Finite Group Databasebeta

← #31914200

Subgroups — #31914200

Order 18176.

396 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 121 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×#17986106C211yesmaximal, characteristic
2C2×#17985822C211yesmaximal, characteristic
3C2×#17985815C211yesmaximal, characteristic
4#22589094C211yesmaximal, characteristic
5#22586445C211yesmaximal, characteristic
6#22586419C211yesmaximal, characteristic
7C71×#2141C211yesmaximal, characteristic
8C2×Q8×Dic71—21no—
9C2×#14556068C2211yescharacteristic
10#17986106—22no—
11#17986106—22no—
12C2×#14556064C2211yescharacteristic
13C2×#14556060C2211yescharacteristic
14#17986026—21no—
15C2×#14556052—21no—
16C2×#14556038C2211yescharacteristic
17#17985822—22no—
18#17985815—22no—
19#17985758—21no—
20C22×Q16×C71C2211yescharacteristic
21C2×M32×C71C2211yescharacteristic
22#427×C71C2211yescharacteristic
23#358×C71C414yescomplemented
24C2×#11765849—21no—
25C22×C71⋊C8D411yescharacteristic
26Q8×Dic71—24no—
27Q8×Dic71—42no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—41no—
30C2×C4×Dic71—41no—
31C2×C4×Dic71D411yescharacteristic
32#14556068—21no—
33#14556068—21no—
34C2×C71⋊C16—21no—
35#14556064—21no—
36#14556064—21no—
37#14556060—21no—
38#14556060D412yes—
39C4⋊(C71⋊C8)—42no—
40#14556052—24no—
41#14556038—21no—
42#14556038—21no—
43C22×Q8×C71—21no—
44C2×Q16×C71—21no—
45C2×Q16×C71—21no—
46C2×Q16×C71—21no—
47C2×Q16×C71C2×C412yes—
48C2×Q16×C71D412yes—
49C2×Q16×C71Q812yes—
50C22×C8×C71C2311yescharacteristic
51#117×C71—21no—
52M32×C71—21no—
53M32×C71—21no—
54C2×C16×C71C2×C412yes—
55#107×C71C2×C412yes—
56#106×C71—21no—
57C22×Dic71—21no—
58C22×Dic71—41no—
59#11765849—41no—
60#11765849—41no—
61C2×C71⋊C8—21no—
62C2×C71⋊C8—21no—
63C2×C71⋊C8—21no—
64C2×C71⋊C8—41no—
65C4⋊Dic71—22no—
66C4⋊Dic71—44no—
67C4⋊Dic71—41no—
68C4×Dic71—21no—
69C4×Dic71—44no—
70C4×Dic71SD1612yes—
71C71⋊C16—41no—
72C2×Q8×C71—22no—
73C2×Q8×C71—22no—
74C2×Q8×C71—22no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71C2×D411yescharacteristic
80Q16×C71—22no—
81Q16×C71—22no—
82Q16×C71—22no—
83Q16×C71C4⋊C414yes—
84C2×C8×C71C22×C411yescharacteristic
85C2×C8×C71C2×D411yescharacteristic
86C2×C8×C71C2×D411yescharacteristic
87C2×C8×C71C2×Q811yescharacteristic
88C2×C8×C71#4111yescharacteristic
89C2×C8×C71#4111yescharacteristic
90C4⋊C4×C71—22no—
91C22⋊C4×C71—41no—
92C42×C71—41no—
93C16×C71—22no—
94C23×C71AGL(1,ℤ/8)11yescharacteristic
95Q8×C71—24no—
96Q8×C71—42no—
97Q8×C71—42no—
98Q8×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71C4×D411yescharacteristic
104C2×C4×C71#12011yescharacteristic
105C2×C4×C71C22⋊Q811yescharacteristic
106C2×C4×C71C2×SD1611yescharacteristic
107C2×C4×C71#13511yescharacteristic
108C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
109C2×Dic71—22no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—82no—
113C8×C71C2×C4⋊C411yescharacteristic
114C8×C71C4×D411yescharacteristic
115C8×C71C22⋊Q811yescharacteristic
116C8×C71#12311yescharacteristic
117C71⋊C8—41no—
118C71⋊C8—81no—
119C22×C71—22no—
120C22×C71#44211yescharacteristic
121C22×C71#45011yescharacteristic
122C22×C71#47811yescharacteristic
123C4×C71—44no—
124C4×C71—82no—
125C4×C71#38511yescharacteristic
126C4×C71C4×SD1611yescharacteristic
127C4×C71#44911yescharacteristic
128C4×C71#47611yescharacteristic
129Dic71—42no—
