Finite Group Databasebeta

← #31886676

Subgroups — #31886676

Order 18176.

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

normal subgroups only

This group has 362 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#22589748C211yesmaximal, characteristic
3#22589748C211yesmaximal, characteristic
4#22586166C211yesmaximal, characteristic
5#22586166C211yesmaximal, characteristic
6#22586163C211yesmaximal, characteristic
7C71×#2806C211yesmaximal, characteristic
8C2×#14556052—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#398×C71—21no—
19#398×C71—21no—
20#395×C71—21no—
21#391×C71—21no—
22#391×C71C2211yescharacteristic
23#391×C71C2211yescharacteristic
24#389×C71—21no—
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#14556052—42no—
34#14556052—42no—
35C22×Q8×C71D411yescharacteristic
36C22×Q8×C71D411yescharacteristic
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×C42×C71—21no—
47C2×C42×C71C2311yescharacteristic
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—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—41no—
68C4⋊Dic71—41no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C2×Q8×C71—21no—
72C2×Q8×C71—21no—
73C2×Q8×C71—21no—
74C2×Q8×C71—41no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71SD1612yes—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—21no—
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×C71—81no—
97C4⋊C4×C71—81no—
98C4⋊C4×C71Q1612yes—
99C42×C71—41no—
100C42×C71—81no—
101C23×C71#12011yescharacteristic
102Q8×C71—42no—
103Q8×C71—42no—
104Q8×C71—81no—
105Q8×C71—81no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
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×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71C2×SD1611yescharacteristic
131C2×C4×C71C2×Q1611yescharacteristic
132C2×C4×C71#13511yescharacteristic
133C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C71#13711yescharacteristic
135C2×C4×C71#13711yescharacteristic
136C2×Dic71—81no—
137C2×Dic71—81no—
138C2×Dic71—81no—
139C71⋊C8—81no—
140C71⋊C8—81no—
141C71⋊C8—81no—
142C22×C71#44811yescharacteristic
143C22×C71#44911yescharacteristic
144C22×C71#45111yescharacteristic
145C22×C71#45411yescharacteristic
146C22×C71#45511yescharacteristic
147C22×C71#45511yescharacteristic
148C22×C71#45811yescharacteristic
149C4×C71—21no—
150C4×C71—21no—
151C4×C71—21no—
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—42no—
156C4×C71—42no—
157C4×C71—81no—
158C4×C71—81no—
159C4×C71—81no—
160C4×C71—81no—
161Dic71—161no—
162#7816—711nomaximal, Sylow
163C2×C71#160811yescharacteristic
164C2×C71#161211yescharacteristic
165C2×C71#161711yescharacteristic
166C2×C71#162611yescharacteristic
167C2×C71#164411yescharacteristic
168C2×C71#164511yescharacteristic
169C2×C71#166211yescharacteristic
170#2806D7111yescharacteristic
171#2099—711no—
172#2072—711no—
173#2072—711no—
174#1272—711no—
175#1272—711no—
176#1271—711no—
177C71#781611yescharacteristic, complemented, Sylow
178C2×#103—1421no—
179C2×#103—1421no—
180#398—21no—
181#398—21no—
182#395—21no—
183#391—21no—
184#391C2×D7111yescharacteristic
185#391C2×D7111yescharacteristic
186#389—21no—
187#389C2×D7111yescharacteristic
188#387—21no—
189#384—711no—
190#340—1421no—
191#340—1421no—
192#340—1421no—
193#340—1421no—
194#336—711no—
195#336—711no—
196#336—711no—
197C22×Q8C22⋊D7111yescharacteristic
198C22×Q8C22⋊D7111yescharacteristic
199C22×C8—1421no—
200C22×C8—1421no—
201C22×C8—1421no—
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×C42—21no—
215C2×C42C22×D7111yescharacteristic
216#103—2842no—
217#103—2842no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8#1176585212yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—2841no—
241C22×C4C2×C22⋊D7111yescharacteristic
242C22×C4C2×C22⋊D7111yescharacteristic
243C22×C4C2×C22⋊D7111yescharacteristic
244C2×C8—2841no—
245C2×C8—2841no—
246C2×C8—2841no—
247C2×C8—2841no—
248C2×C8—2841no—
249C2×C8—2841no—
250C2×C8—2841no—
251C2×C8—2841no—
252C2×C8—2841no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—2842no—
264C4⋊C4—2841no—
265C4⋊C4—2841no—
266C4⋊C4—2841no—
267C4⋊C4—2841no—
268C4⋊C4#1176585512yes—
269C42—41no—
270C42—81no—
271C23#1455609711yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—5681no—
301C2×C4—5681no—
302C2×C4—5681no—
303C2×C4C2×#1176585211yescharacteristic
304C2×C4C2×#1176585511yescharacteristic
305C2×C4#1455614811yescharacteristic
306C2×C4#1455615311yescharacteristic
307C2×C4#1455615911yescharacteristic
308C2×C4#1455615911yescharacteristic
309C8—5681no—
310C8—5681no—
311C8—5681no—
312C22#1798613811yescharacteristic
313C22#1798614411yescharacteristic
314C22#1798615511yescharacteristic
315C22#1798617211yescharacteristic
316C22#1798617711yescharacteristic
317C22#1798617711yescharacteristic
318C22#1798618911yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—42no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—11361no—
332C2#2258733111yescharacteristic
333C2#2258735411yescharacteristic
334C2#2258738211yescharacteristic
335C2#2258743611yescharacteristic
336C2#2258755411yescharacteristic
337C2#2258756111yescharacteristic
338C2#2258766311yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2806C2
socle, prime radicalC23×C71#120
hypercenter#2806D71
hypocenterC71#7816
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2806

Complements

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

NH ≅ G/Ntype
177. C71#7816semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225897482
#225861662
#225861632
C71×#28062
#781671