Finite Group Databasebeta

← #37546652

Subgroups — #37546652

Order 23168.

326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 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#26042004C211yesmaximal, characteristic
3#26041686C212yesmaximal
4#26041675C211yesmaximal, characteristic
5#26041588C212yesmaximal
6#26041575C211yesmaximal, characteristic
7#26041563C212yesmaximal
8#26041406C211yesmaximal, characteristic
9#26041383C212yesmaximal
10C4×#15668439C211yesmaximal, characteristic
11#475×C181C211yesmaximal, characteristic
12C2×#15668439C2211yescharacteristic
13#19153816—21no—
14#19153816C2211yescharacteristic
15#19153815—22no—
16#19153815C2212yes—
17#19153814—21no—
18#19153807C2212yes—
19#19153806—22no—
20Q8×Dic181—21no—
21Q8×Dic181C2212yes—
22C4×Dic362—21no—
23C4×Dic362C2211yescharacteristic
24C4×Dic362C2211yescharacteristic
25#19153781C2212yes—
26#19153750C2212yes—
27#19153749C2211yescharacteristic
28#19153747C2212yes—
29#19153747C2212yes—
30#19153745C2212yes—
31#19153741C2212yes—
32#19153740C2211yescharacteristic
33#19153739C2211yescharacteristic
34#19153726C2212yes—
35C4×#13089678C2211yescharacteristic
36C4⋊Q8×C181C2211yescharacteristic
37C4×Q8×C181C2211yescharacteristic
38#106×C181C2212yes—
39C4⋊C8×C181C2211yescharacteristic
40#103×C181C2212yes—
41C2×Dic362—22no—
42C2×Dic362—21no—
43C2×Dic362D411yescharacteristic
44C2×Dic362C2311yescharacteristic
45#15668439Q814yescomplemented
46C2×#13089678C2311yescharacteristic
47C2×#13089678C2311yescharacteristic
48C4⋊Dic181—22no—
49C4⋊Dic181—22no—
50C4⋊Dic181—21no—
51C4⋊Dic181D411yescharacteristic
52C4⋊Dic181C2311yescharacteristic
53C4⋊Dic181C2312yes—
54#15668428—22no—
55#15668428—22no—
56#15668428—22no—
57#15668428—21no—
58#15668428D412yes—
59C4×Dic181—22no—
60C4×Dic181—22no—
61C4×Dic181—21no—
62C4×Dic181C2312yes—
63C2×Q8×C181—21no—
64C2×Q8×C181C2311yescharacteristic
65C2×C8×C181C2312yes—
66C4⋊C4×C181—21no—
67C4⋊C4×C181—21no—
68C4⋊C4×C181C2312yes—
69C4⋊C4×C181C2311yescharacteristic
70C42×C181—21no—
71C42×C181C2311yescharacteristic
72Dic362—22no—
73Dic362—21no—
74Dic362—82no—
75Dic362—81no—
76Dic362C2×Q812yes—
77C2×Dic181—21no—
78C2×Dic181—21no—
79C2×Dic181—22no—
80C2×Dic181C2×D412yes—
81C2×Dic181C2×D412yes—
82#13089678—21no—
83#13089678C2×Q812yes—
84Q8×C181—21no—
85Q8×C181—42no—
86Q8×C181C2×Q812yes—
87C2×C4×C181—21no—
88C2×C4×C181—21no—
89C2×C4×C181—22no—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2411yescharacteristic
93C8×C181—22no—
94C22×C181C22×D411yescharacteristic
95C4×C181—22no—
96C4×C181—21no—
97C4×C181—42no—
98C4×C181—41no—
99C4×C181#13712yes—
100C4×C181C22×Q811yescharacteristic
101C4×C1812−1+411yescharacteristic
102Dic181—22no—
103Dic181—22no—
104Dic181—22no—
105Dic181—41no—
106Dic181—42no—
107C2×C181D4×Q811yescharacteristic
108C2×C181C2×#13711yescharacteristic
109C2×C181#57711yescharacteristic
110C181#338711yescharacteristic, complemented, Sylow
111#3387—1811nomaximal, Sylow
112Q82—1811no—
113#557—1811no—
114#501—1812no—
115#498—1811no—
116#479—1812no—
117#477—1811no—
118#475—1812no—
119#475D18111yescharacteristic
120#446—1811no—
121#441—1812no—
122C4×Q16—1811no—
123C2×Q16—1811no—
124C4⋊Q8—3621no—
125C4⋊Q8—3622no—
126C4⋊Q8—3621no—
127C4⋊Q8—1812no—
128C4⋊Q8—1811no—
129C4⋊Q8C2×D18111yescharacteristic
130#125—3622no—
131#125—1812no—
132C4×Q8—3621no—
133C4×Q8—3621no—
134C4×Q8—1812no—
135C4×Q8—1812no—
136C4×Q8—1811no—
137C4×Q8—1811no—
138C4×Q8C2×D18111yescharacteristic
139#107—1812no—
140#107—1811no—
141#106—1812no—
142#106—1812no—
143#106C2×D18112yes—
144C4⋊C8—1812no—
145C4⋊C8C2×D18111yescharacteristic
146#103—1811no—
147#103—1812no—
148#103—1811no—
149#103C2×D18112yes—
150#97—1812no—
151C4×C8—1811no—
152C2×Q8—21no—
153C2×Q8—3622no—
154C2×Q8—3621no—
155C2×Q8—1811no—
156C2×Q8—1811no—
157C2×Q8C22×D18111yescharacteristic
158Q16—1814no—
159C2×C8—1811no—
160C2×C8—1811no—
161C2×C8C22×D18112yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—3622no—
165C4⋊C4—3622no—
166C4⋊C4—3622no—
167C4⋊C4—3622no—
168C4⋊C4—3622no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4—1812no—
172C4⋊C4—1811no—
173C4⋊C4—1811no—
174C4⋊C4—1812no—
175C4⋊C4C22×D18111yescharacteristic
176C4⋊C4C22×D18112yes—
177C42—21no—
178C42—3622no—
179C42—3622no—
180C42—3621no—
181C42—1812no—
182C42C22×D18111yescharacteristic
183Q8—21no—
184Q8—42no—
185Q8—3622no—
186Q8—3621no—
187Q8—14481no—
188Q8—14482no—
189Q8—1812no—
190Q8Q8×D18112yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—3621no—
195C2×C4—3621no—
196C2×C4—3622no—
197C2×C4—1812no—
198C2×C4—1812no—
199C2×C4D4×D18111yescharacteristic
200C2×C4D4×D18111yescharacteristic
201C2×C4C23×D18111yescharacteristic
202C8—22no—
203C8—3621no—
204C8—1812no—
205C22C2×D4×D18111yescharacteristic
206C4—22no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—3622no—
211C4—3622no—
212C4—3622no—
213C4—7242no—
214C4—7241no—
215C4#1915384512yes—
216C4C2×Q8×D18111yescharacteristic
217C4#1915386311yescharacteristic
218C2#2604194211yescharacteristic
219C2C2×#1915384511yescharacteristic
220C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#475×C181C2
socle, prime radicalC22×C181C22×D4
hypercenter#475D181
hypocenterC181#3387
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
45. #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
#260420042
#260416862
#260416752
#260415882
#260415752
#260415632
#260414062
#260413832
C4×#156684392
#475×C1812
#3387181