Finite Group Databasebeta

← #31886556

Subgroups — #31886556

Order 18176.

374 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 91 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#22589811C211yesmaximal, characteristic
4#22586163C211yesmaximal, characteristic
5#22586163C211yesmaximal, characteristic
6#22586163C211yesmaximal, characteristic
7C71×#2726C211yesmaximal, characteristic
8C2×#14556061—21no—
9C2×#14556059—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18C2×#125×C71—21no—
19#398×C71—21no—
20#398×C71—21no—
21#398×C71—21no—
22#389×C71—21no—
23#389×C71C2211yescharacteristic
24#389×C71C2211yescharacteristic
25#389×C71C2211yescharacteristic
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#14556061—42no—
34#14556059—42no—
35#125×C71—42no—
36#125×C71—42no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C42×C71—21no—
51C2×C42×C71C2311yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58C22×Dic71—41no—
59C2×C71⋊C8—22no—
60C2×C71⋊C8—22no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C4⋊Dic71—42no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—21no—
83C4⋊C4×C71—21no—
84C4⋊C4×C71—21no—
85C4⋊C4×C71—21no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—42no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—81no—
99C4⋊C4×C71D812yes—
100C4⋊C4×C71SD1612yes—
101C42×C71—41no—
102C42×C71—42no—
103C23×C71#12011yescharacteristic
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
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×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71C2×D811yescharacteristic
129C2×C4×C71C2×SD1611yescharacteristic
130C2×C4×C71#13511yescharacteristic
131C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C71#13711yescharacteristic
133C2×C4×C71#13711yescharacteristic
134C2×Dic71—81no—
135C2×Dic71—81no—
136C2×Dic71—81no—
137C71⋊C8—42no—
138C71⋊C8—42no—
139C71⋊C8—81no—
140C22×C71#44711yescharacteristic
141C22×C71#44811yescharacteristic
142C22×C71#45211yescharacteristic
143C22×C71#45411yescharacteristic
144C22×C71#45611yescharacteristic
145C22×C71#45611yescharacteristic
146C22×C71#45811yescharacteristic
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—21no—
150C4×C71—21no—
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—42no—
154C4×C71—42no—
155C4×C71—81no—
156C4×C71—81no—
157C4×C71—81no—
158C4×C71—81no—
159Dic71—161no—
160#7798—711nomaximal, Sylow
161C2×C71#161011yescharacteristic
162C2×C71#161111yescharacteristic
163C2×C71#161511yescharacteristic
164C2×C71#162711yescharacteristic
165C2×C71#164711yescharacteristic
166C2×C71#165311yescharacteristic
167C2×C71#166311yescharacteristic
168#2726D7111yescharacteristic
169#2099—711no—
170#2084—711no—
171#2083—711no—
172#1271—711no—
173#1271—711no—
174#1271—711no—
175C71#779811yescharacteristic, complemented, Sylow
176C2×#125—21no—
177C2×#107—1421no—
178C2×#106—1421no—
179#398—21no—
180#398—21no—
181#398—21no—
182#389—21no—
183#389C2×D7111yescharacteristic
184#389C2×D7111yescharacteristic
185#389C2×D7111yescharacteristic
186#387—21no—
187#384—711no—
188#340—1421no—
189#340—1421no—
190#340—1421no—
191#340—1421no—
192#336—711no—
193#336—711no—
194#336—711no—
195C22×C8—1421no—
196C22×C8—1421no—
197C22×C8—1421no—
198#125—42no—
199#125—42no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—1421no—
208C2×C4⋊C4—1421no—
209C2×C4⋊C4—1421no—
210C2×C4⋊C4C22⋊D7111yescharacteristic
211C2×C4⋊C4C22⋊D7111yescharacteristic
212C2×C4⋊C4C22⋊D7111yescharacteristic
213C2×C4⋊C4C22⋊D7111yescharacteristic
214C2×C4⋊C4C22⋊D7111yescharacteristic
215C2×C4⋊C4C22⋊D7111yescharacteristic
216C2×C42—21no—
217C2×C42C22×D7111yescharacteristic
218#107—2842no—
219#106—2842no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
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—1422no—
244C2×C8—1422no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—2842no—
263C4⋊C4—2841no—
264C4⋊C4—2841no—
265C4⋊C4—2841no—
266C4⋊C4—2841no—
267C4⋊C4#1176585012yes—
268C4⋊C4#1176585212yes—
269C42—41no—
270C42—42no—
271C23#1455609711yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
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—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—5681no—
297C2×C4—5681no—
298C2×C4—5681no—
299C2×C4C2×#1176585011yescharacteristic
300C2×C4C2×#1176585211yescharacteristic
301C2×C4#1455614811yescharacteristic
302C2×C4#1455615411yescharacteristic
303C2×C4#1455615811yescharacteristic
304C2×C4#1455615911yescharacteristic
305C8—2842no—
306C8—2842no—
307C8—5681no—
308C22#1798613311yescharacteristic
309C22#1798613811yescharacteristic
310C22#1798616111yescharacteristic
311C22#1798617211yescharacteristic
312C22#1798618211yescharacteristic
313C22#1798618211yescharacteristic
314C22#1798618911yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—11361no—
328C2#2258734311yescharacteristic
329C2#2258734811yescharacteristic
330C2#2258737111yescharacteristic
331C2#2258744311yescharacteristic
332C2#2258757511yescharacteristic
333C2#2258761711yescharacteristic
334C2#2258766811yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2726C2
socle, prime radicalC23×C71#120
hypercenter#2726D71
hypocenterC71#7798
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2726

Complements

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

NH ≅ G/Ntype
175. C71#7798semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225898162
#225898112
#225861632
C71×#27262
#779871