Finite Group Databasebeta

← #31913487

Subgroups — #31913487

Order 18176.

412 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 97 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#22597058C211yesmaximal, characteristic
2#22590025C211yesmaximal, characteristic
3#22588388C212yesmaximal, complemented
4#22586462C212yesmaximal, complemented
5C71×#2132C211yesmaximal, characteristic
6C2×#14556135C411yescharacteristic
7C2×#14556135C2211yescharacteristic
8#17986071—22no—
9#17986071—22no—
10C2×#14556063—22no—
11#17986023C411yescharacteristic
12#17985983C2212yes—
13#17985966C411yescharacteristic
14#17985966C2211yescharacteristic
15C2×#14556039C411yescharacteristic
16#17985798—21no—
17#17985796—21no—
18C71⋊#338—22no—
19#532×C71C2211yescharacteristic
20#413×C71C2212yes—
21#356×C71—21no—
22#355×C71—21no—
23#14556135—42no—
24#14556135—42no—
25C2×#11765849—21no—
26C2×#11765849—22no—
27C2×#11765849C2×C411yescharacteristic
28C2×#11765849C2×C411yescharacteristic
29C2×#11765849D412yes—
30C22×C71⋊C8—21no—
31C22×C71⋊C8C2×C411yescharacteristic
32C22×C71⋊C8C2×C411yescharacteristic
33C22×C71⋊C8D412yes—
34#14556063—44no—
35C4⋊(C71⋊C8)—22no—
36#14556047—22no—
37#14556045—22no—
38C22⋊(C71⋊C8)—22no—
39C22⋊(C71⋊C8)—22no—
40#14556039—21no—
41#14556039—21no—
42C2×#41×C71C2×C411yescharacteristic
43C2×#41×C71C2311yescharacteristic
44C2×M16×C71—22no—
45C4⋊Q8×C71D411yescharacteristic
46#127×C71Q811yescharacteristic
47#124×C71D411yescharacteristic
48#124×C71Q811yescharacteristic
49#121×C71—22no—
50C4×D4×C71—22no—
51C2×C42×C71C2×C411yescharacteristic
52#101×C71—22no—
53#100×C71—22no—
54#11765849—41no—
55#11765849—42no—
56#11765849—42no—
57#11765849—41no—
58#11765849—42no—
59#11765849—42no—
60C2×C71⋊C8—24no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—21no—
63C2×C71⋊C8—21no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8C22⋊C411yescharacteristic
68C2×C71⋊C8C22⋊C412yes—
69C2×C71⋊C8C22⋊C411yescharacteristic
70#41×C71—41no—
71#41×C71—41no—
72#41×C71—41no—
73#41×C71—41no—
74C2×Q8×C71C4⋊C411yescharacteristic
75C2×Q8×C71C2×D411yescharacteristic
76C2×D4×C71—21no—
77C2×D4×C71—22no—
78C2×D4×C71C4⋊C411yescharacteristic
79C2×D4×C71C2×Q811yescharacteristic
80C2×D4×C71#4112yes—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85M16×C71—42no—
86M16×C71—42no—
87C2×C8×C71—42no—
88C4⋊C4×C71—22no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—22no—
91C42×C71—21no—
92C42×C71C4⋊C411yescharacteristic
93C42×C71C4⋊C411yescharacteristic
94C23×C71—21no—
95C23×C71#11711yescharacteristic
96C23×C71C22⋊Q812yes—
97Q8×C71—41no—
98Q8×C71—41no—
99D4×C71—41no—
100D4×C71—42no—
101D4×C71—41no—
102D4×C71—42no—
103D4×C71—82no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C2×C4×C71—22no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—81no—
114C2×C4×C71C2×C22⋊C411yescharacteristic
115C2×C4×C71C2×C4⋊C411yescharacteristic
116C2×C4×C71#12112yes—
117C8×C71—82no—
118C71⋊C8—21no—
119C71⋊C8—22no—
120C71⋊C8—41no—
121C71⋊C8—41no—
122C71⋊C8—82no—
123C71⋊C8#9912yes—
124C22×C71—21no—
125C22×C71—22no—
126C22×C71—41no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71—41no—
130C22×C71#38011yescharacteristic
131C4×C71—22no—
132C4×C71—21no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71C2×#9911yescharacteristic
