Finite Group Databasebeta

← #31886530

Subgroups — #31886530

Order 18176.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#22589811C211yesmaximal, characteristic
3#22589748C211yesmaximal, characteristic
4#22586166C211yesmaximal, characteristic
5#22586163C211yesmaximal, characteristic
6#22586163C211yesmaximal, characteristic
7C71×#2734C211yesmaximal, characteristic
8C2×#14556061—21no—
9C2×#14556052—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18C2×C4⋊Q8×C71—21no—
19#398×C71—21no—
20#395×C71—21no—
21#395×C71—21no—
22#391×C71C2211yescharacteristic
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#14556061—42no—
34#14556052—42no—
35C22×Q8×C71—21no—
36C22×Q8×C71D411yescharacteristic
37C4⋊Q8×C71—42no—
38C4⋊Q8×C71—42no—
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×C42×C71—21no—
50C2×C42×C71C2311yescharacteristic
51#95×C71—21no—
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—41no—
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—
73C2×Q8×C71—21no—
74C2×Q8×C71—42no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—81no—
78C2×Q8×C71—81no—
79C2×Q8×C71D812yes—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—21no—
90C4⋊C4×C71—21no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—42no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—81no—
101C4⋊C4×C71SD1612yes—
102C4⋊C4×C71Q1612yes—
103C42×C71—41no—
104C42×C71—42no—
105C23×C71#12011yescharacteristic
106Q8×C71—42no—
107Q8×C71—81no—
108Q8×C71—84no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—22no—
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×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71C2×D811yescharacteristic
134C2×C4×C71C2×SD1611yescharacteristic
135C2×C4×C71C2×Q1611yescharacteristic
136C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C71#13711yescharacteristic
138C2×C4×C71#13711yescharacteristic
139C2×Dic71—81no—
140C2×Dic71—81no—
141C2×Dic71—81no—
142C71⋊C8—42no—
143C71⋊C8—81no—
144C71⋊C8—81no—
145C22×C71#44711yescharacteristic
146C22×C71#44811yescharacteristic
147C22×C71#45111yescharacteristic
148C22×C71#45411yescharacteristic
149C22×C71#45511yescharacteristic
150C22×C71#45611yescharacteristic
151C22×C71#45811yescharacteristic
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—21no—
156C4×C71—21no—
157C4×C71—21no—
158C4×C71—42no—
159C4×C71—42no—
160C4×C71—42no—
161C4×C71—81no—
162C4×C71—81no—
163C4×C71—81no—
164Dic71—161no—
165#7794—711nomaximal, Sylow
166C2×C71#160711yescharacteristic
167C2×C71#161111yescharacteristic
168C2×C71#161711yescharacteristic
169C2×C71#162711yescharacteristic
170C2×C71#164511yescharacteristic
171C2×C71#164911yescharacteristic
172C2×C71#166411yescharacteristic
173#2734D7111yescharacteristic
174#2099—711no—
175#2083—711no—
176#2072—711no—
177#1272—711no—
178#1271—711no—
179#1271—711no—
180C71#779411yescharacteristic, complemented, Sylow
181C2×C4⋊Q8—21no—
182C2×#107—1421no—
183C2×#103—1421no—
184#398—21no—
185#395—21no—
186#395—21no—
187#391C2×D7111yescharacteristic
188#389C2×D7111yescharacteristic
189#389C2×D7111yescharacteristic
190#387—21no—
191#387—21no—
192#384—711no—
193#340—1421no—
194#340—1421no—
195#340—1421no—
196#340—1421no—
197#336—711no—
198#336—711no—
199#336—711no—
200C22×Q8—21no—
201C22×Q8C22⋊D7111yescharacteristic
202C22×C8—1421no—
203C22×C8—1421no—
204C22×C8—1421no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1421no—
213C2×C4⋊C4—1421no—
214C2×C4⋊C4—1421no—
215C2×C4⋊C4C22⋊D7111yescharacteristic
216C2×C4⋊C4C22⋊D7111yescharacteristic
217C2×C4⋊C4C22⋊D7111yescharacteristic
218C2×C4⋊C4C22⋊D7111yescharacteristic
219C2×C4⋊C4C22⋊D7111yescharacteristic
220C2×C42—21no—
221C2×C42C22×D7111yescharacteristic
222#107—2842no—
223#103—2842no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C2×Q8—21no—
232C2×Q8—42no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—81no—
236C2×Q8—81no—
237C2×Q8#1176585012yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—2841no—
245C22×C4C2×C22⋊D7111yescharacteristic
246C22×C4C2×C22⋊D7111yescharacteristic
247C22×C4C2×C22⋊D7111yescharacteristic
248C2×C8—2841no—
249C2×C8—2841no—
250C2×C8—2841no—
251C2×C8—2841no—
252C2×C8—2841no—
253C2×C8—2841no—
254C2×C8—2841no—
255C2×C8—2841no—
256C2×C8—1422no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—81no—
269C4⋊C4—2842no—
270C4⋊C4—2841no—
271C4⋊C4—2841no—
272C4⋊C4—2841no—
273C4⋊C4—2841no—
274C4⋊C4#1176585212yes—
275C4⋊C4#1176585512yes—
276C42—41no—
277C42—42no—
278C23#1455609711yescharacteristic
279Q8—42no—
280Q8—81no—
281Q8—84no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—5681no—
307C2×C4—5681no—
308C2×C4—5681no—
309C2×C4C2×#1176585011yescharacteristic
310C2×C4C2×#1176585211yescharacteristic
311C2×C4C2×#1176585511yescharacteristic
312C2×C4#1455615411yescharacteristic
313C2×C4#1455615911yescharacteristic
314C2×C4#1455615911yescharacteristic
315C8—2842no—
316C8—5681no—
317C8—5681no—
318C22#1798613311yescharacteristic
319C22#1798613811yescharacteristic
320C22#1798615511yescharacteristic
321C22#1798617211yescharacteristic
322C22#1798617711yescharacteristic
323C22#1798618211yescharacteristic
324C22#1798618911yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—11361no—
338C2#2258732611yescharacteristic
339C2#2258734811yescharacteristic
340C2#2258738211yescharacteristic
341C2#2258744311yescharacteristic
342C2#2258756111yescharacteristic
343C2#2258758911yescharacteristic
344C2#2258767411yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2734C2
socle, prime radicalC23×C71#120
hypercenter#2734D71
hypocenterC71#7794
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
180. C71#7794semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225898112
#225897482
#225861662
#225861632
C71×#27342
#779471