Finite Group Databasebeta

← #31901057

Subgroups — #31901057

Order 18176.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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#22588336C211yesmaximal, characteristic
2#22587114C211yesmaximal, characteristic
3#22587012C211yesmaximal, characteristic
4#22587012C211yesmaximal, characteristic
5#22586735C211yesmaximal, characteristic
6#22586237C211yesmaximal, characteristic
7C2×#462×C71C211yesmaximal, characteristic
8#17986052C2211yescharacteristic
9#17986052C2211yescharacteristic
10#17986023C2211yescharacteristic
11#17986023C2211yescharacteristic
12C2×#14556047—21no—
13#17985768—21no—
14#17985751—22no—
15#17985745—22no—
16#17985745—22no—
17Q8⋊(C71⋊C8)—22no—
18C22×Q16×C71—21no—
19C2×C4⋊Q8×C71C2211yescharacteristic
20C2×C4×Q8×C71C2211yescharacteristic
21#462×C71—22no—
22#462×C71—22no—
23C2×C4⋊C8×C71C2211yescharacteristic
24C2×#103×C71—21no—
25C2×#11765849—21no—
26C2×#11765849—21no—
27C4⋊(C71⋊C8)—21no—
28C4⋊(C71⋊C8)—21no—
29C4⋊(C71⋊C8)—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)—21no—
32C4⋊(C71⋊C8)D412yes—
33#14556047—42no—
34C22⋊(C71⋊C8)—21no—
35C22⋊(C71⋊C8)—21no—
36C4×C71⋊C8—21no—
37C4×C71⋊C8—21no—
38C22×Q8×C71—21no—
39C22×Q8×C71D411yescharacteristic
40C2×Q16×C71—24no—
41C2×Q16×C71—42no—
42C2×Q16×C71—42no—
43C22×C8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71—21no—
46C4⋊Q8×C71—21no—
47C4⋊Q8×C71D412yes—
48C4×Q8×C71—21no—
49C4×Q8×C71—21no—
50C4×Q8×C71—21no—
51C4×Q8×C71—21no—
52C2×C4⋊C4×C71—21no—
53C2×C4⋊C4×C71—21no—
54C2×C4⋊C4×C71—21no—
55C2×C4⋊C4×C71D411yescharacteristic
56C2×C42×C71—21no—
57C2×C42×C71C2311yescharacteristic
58C4⋊C8×C71—21no—
59C4⋊C8×C71D412yes—
60#103×C71—24no—
61#11765849—41no—
62#11765849—41no—
63#11765849—41no—
64#11765849—41no—
65C2×C71⋊C8—22no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8—41no—
69C2×Q8×C71—21no—
70C2×Q8×C71—21no—
71C2×Q8×C71—22no—
72C2×Q8×C71—21no—
73C2×Q8×C71—21no—
74C2×Q8×C71—21no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71—41no—
79C2×Q8×C71—41no—
80C2×Q8×C71D812yes—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—41no—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71#4111yescharacteristic
86C22×C4×C71#4111yescharacteristic
87Q16×C71—48no—
88C2×C8×C71—22no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C4⋊C4×C71—21no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—21no—
94C4⋊C4×C71—22no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C42×C71—41no—
100C42×C71—41no—
101C42×C71C2×D411yescharacteristic
102C42×C71C2×D411yescharacteristic
103C42×C71C2×D411yescharacteristic
104C42×C71#4111yescharacteristic
105C23×C71#12311yescharacteristic
106Q8×C71—22no—
107Q8×C71—22no—
108Q8×C71—41no—
109Q8×C71—41no—
110Q8×C71—42no—
111Q8×C71—42no—
112Q8×C71—81no—
113Q8×C71—81no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—42no—
121C2×C4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71—81no—
124C2×C4×C71#12011yescharacteristic
125C2×C4×C71#12111yescharacteristic
126C2×C4×C71#12111yescharacteristic
127C2×C4×C71#12111yescharacteristic
128C2×C4×C71#12311yescharacteristic
129C2×C4×C71#12411yescharacteristic
130C2×C4×C71C2×D811yescharacteristic
131C2×C4×C71#13511yescharacteristic
132C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C71#13711yescharacteristic
134C8×C71—42no—
135C71⋊C8—42no—
136C71⋊C8—81no—
137C71⋊C8—81no—
138C71⋊C8—81no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71#39411yescharacteristic
142C22×C71#48011yescharacteristic
143C22×C71#48211yescharacteristic
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—42no—
