Finite Group Databasebeta

← #31895221

Subgroups — #31895221

Order 18176.

578 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 135 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#22592657C211yesmaximal, characteristic
2#22588431C212yesmaximal, complemented
3#22588328C211yesmaximal, characteristic
4#22586208C212yesmaximal, complemented
5C71×#2972C211yesmaximal, characteristic
6C22×C4⋊Dic71C2211yescharacteristic
7C2×#14556043—21no—
8#17985938C412yescomplemented
9#17985935C412yescomplemented
10#17985897—21no—
11#17985879—21no—
12#17985871—21no—
13#17985859—21no—
14#17985859C2212yes—
15#17985859C2211yescharacteristic
16#17985766—21no—
17#17985762—24no—
18C2×C4×D4×C71C2211yescharacteristic
19C8×D4×C71—24no—
20#431×C71—21no—
21C2×C22⋊C8×C71—21no—
22C2×C22⋊C8×C71C2212yes—
23C23×Dic71—21no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—22no—
28C2×C4⋊Dic71—22no—
29C2×C4⋊Dic71C2×C412yes—
30C2×Dic71⋊C4—22no—
31C2×C22⋊Dic71—22no—
32C2×C22⋊Dic71—21no—
33C2×C22⋊Dic71—22no—
34#14556043—42no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—22no—
38C71⋊#95C2×C412yes—
39C22×D4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C23×C4×C71C2311yescharacteristic
42C22×C8×C71—22no—
43C22×C8×C71D414yes—
44C4×D4×C71—22no—
45C4×D4×C71—22no—
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C4⋊C8×C71—22no—
51C22⋊C8×C71—22no—
52C22⋊C8×C71—22no—
53C22⋊C8×C71—22no—
54C4×C8×C71—22no—
55C22×Dic71—22no—
56C22×Dic71—22no—
57C22×Dic71—42no—
58C22×Dic71—42no—
59C22×Dic71—82no—
60C4⋊Dic71—24no—
61C4⋊Dic71—42no—
62C4⋊Dic71—41no—
63C4⋊Dic71—41no—
64C4⋊Dic71—44no—
65Dic71⋊C4—82no—
66C22⋊Dic71—42no—
67C22⋊Dic71—42no—
68C22⋊Dic71—42no—
69C22⋊Dic71—82no—
70C24×C71C22⋊C411yescharacteristic
71C24×C71C2×D411yescharacteristic
72C2×D4×C71—22no—
73C2×D4×C71—22no—
74C2×D4×C71SD1612yes—
75C2×D4×C71Q1612yes—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—42no—
79C22×C4×C71—42no—
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4112yes—
87C2×C8×C71—24no—
88C2×C8×C71—44no—
89C2×C8×C71—82no—
90C4⋊C4×C71—22no—
91C22⋊C4×C71—22no—
92C22⋊C4×C71—22no—
93C42×C71—21no—
94C42×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71#10312yes—
102C23×C71C2×C22⋊C411yescharacteristic
103C23×C71#12112yes—
104C23×C71C2×SD1611yescharacteristic
105C23×C71C2×Q1611yescharacteristic
106C23×C71#13512yes—
107D4×C71—24no—
108D4×C71—24no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—24no—
112C2×C4×C71—22no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—41no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—81no—
121C2×C4×C71#10312yes—
122C2×C4×C71C4×D412yes—
123C2×C4×C71#12011yescharacteristic
124C2×C4×C71#12311yescharacteristic
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—42no—
128C2×Dic71—42no—
129C2×Dic71—42no—
130C2×Dic71—42no—
131C2×Dic71—84no—
132C2×Dic71—82no—
133C2×Dic71—82no—
134C2×Dic71—82no—
135C8×C71—44no—
136C22×C71—21no—
137C22×C71—22no—
138C22×C71—21no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—42no—
144C22×C71—42no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71#38611yescharacteristic
149C22×C71C2×#10311yescharacteristic
150C22×C71#41611yescharacteristic
151C22×C71#46512yes—
152C22×C71#46712yes—
153C4×C71—21no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71#45812yes—
159Dic71—82no—
160Dic71—82no—
161Dic71—162no—
162#9469—711nomaximal, Sylow
163C2×C71—21no—
164C2×C71—22no—
165C2×C71—21no—
166C2×C71—22no—
167C2×C71—42no—
168C2×C71#189011yescharacteristic
169C2×C71#189211yescharacteristic
