22no—
97C4×C181—22no—
98C4×C181—21no—
99C4×C181—42no—
100C4×C181—41no—
101C4×C181#13712yes—
102C4×C181C22×Q811yescharacteristic
103C4×C1812−1+411yescharacteristic
104Dic181—22no—
105Dic181—42no—
106Dic181—41no—
107C2×C181D4×Q811yescharacteristic
108C2×C181C2×#13711yescharacteristic
109C2×C181#57711yescharacteristic
110C181#338711yescharacteristic, complemented, Sylow
111#3387—1811nomaximal, Sylow
112Q82—1811no—
113#557D18111yescharacteristic
114#501—1812no—
115#498—1811no—
116#479—1812no—
117#477—1811no—
118#475—1811no—
119#475—1812no—
120#446—1811no—
121#441—1812no—
122C4×Q16—1811no—
123C2×Q16—1811no—
124C4⋊Q8—21no—
125C4⋊Q8—3621no—
126C4⋊Q8—3622no—
127C4⋊Q8—1812no—
128C4⋊Q8—1811no—
129C4⋊Q8C2×D18111yescharacteristic
130#125—22no—
131#125C2×D18112yes—
132C4×Q8—21no—
133C4×Q8—3621no—
134C4×Q8—1811no—
135C4×Q8—1812no—
136C4×Q8C2×D18111yescharacteristic
137C4×Q8C2×D18112yes—
138C4×Q8C2×D18111yescharacteristic
139#107—1811no—
140#107—1812no—
141#106—1812no—
142#106—1812no—
143#106—1812no—
144C4⋊C8—1811no—
145C4⋊C8—1812no—
146#103—1811no—
147#103—1812no—
148#103—1811no—
149#103—1812no—
150#97—1812no—
151C4×C8—1811no—
152C2×Q8—21no—
153C2×Q8—3622no—
154C2×Q8—3621no—
155C2×Q8—1811no—
156C2×Q8C22⋊D18111yescharacteristic
157C2×Q8C22×D18111yescharacteristic
158Q16—1814no—
159C2×C8—1811no—
160C2×C8—1812no—
161C2×C8—1811no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—3622no—
168C4⋊C4—3621no—
169C4⋊C4—3622no—
170C4⋊C4—3621no—
171C4⋊C4—1811no—
172C4⋊C4—1812no—
173C4⋊C4C22⋊D18111yescharacteristic
174C4⋊C4C22⋊D18112yes—
175C4⋊C4C22×D18112yes—
176C4⋊C4C22×D18111yescharacteristic
177C42—22no—
178C42—21no—
179C42—3622no—
180C42—3621no—
181C42C22×D18111yescharacteristic
182C42C22×D18112yes—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—3621no—
187Q8—14482no—
188Q8—14481no—
189Q8—1812no—
190Q8Q8×D18112yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—3621no—
195C2×C4—3622no—
196C2×C4—3621no—
197C2×C4C2×C22⋊D18112yes—
198C2×C4C2×C22⋊D18112yes—
199C2×C4C2×C22⋊D18111yescharacteristic
200C2×C4C2×C22⋊D18111yescharacteristic
201C2×C4C23×D18111yescharacteristic
202C8—3621no—
203C8—3622no—
204C8—1812no—
205C22C22×C22⋊D18111yescharacteristic
206C4—22no—
207C4—22no—
208C4—22no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—3622no—
213C4—7242no—
214C4—7241no—
215C4#1915384612yes—
216C4C2×Q8×D18111yescharacteristic
217C4#1915386411yescharacteristic
218C2Q8×C22⋊D18111yescharacteristic
219C2C2×#1915384611yescharacteristic
220C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#557×C181C2
socle, prime radicalC22×C181C22×D4
hypercenter#557D181
hypocenterC181#3387
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
44. #15668439Q8semidirect
110. C181#3387semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3622
#260416892
#260416752
#260415832
#260415742
#260415642
#260415612
#260414102
#260413842
C4×#156684392
#557×C1812
#3387181