Finite Group Databasebeta

← #31886688

Subgroups — #31886688

Order 18176.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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#22589898C211yesmaximal, characteristic
2#22589816C211yesmaximal, characteristic
3#22589748C211yesmaximal, characteristic
4#22586166C211yesmaximal, characteristic
5#22586163C211yesmaximal, characteristic
6#22586163C211yesmaximal, characteristic
7C71×#2805C211yesmaximal, characteristic
8C2×#14556059—21no—
9C2×#14556052—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18#400×C71—21no—
19#400×C71—21no—
20#395×C71—21no—
21#391×C71C2211yescharacteristic
22#389×C71—21no—
23#389×C71C2211yescharacteristic
24#389×C71C2211yescharacteristic
25#387×C71—21no—
26#387×C71—21no—
27C22×C71⋊C8—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33#14556059—42no—
34#14556052—42no—
35C22×Q8×C71D411yescharacteristic
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C42×C71—21no—
46C2×C42×C71C2311yescharacteristic
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56C22×Dic71—41no—
57C2×C71⋊C8—22no—
58C2×C71⋊C8—41no—
59C2×C71⋊C8—41no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C4⋊Dic71—42no—
67C4⋊Dic71—41no—
68C4⋊Dic71—41no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C2×Q8×C71—21no—
72C2×Q8×C71—21no—
73C2×Q8×C71—41no—
74C2×Q8×C71—41no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C4⋊C4×C71—21no—
85C4⋊C4×C71—21no—
86C4⋊C4×C71—21no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—81no—
95C4⋊C4×C71—81no—
96C4⋊C4×C71SD1612yes—
97C42×C71—41no—
98C42×C71—81no—
99C23×C71#12011yescharacteristic
100Q8×C71—42no—
101Q8×C71—81no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—22no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71C2×SD1611yescharacteristic
127C2×C4×C71#13511yescharacteristic
128C2×C4×C71#13511yescharacteristic
129C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C71#13711yescharacteristic
131C2×C4×C71#13711yescharacteristic
132C2×Dic71—81no—
133C2×Dic71—81no—
134C2×Dic71—81no—
135C71⋊C8—42no—
136C71⋊C8—81no—
137C71⋊C8—81no—
138C22×C71#44811yescharacteristic
139C22×C71#44911yescharacteristic
140C22×C71#45211yescharacteristic
141C22×C71#45411yescharacteristic
142C22×C71#45511yescharacteristic
143C22×C71#45611yescharacteristic
144C22×C71#45811yescharacteristic
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—21no—
150C4×C71—21no—
151C4×C71—42no—
152C4×C71—81no—
153C4×C71—81no—
154C4×C71—81no—
155C4×C71—81no—
156C4×C71—81no—
157Dic71—161no—
158#7818—711nomaximal, Sylow
159C2×C71#161111yescharacteristic
160C2×C71#161211yescharacteristic
161C2×C71#161511yescharacteristic
162C2×C71#162611yescharacteristic
163C2×C71#164311yescharacteristic
164C2×C71#165211yescharacteristic
165C2×C71#166411yescharacteristic
166#2805D7111yescharacteristic
167#2099—711no—
168#2084—711no—
169#2072—711no—
170#1272—711no—
171#1271—711no—
172#1271—711no—
173C71#781811yescharacteristic, complemented, Sylow
174C2×#106—1421no—
175C2×#103—1421no—
176#400—21no—
177#400—21no—
178#395—21no—
179#391C2×D7111yescharacteristic
180#389—21no—
181#389C2×D7111yescharacteristic
182#389C2×D7111yescharacteristic
183#387—21no—
184#387—21no—
185#384—711no—
186#340—1421no—
187#340—1421no—
188#340—1421no—
189#340—1421no—
190#336—711no—
191#336—711no—
192#336—711no—
193C22×Q8C22⋊D7111yescharacteristic
194C22×C8—1421no—
195C22×C8—1421no—
196C22×C8—1421no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1421no—
202C2×C4⋊C4—1421no—
203C2×C4⋊C4—1421no—
204C2×C4⋊C4C22⋊D7111yescharacteristic
205C2×C4⋊C4C22⋊D7111yescharacteristic
206C2×C4⋊C4C22⋊D7111yescharacteristic
207C2×C4⋊C4C22⋊D7111yescharacteristic
208C2×C4⋊C4C22⋊D7111yescharacteristic
209C2×C42—21no—
210C2×C42C22×D7111yescharacteristic
211#106—2842no—
212#103—2842no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—2841no—
233C22×C4C2×C22⋊D7111yescharacteristic
234C22×C4C2×C22⋊D7111yescharacteristic
235C22×C4C2×C22⋊D7111yescharacteristic
236C2×C8—2841no—
237C2×C8—2841no—
238C2×C8—2841no—
239C2×C8—2841no—
240C2×C8—2841no—
241C2×C8—2841no—
242C2×C8—2841no—
243C2×C8—2841no—
244C2×C8—1422no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—2842no—
258C4⋊C4—2841no—
259C4⋊C4—2841no—
260C4⋊C4—2841no—
261C4⋊C4—2841no—
262C4⋊C4#1176585212yes—
263C42—41no—
264C42—81no—
265C23#1455609711yescharacteristic
266Q8—42no—
267Q8—81no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—5681no—
293C2×C4—5681no—
294C2×C4—5681no—
295C2×C4C2×#1176585211yescharacteristic
296C2×C4#1455614811yescharacteristic
297C2×C4#1455614811yescharacteristic
298C2×C4#1455615311yescharacteristic
299C2×C4#1455615811yescharacteristic
300C2×C4#1455615911yescharacteristic
301C8—2842no—
302C8—5681no—
303C8—5681no—
304C22#1798613811yescharacteristic
305C22#1798614411yescharacteristic
306C22#1798616111yescharacteristic
307C22#1798617211yescharacteristic
308C22#1798617711yescharacteristic
309C22#1798618211yescharacteristic
310C22#1798618911yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—11361no—
324C2#2258734811yescharacteristic
325C2#2258735411yescharacteristic
326C2#2258737111yescharacteristic
327C2#2258743611yescharacteristic
328C2#2258754711yescharacteristic
329C2#2258761011yescharacteristic
330C2#2258767411yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2805C2
socle, prime radicalC23×C71#120
hypercenter#2805D71
hypocenterC71#7818
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
173. C71#7818semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225898162
#225897482
#225861662
#225861632
C71×#28052
#781871