170C2×C71#191011yescharacteristic
171#2972D7111yescharacteristic
172#2487—711no—
173#1824—712no—
174#1799—711no—
175#1289—712no—
176C71#946911yescharacteristic, complemented, Sylow
177C2×C4×D4C2×D7111yescharacteristic
178C22×C4⋊C4—711no—
179C8×D4—24no—
180#431—21no—
181C2×#99—1421no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8C2×D7112yes—
184#400—712no—
185#399—712no—
186#389—1421no—
187#386—1421no—
188#385—1421no—
189#380—711no—
190#380—712no—
191#380—1421no—
192#342—1421no—
193#340—1424no—
194C22×D4D28411yescharacteristic
195C23×C4—1421no—
196C23×C4C22⋊D7111yescharacteristic
197C23×C4C22×D7111yescharacteristic
198C22×C8—22no—
199C22×C8C22⋊D7114yes—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—712no—
203C2×C4⋊C4—1422no—
204C2×C4⋊C4—1421no—
205C2×C4⋊C4—1421no—
206C2×C4⋊C4—1422no—
207C2×C4⋊C4—1422no—
208C2×C4⋊C4—1422no—
209C2×C4⋊C4C4×D7111yescharacteristic
210C2×C22⋊C4—1422no—
211C2×C22⋊C4—1421no—
212C2×C22⋊C4—1422no—
213C2×C22⋊C4C4×D7111yescharacteristic
214C2×C22⋊C4D28411yescharacteristic
215C2×C42C22⋊D7111yescharacteristic
216C4⋊C8—22no—
217#99—2842no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C4×C8—22no—
222#95—712no—
223#95—1422no—
224#95—1422no—
225#95—1422no—
226C24(C2×D71)⋊C411yescharacteristic
227C24C2×D28411yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#1176585412yes—
231C2×D4Dic28412yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—2842no—
237C22×C4—2842no—
238C22×C4—5682no—
239C22×C4—1422no—
240C22×C4—1422no—
241C22×C4(C2×D71)⋊C411yescharacteristic
242C22×C4(C2×D71)⋊C411yescharacteristic
243C22×C4(C2×D71)⋊C411yescharacteristic
244C22×C4C2×C4×D7111yescharacteristic
245C22×C4C2×C22⋊D7112yes—
246C22×C4C2×C22⋊D7111yescharacteristic
247C22×C4#1176586512yes—
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—82no—
251C4⋊C4—22no—
252C4⋊C4—2842no—
253C4⋊C4—2844no—
254C4⋊C4—2841no—
255C4⋊C4—2841no—
256C4⋊C4—5682no—
257C4⋊C4—1424no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—2842no—
261C22⋊C4—2842no—
262C22⋊C4—2842no—
263C22⋊C4—5682no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1455605312yes—
273C23C2×(C2×D71)⋊C411yescharacteristic
274C23#1455610212yes—
275C23C2×#1176585411yescharacteristic
276C23C2×Dic28411yescharacteristic
277C23#1455615012yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—2842no—
293C2×C4—2842no—
294C2×C4—2842no—
295C2×C4—2842no—
296C2×C4—2842no—
297C2×C4—2842no—
298C2×C4—5684no—
299C2×C4—5682no—
300C2×C4—5682no—
301C2×C4—5682no—
302C2×C4#1455605312yes—
303C2×C4C4×C22⋊D7112yes—
304C2×C4#1455609711yescharacteristic
305C2×C4#1455611111yescharacteristic
306C8—44no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1798587811yescharacteristic
320C22C2×#1455605311yescharacteristic
321C22#1798599611yescharacteristic
322C22#1798622812yes—
323C22#1798624012yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—5682no—
330C4—5682no—
331C4—11362no—
332C4#1798618912yes—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#2258875311yescharacteristic
339C2#2258876311yescharacteristic
340C2#2258886711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2972C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2972D71
hypocenterC71#9469
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #22588431C2semidirect
4. #22586208C2semidirect
8. #17985938C4semidirect
9. #17985935C4semidirect
176. C71#9469semidirect

Maximal subgroups

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

subgroupindex
#225926572
#225884312
#225883282
#225862082
C71×#29722
#946971