148C4×C71—81no—
149C4×C71—82no—
150C4×C71#44711yescharacteristic
151C4×C71#45211yescharacteristic
152C4×C71#46611yescharacteristic
153C4×C71#46811yescharacteristic
154C4×C71#47312yes—
155#10446—711nomaximal, Sylow
156C2×C71—21no—
157C2×C71—21no—
158C2×C71#203311yescharacteristic
159C2×C71#204711yescharacteristic
160C2×C71#204911yescharacteristic
161C2×#462D7111yescharacteristic
162#1801—711no—
163#1571—711no—
164#1546—711no—
165#1546—711no—
166#1479—711no—
167#1301—711no—
168C71#1044611yescharacteristic, complemented, Sylow
169C22×Q16—21no—
170C2×C4⋊Q8C2×D7111yescharacteristic
171C2×C4×Q8C2×D7111yescharacteristic
172#462—22no—
173#462—22no—
174#431—711no—
175#431—711no—
176#423—711no—
177#423—711no—
178C2×C4⋊C8C2×D7111yescharacteristic
179C2×#103—21no—
180C2×#101—1421no—
181#343—1421no—
182#335—1422no—
183#332—1422no—
184#332—1422no—
185Q8⋊C8—1422no—
186C22×Q8—21no—
187C22×Q8C22⋊D7111yescharacteristic
188C2×Q16—24no—
189C2×Q16—42no—
190C2×Q16—42no—
191C2×M16—1421no—
192C2×M16—1421no—
193C22×C8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22⋊D7112yes—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D7111yescharacteristic
206C2×C42—21no—
207C2×C42C22×D7111yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—712no—
210C4⋊C8—1421no—
211C4⋊C8—1421no—
212C4⋊C8—1421no—
213C4⋊C8—1421no—
214C4⋊C8—1421no—
215C4⋊C8C22⋊D7112yes—
216#103—24no—
217#101—2842no—
218C22⋊C8—1421no—
219C22⋊C8—1421no—
220C4×C8—1421no—
221C4×C8—1421no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8#1176585012yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4C2×C22⋊D7111yescharacteristic
238C22×C4#1176586411yescharacteristic
239C22×C4#1176586411yescharacteristic
240Q16—48no—
241M16—2841no—
242M16—2841no—
243M16—2841no—
244M16—2841no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—2841no—
249C2×C8—2841no—
250C2×C8—2841no—
251C2×C8—1422no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C42—41no—
261C42—41no—
262C42D4×D7111yescharacteristic
263C42C2×C22⋊D7111yescharacteristic
264C42C2×C22⋊D7111yescharacteristic
265C42#1176586411yescharacteristic
266C23#1455610911yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—41no—
270Q8—41no—
271Q8—42no—
272Q8—42no—
273Q8—81no—
274Q8—81no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4#1455609711yescharacteristic
286C2×C4#1455609911yescharacteristic
287C2×C4#1455610011yescharacteristic
288C2×C4#1455610011yescharacteristic
289C2×C4#1455610911yescharacteristic
290C2×C4#1455611511yescharacteristic
291C2×C4C2×#1176585011yescharacteristic
292C2×C4#1455614811yescharacteristic
293C2×C4#1455615411yescharacteristic
294C2×C4#1455615811yescharacteristic
295C8—42no—
296C8—2842no—
297C8—5681no—
298C8—5681no—
299C8—5681no—
300C22—21no—
301C22—21no—
302C22#1798591311yescharacteristic
303C22#1798631011yescharacteristic
304C22#1798632011yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—82no—
310C4—81no—
311C4#1798613311yescharacteristic
312C4#1798616111yescharacteristic
313C4#1798623311yescharacteristic
314C4#1798624311yescharacteristic
315C4#1798627112yes—
316C2—21no—
317C2—21no—
318C2#2258951911yescharacteristic
319C2#2258960511yescharacteristic
320C2#2258961711yescharacteristic
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC2×#462×C71C2
socleC22×C71#394
prime radicalC23×C71#123
hypercenterC2×#462D71
hypocenterC71#10446
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
168. C71#10446semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225871142
#225870122
#225867352
#225862372
C2×#462×C712
#1044671