Finite Group Databasebeta

← #31914281

Subgroups — #31914281

Order 18176.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#17985798C211yesmaximal, characteristic
2#22588734C211yesmaximal, characteristic
3#22588554C212yesmaximal
4#22586467C212yesmaximal
5C71×#2381C211yesmaximal, characteristic
6#17986477C2211yescharacteristic
7#17986039—22no—
8#17986008—21no—
9C2×#14556047C2211yescharacteristic
10C2×#14556047C2212yes—
11#17985798C414yescomplemented
12#17985759—21no—
13#17985759—22no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15#391×C71—21no—
16#391×C71C2212yes—
17#387×C71—22no—
18#387×C71—22no—
19#383×C71—21no—
20C2×#11765849—22no—
21C2×#11765849—21no—
22Q8×Dic71—42no—
23#14556087—21no—
24#14556087—22no—
25#14556061—42no—
26#14556059—42no—
27#14556052—42no—
28#14556047—22no—
29#14556047—22no—
30#14556047C2×C414yes—
31C22×Q8×C71D411yescharacteristic
32C22×Q8×C71C2311yescharacteristic
33C4⋊Q8×C71—22no—
34C4⋊Q8×C71C2×C412yes—
35C4⋊Q8×C71Q812yes—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71D412yes—
39C2×C42×C71—21no—
40C2×C42×C71D411yescharacteristic
41C2×C42×C71D412yes—
42#95×C71—21no—
43#95×C71—22no—
44#95×C71—21no—
45#95×C71—22no—
46#95×C71—22no—
47#95×C71—22no—
48#11765849—24no—
49#11765849—42no—
50#11765849—41no—
51#11765849—41no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—41no—
54C4⋊Dic71—22no—
55C4⋊Dic71—22no—
56C4⋊Dic71—22no—
57C22⋊Dic71—41no—
58C4×Dic71—42no—
59C4×Dic71—41no—
60C2×Q8×C71—21no—
61C2×Q8×C71—41no—
62C2×Q8×C71—41no—
63C2×Q8×C71—41no—
64C2×Q8×C71—41no—
65C2×Q8×C71C4⋊C412yes—
66C2×Q8×C71C22×C411yescharacteristic
67C2×Q8×C71C2×Q811yescharacteristic
68C2×Q8×C71#4112yes—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D412yes—
75C4⋊C4×C71—24no—
76C4⋊C4×C71—24no—
77C4⋊C4×C71—42no—
78C4⋊C4×C71—81no—
79C42×C71—21no—
80C42×C71—42no—
81C42×C71—81no—
82C42×C71C4⋊C412yes—
83C23×C71#12011yescharacteristic
84Q8×C71—41no—
85Q8×C71—41no—
86Q8×C71—42no—
87Q8×C71—42no—
88Q8×C71—81no—
89C2×C4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—21no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—42no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—44no—
99C2×C4×C71—44no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71C2×C4⋊C411yescharacteristic
103C2×C4×C71C4×D412yes—
104C2×C4×C71C22⋊Q812yes—
105C2×C4×C71#12311yescharacteristic
106C2×Dic71—42no—
107C71⋊C8—44no—
108C71⋊C8—81no—
109C22×C71—22no—
110C22×C71#38511yescharacteristic
111C22×C71B(4,𝔽2)11yescharacteristic
112C22×C71#45811yescharacteristic
113C4×C71—22no—
114C4×C71—21no—
115C4×C71—21no—
116C4×C71—22no—
117C4×C71—42no—
118C4×C71—42no—
119C4×C71—82no—
120C4×C71—82no—
121Dic71—82no—
122#12971—711nomaximal, Sylow
123C2×C71—21no—
124C2×C71—21no—
125C2×C71#186711yescharacteristic
126C2×C71#220312yes—
127#2381D7111yescharacteristic
128C2×#356—711no—
129#1886—711no—
130#1857—712no—
131#1394—712no—
132C71#1297111yescharacteristic, complemented, Sylow
133C2×C4⋊Q8C2×D7111yescharacteristic
134#519—711no—
135#428—1422no—
136#419—1421no—
137C2×#101—712no—
138C2×#101—711no—
139#391—21no—
140#391C2×D7112yes—
141#387—22no—
142#387—22no—
143#383—21no—
144#356—714no—
145#339—1422no—
146#339—1421no—
147C22×Q8C22⋊D7111yescharacteristic
148C22×Q8C22×D7111yescharacteristic
149C2×M16—1421no—
150C2×M16—1422no—
151C4⋊Q8—22no—
152C4⋊Q8Dic14212yes—
153C4⋊Q8C4×D7112yes—
154C4×Q8—2842no—
155#117—1421no—
156#117—1422no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C22⋊D7112yes—
160C2×C42—21no—
161C2×C42C22⋊D7111yescharacteristic
162C2×C42C22⋊D7112yes—
163#107—2842no—
164#106—2842no—
165#103—2842no—
166#101—714no—
167#101—1422no—
168#101—1422no—
169#95—21no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95—22no—
174#95—22no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8Dic71⋊C412yes—
181C2×Q8C2×C4×D7111yescharacteristic
182C2×Q8C2×Dic14211yescharacteristic
183C2×Q8#1176586512yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4C2×C22⋊D7111yescharacteristic
189C22×C4C2×C22⋊D7112yes—
190M16—2842no—
191M16—2841no—
192M16—2841no—
193M16—1424no—
194C2×C8—2842no—
195C2×C8—2841no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—1422no—
201C4⋊C4—1422no—
202C4⋊C4—1422no—
203C22⋊C4—2841no—
204C42—21no—
205C42—42no—
206C42—81no—
207C42—2842no—
208C42—2841no—
209C42Dic71⋊C412yes—
210C23#1455609711yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—42no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—2842no—
230C2×C4C2×Dic71⋊C411yescharacteristic
231C2×C4C4×C22⋊D7112yes—
232C2×C4#1455610712yes—
233C2×C4#1455611111yescharacteristic
234C8—2844no—
235C8—5681no—
236C22—22no—
237C22#1798587211yescharacteristic
238C22#1798618611yescharacteristic
239C22#1798618911yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4—5682no—
249C2—21no—
250C2—21no—
251C2#2258862111yescharacteristic
252C2#2259036112yes—
subgroupG/H
centerC22#17986186
commutatorC2×Q8×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2381C2
socleC22×C71B(4,𝔽2)
prime radicalC23×C71#120
cosocleC22×Q8×C71C23
hypercenter#2381D71
hypocenterC71#12971
LCSG ▷ C2×Q8×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2381

Complements

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

NH ≅ G/Ntype
11. #17985798C4semidirect
132. C71#12971semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225887342
#225885542
#225864672
C71×#23812
#1297171