137C4×C71#41011yescharacteristic
138#12810—711nomaximal, Sylow
139C2×C71—21no—
140C2×C71—41no—
141C2×C71—42no—
142C2×C71—81no—
143C2×C71#179911yescharacteristic
144#2932—711no—
145#2132—711no—
146#2132D7111yescharacteristic
147#1813—712no—
148#1392—712no—
149C71#1281011yescharacteristic, complemented, Sylow
150C2×#131—711no—
151C2×#131—711no—
152#532C2×D7111yescharacteristic
153#435—1422no—
154#435—1422no—
155C2×#108—1422no—
156#423—711no—
157#413—712no—
158#413C2×D7112yes—
159#408—711no—
160#408—711no—
161C2×#97—711no—
162#356—21no—
163#356—1421no—
164#355—21no—
165#355—1421no—
166#338—1422no—
167C2×#41C4×D7111yescharacteristic
168C2×#41C22×D7111yescharacteristic
169#131—2842no—
170#131—2842no—
171C2×M16—22no—
172C2×M16—711no—
173C2×M16—712no—
174C2×M16—711no—
175C2×M16—1422no—
176C2×M16—1421no—
177C22×C8—711no—
178C22×C8—711no—
179C22×C8—712no—
180C22×C8—1421no—
181C4⋊Q8C22⋊D7111yescharacteristic
182#127Dic14211yescharacteristic
183#124Dic14211yescharacteristic
184#124C22⋊D7111yescharacteristic
185#121—22no—
186C4×D4—22no—
187C2×C42C4×D7111yescharacteristic
188#108—2844no—
189C4⋊C8—1422no—
190#101—22no—
191#101—1422no—
192#100—22no—
193#100—1422no—
194C22⋊C8—1422no—
195C22⋊C8—1422no—
196#97—1421no—
197#97—1421no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202C2×Q8Dic71⋊C411yescharacteristic
203C2×Q8C2×C22⋊D7111yescharacteristic
204C2×D4—21no—
205C2×D4—22no—
206C2×D4Dic71⋊C411yescharacteristic
207C2×D4C2×Dic14211yescharacteristic
208C2×D4#1176586412yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C2×C4×D7111yescharacteristic
212C22×C4D4×D7112yes—
213M16—42no—
214M16—42no—
215M16—2842no—
216M16—2841no—
217M16—2842no—
218M16—2842no—
219M16—2842no—
220M16—2841no—
221C2×C8—42no—
222C2×C8—2842no—
223C2×C8—2841no—
224C2×C8—2842no—
225C2×C8—711no—
226C2×C8—711no—
227C2×C8—712no—
228C2×C8—1421no—
229C2×C8—1421no—
230C2×C8—1424no—
231C2×C8—1422no—
232C4⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C42—21no—
236C42Dic71⋊C411yescharacteristic
237C42Dic71⋊C411yescharacteristic
238C23—21no—
239C23#1455608411yescharacteristic
240C23#1455610512yes—
241Q8—41no—
242Q8—41no—
243D4—41no—
244D4—42no—
245D4—41no—
246D4—42no—
247D4—82no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4C22⋊C4×D7111yescharacteristic
259C2×C4C2×Dic71⋊C411yescharacteristic
260C2×C4#1455610012yes—
261C8—82no—
262C8—2841no—
263C8—2841no—
264C8—5682no—
265C8—712no—
266C8—1421no—
267C8—1422no—
268C22—21no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22#1798585711yescharacteristic
275C4—22no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4#99×D7111yescharacteristic
281C4#1798597111yescharacteristic
282C2—21no—
283C2—41no—
284C2—42no—
285C2—81no—
286C2#2258832611yescharacteristic
subgroupG/H
centerC4#17985971
commutatorC22×C4×C71C22×C4
FrattiniC2×#41C22×D71
FittingC71×#2132C2
socleC2×C71#1799
prime radical#532×C71C22
cosocleC2×#41×C71C23
hypercenter#2132D71
hypocenterC71#12810
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #22588388C2semidirect
4. #22586462C2semidirect
149. C71#12810semidirect

Maximal subgroups

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

subgroupindex
#225970582
#225900252
#225883882
#225864622
C71×#21322
#1281071