Finite Group Databasebeta

← #31886739

Subgroups — #31886739

Order 18176.

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

normal subgroups only

This group has 358 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#22589903C211yesmaximal, characteristic
2#22589903C211yesmaximal, characteristic
3#22589903C211yesmaximal, characteristic
4#22586163C211yesmaximal, characteristic
5#22586163C211yesmaximal, characteristic
6#22586163C211yesmaximal, characteristic
7C71×#2726C211yesmaximal, characteristic
8#17985869C2211yescharacteristic
9#17985761—21no—
10#17985761—21no—
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#125×C71—42no—
34#125×C71—42no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
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×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C42×C71—21no—
49C2×C42×C71C2311yescharacteristic
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—41no—
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—42no—
68C4⋊Dic71—42no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C4⋊C4×C71—21no—
79C4⋊C4×C71—21no—
80C4⋊C4×C71—21no—
81C4⋊C4×C71—21no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
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×C71SD1612yes—
96C4⋊C4×C71Q1612yes—
97C42×C71—41no—
98C42×C71—42no—
99C23×C71#12011yescharacteristic
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—41no—
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×C71C2×SD1611yescharacteristic
125C2×C4×C71C2×Q1611yescharacteristic
126C2×C4×C71#13511yescharacteristic
127C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C71#13711yescharacteristic
129C2×C4×C71#13711yescharacteristic
130C2×Dic71—81no—
131C2×Dic71—81no—
132C2×Dic71—81no—
133C71⋊C8—81no—
134C71⋊C8—81no—
135C71⋊C8—81no—
136C22×C71#45011yescharacteristic
137C22×C71#45111yescharacteristic
138C22×C71#45211yescharacteristic
139C22×C71#45411yescharacteristic
140C22×C71#45411yescharacteristic
141C22×C71#45411yescharacteristic
142C22×C71#45811yescharacteristic
143C4×C71—21no—
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—42no—
150C4×C71—42no—
151C4×C71—81no—
152C4×C71—81no—
153C4×C71—81no—
154C4×C71—81no—
155Dic71—161no—
156#7827—711nomaximal, Sylow
157C2×C71#161511yescharacteristic
158C2×C71#161611yescharacteristic
159C2×C71#161711yescharacteristic
160C2×C71#163511yescharacteristic
161C2×C71#163611yescharacteristic
162C2×C71#163711yescharacteristic
163C2×C71#165811yescharacteristic
164#2726D7111yescharacteristic
165#2100—711no—
166#2100—711no—
167#2100—711no—
168#1271—711no—
169#1271—711no—
170#1271—711no—
171C71#782711yescharacteristic, complemented, Sylow
172C2×#125—21no—
173#398—21no—
174#398—21no—
175#398—21no—
176#389—21no—
177#389C2×D7111yescharacteristic
178#389C2×D7111yescharacteristic
179#389C2×D7111yescharacteristic
180#387—21no—
181#384—711no—
182#340—1421no—
183#340—1421no—
184#340—1421no—
185#340—1421no—
186#340—1421no—
187#340—1421no—
188#336—711no—
189#336—711no—
190#336—711no—
191C22×C8—1421no—
192C22×C8—1421no—
193C22×C8—1421no—
194#125—42no—
195#125—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1421no—
204C2×C4⋊C4—1421no—
205C2×C4⋊C4—1421no—
206C2×C4⋊C4C22⋊D7111yescharacteristic
207C2×C4⋊C4C22⋊D7111yescharacteristic
208C2×C4⋊C4C22⋊D7111yescharacteristic
209C2×C4⋊C4C22⋊D7111yescharacteristic
210C2×C4⋊C4C22⋊D7111yescharacteristic
211C2×C4⋊C4C22⋊D7111yescharacteristic
212C2×C42—21no—
213C2×C42C22×D7111yescharacteristic
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—2841no—
227C22×C4C2×C22⋊D7111yescharacteristic
228C22×C4C2×C22⋊D7111yescharacteristic
229C22×C4C2×C22⋊D7111yescharacteristic
230C2×C8—2841no—
231C2×C8—2841no—
232C2×C8—2841no—
233C2×C8—2841no—
234C2×C8—2841no—
235C2×C8—2841no—
236C2×C8—2841no—
237C2×C8—2841no—
238C2×C8—2841no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—2842no—
257C4⋊C4—2842no—
258C4⋊C4—2842no—
259C4⋊C4#1176585312yes—
260C4⋊C4#1176585512yes—
261C42—41no—
262C42—42no—
263C23#1455609711yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
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—5681no—
289C2×C4—5681no—
290C2×C4—5681no—
291C2×C4C2×#1176585311yescharacteristic
292C2×C4C2×#1176585511yescharacteristic
293C2×C4#1455614811yescharacteristic
294C2×C4#1455615411yescharacteristic
295C2×C4#1455615811yescharacteristic
296C2×C4#1455615911yescharacteristic
297C8—5681no—
298C8—5681no—
299C8—5681no—
300C22#1798615011yescharacteristic
301C22#1798615511yescharacteristic
302C22#1798616111yescharacteristic
303C22#1798617211yescharacteristic
304C22#1798617211yescharacteristic
305C22#1798617211yescharacteristic
306C22#1798618911yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—11361no—
320C2#2258737111yescharacteristic
321C2#2258737711yescharacteristic
322C2#2258738211yescharacteristic
323C2#2258749111yescharacteristic
324C2#2258749811yescharacteristic
325C2#2258750511yescharacteristic
326C2#2258764211yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2726C2
socle, prime radicalC23×C71#120
hypercenter#2726D71
hypocenterC71#7827
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
171. C71#7827semidirect

Maximal subgroups

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

subgroupindex
#225899032
#225861632
C71×#27262
#782771