130Dic71—82no—
131#12957—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71#186411yescharacteristic
135C2×C71#218012yes—
136C2×#441—711no—
137C2×#365—711no—
138C2×#362—711no—
139#2141D7111yescharacteristic
140#1946—711no—
141#1386—711no—
142#1375—711no—
143C71#1295711yescharacteristic, complemented, Sylow
144C22×Q16C2×D7111yescharacteristic
145C2×C4×Q8—1421no—
146C2×M32—711no—
147C2×M32C2×D7111yescharacteristic
148#441—1422no—
149#441—1422no—
150C2×#108—711no—
151#427C2×D7111yescharacteristic
152C2×#106—711no—
153#424—1421no—
154C2×#103—1421no—
155C2×#97—711no—
156#365—1422no—
157#362—1422no—
158#358Dic7114yescomplemented
159#339—1421no—
160C22×Q8—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16Dic14212yes—
165C2×Q16D28412yes—
166C2×Q16C2×Dic7112yes—
167C2×M16—1421no—
168C22×C8—711no—
169C22×C8C22×D7111yescharacteristic
170C4×Q8—2842no—
171C4×Q8—1424no—
172#117—21no—
173C2×C4⋊C4—2841no—
174C2×C4⋊C4—1421no—
175C2×C42—2841no—
176C2×C42—711no—
177M32—21no—
178M32—21no—
179M32—1421no—
180M32—1421no—
181C2×C16—1421no—
182C2×C16C2×Dic7112yes—
183#108—1421no—
184#108—1421no—
185#107C2×Dic7112yes—
186#106—21no—
187#106—712no—
188#106—1421no—
189C4⋊C8—2842no—
190#103—1424no—
191#97—1421no—
192#97—1421no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C22×C4—41no—
200C22×C4—2841no—
201C22×C4—1421no—
202C22×C4D4×D7111yescharacteristic
203Q16—22no—
204Q16—22no—
205Q16—22no—
206Q16C4⋊Dic7114yes—
207M16—2841no—
208M16—2841no—
209C2×C8—2841no—
210C2×C8—1421no—
211C2×C8—1421no—
212C2×C8—1421no—
213C2×C8C22×Dic7111yescharacteristic
214C2×C8D4×D7111yescharacteristic
215C2×C8C2×D28411yescharacteristic
216C2×C8C2×Dic14211yescharacteristic
217C2×C8#1176586411yescharacteristic
218C2×C8#1176586411yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—2844no—
221C4⋊C4—2841no—
222C4⋊C4—1422no—
223C22⋊C4—41no—
224C42—41no—
225C42—2844no—
226C42—712no—
227C42—1421no—
228C16—22no—
229C16—2841no—
230C23#1455615211yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—2842no—
240C2×C4—2842no—
241C2×C4—5682no—
242C2×C4—1422no—
243C2×C4D4×Dic7111yescharacteristic
244C2×C4#1455609811yescharacteristic
245C2×C4#1455610511yescharacteristic
246C2×C4SD16×D7111yescharacteristic
247C2×C4#1455614911yescharacteristic
248C2×C4#1455615211yescharacteristic
249C8—2841no—
250C8—5681no—
251C8C2×C4⋊Dic7111yescharacteristic
252C8D4×Dic7111yescharacteristic
253C8#1455610511yescharacteristic
254C8#1455611211yescharacteristic
255C22—22no—
256C22#1798611111yescharacteristic
257C22#1798615111yescharacteristic
258C22#1798630111yescharacteristic
259C4—44no—
260C4—82no—
261C4—2842no—
262C4—5682no—
263C4#1798587411yescharacteristic
264C4SD16×Dic7111yescharacteristic
265C4#1798614511yescharacteristic
266C4#1798628711yescharacteristic
267C2—21no—
268C2—21no—
269C2#2258860311yescharacteristic
270C2#2259023212yes—
subgroupG/H
centerC22#17986111
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#2141C2
socleC22×C71#442
prime radicalC23×C71AGL(1,ℤ/8)
cosocleC22×C8×C71C23
hypercenter#2141D71
hypocenterC71#12957
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
23. #358×C71C4semidirect
143. C71#12957semidirect
158. #358Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179861062
C2×#179858222
C2×#179858152
#225890942
#225864452
#225864192
C71×#21412
#1295771