Finite Group Databasebeta

← #31893210

Subgroups — #31893210

Order 18176.

630 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 143 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#22596679C211yesmaximal, characteristic
2C2×#17985865C211yesmaximal, characteristic
3C2×#17985737C212yesmaximal
4#22586220C212yesmaximal
5C71×#1817C211yesmaximal, characteristic
6C2×#14556107C2211yescharacteristic
7C22×C4⋊Dic71—21no—
8#17985905C412yescomplemented
9#17985897C412yescomplemented
10#17985883—22no—
11#17985872—21no—
12#17985865—24no—
13#17985862—21no—
14#17985859C2211yescharacteristic
15C2×C71⋊#95—21no—
16C2×C71⋊#95C2212yes—
17#17985737—24no—
18C22×C42×C71C2211yescharacteristic
19C2×C22⋊C8×C71C2212yes—
20#336×C71—22no—
21C22×Dic142C2×C411yescharacteristic
22C23×Dic71—22no—
23#14556107—22no—
24#14556107—22no—
25C2×C4⋊Dic71—24no—
26C2×C4⋊Dic71—42no—
27C2×C4⋊Dic71—42no—
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×C4⋊Dic71C2×C412yes—
30C2×Dic71⋊C4—22no—
31C2×Dic71⋊C4—21no—
32C2×C22⋊Dic71—21no—
33C2×C22⋊Dic71—21no—
34C71⋊#95—24no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—22no—
38C71⋊#95—22no—
39C71⋊#95—21no—
40C71⋊#95D414yes—
41C23×C4×C71D411yescharacteristic
42C23×C4×C71D411yescharacteristic
43C23×C4×C71C2311yescharacteristic
44C22×C8×C71—22no—
45C2×C42×C71—22no—
46C2×C42×C71—22no—
47C2×C42×C71C2×C412yes—
48C2×C42×C71D412yes—
49C22⋊C8×C71—22no—
50C22⋊C8×C71—22no—
51C2×Dic142—22no—
52C2×Dic142—42no—
53C22×Dic71—21no—
54C22×Dic71—22no—
55C22×Dic71—21no—
56C22×Dic71—24no—
57C22×Dic71—44no—
58C22×Dic71—44no—
59C22×Dic71—42no—
60C22×Dic71—42no—
61C4⋊Dic71—22no—
62C4⋊Dic71—48no—
63C4⋊Dic71—42no—
64Dic71⋊C4—44no—
65Dic71⋊C4—42no—
66C22⋊Dic71—41no—
67C22⋊Dic71—41no—
68C22⋊Dic71—41no—
69C22⋊Dic71—41no—
70C24×C71C2×D411yescharacteristic
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71C22⋊C412yes—
80C22×C4×C71C22⋊C412yes—
81C22×C4×C71SD1612yes—
82C22×C4×C71Q1612yes—
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4112yes—
87C2×C8×C71—44no—
88C2×C8×C71—42no—
89C42×C71—24no—
90C42×C71—24no—
91C23×C71—22no—
92C23×C71—22no—
93C23×C71C2×C22⋊C411yescharacteristic
94C23×C71#12011yescharacteristic
95C23×C71#12311yescharacteristic
96C23×C71C2×SD1611yescharacteristic
97C23×C71C2×Q1611yescharacteristic
98C23×C71AGL(1,ℤ/8)11yescharacteristic
99C23×C71#13711yescharacteristic
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71#10314yes—
116C2×C4×C71C4×D412yes—
117C2×C4×C71#12112yes—
118C2×Dic71—44no—
119C2×Dic71—44no—
120C2×Dic71—41no—
121C2×Dic71—41no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—44no—
128C2×Dic71—84no—
129C2×Dic71—84no—
130Dic142—82no—
131C8×C71—82no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—22no—
137C22×C71—22no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71#38611yescharacteristic
141C22×C71#41012yes—
142C22×C71C2×#10311yescharacteristic
143C22×C71#41711yescharacteristic
144C22×C71#42112yes—
145C22×C71#45011yescharacteristic
146C22×C71#45111yescharacteristic
147C22×C71#48311yescharacteristic
148C22×C71#48411yescharacteristic
149C4×C71—24no—
150C4×C71—22no—
151C4×C71—22no—
152C4×C71—42no—
153Dic71—84no—
154Dic71—81no—
155Dic71—81no—
156Dic71—82no—
157#9152—711nomaximal, Sylow
158C2×C71—21no—
159C2×C71—21no—
160C2×C71—21no—
161C2×C71—21no—
162C2×C71#151212yes—
163C2×C71#151812yes—
164C2×C71#187211yescharacteristic
165C2×C71#189711yescharacteristic
166C2×C71#189911yescharacteristic
167#2855—711no—
168C2×#382—711no—
169#1817D7111yescharacteristic
170C2×#328—712no—
171#1294—712no—
172C71#915211yescharacteristic, complemented, Sylow
173C2×C22⋊Q8—711no—
174C22×C4⋊C4—1421no—
175C22×C42C2×D7111yescharacteristic
176C2×C22⋊C8C2×D7112yes—
177#391—712no—
178#389—712no—
179#387—1422no—
180#385—1421no—
181#382—1424no—
182#381—1421no—
183#380—711no—
184C2×#95—712no—
185C2×#95—1421no—
186#336—22no—
187#328—1424no—
188C22×Q8—711no—
189C23×C4—1422no—
190C23×C4D28411yescharacteristic
191C23×C4D28411yescharacteristic
192C23×C4C22×D7111yescharacteristic
193C22×C8—22no—
194C22⋊Q8—1422no—
195C22⋊Q8—1422no—
196C2×C4⋊C4—2842no—
197C2×C4⋊C4—2842no—
198C2×C4⋊C4—712no—
199C2×C4⋊C4—711no—
200C2×C4⋊C4—1424no—
201C2×C4⋊C4—1422no—
202C2×C4⋊C4—1421no—
203C2×C22⋊C4—1421no—
204C2×C22⋊C4—1421no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D7112yes—
208C2×C42C22⋊D7112yes—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—714no—
212#95—1424no—
213#95—1422no—
214#95—1422no—
215#95—1422no—
216#95—1422no—
217#95—1421no—
218C24C2×D28411yescharacteristic
219C2×Q8—2842no—
220C2×Q8—1422no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—2844no—
230C22×C4—2844no—
231C22×C4—2842no—
232C22×C4—2842no—
233C22×C4—1422no—
234C22×C4—1421no—
235C22×C4—1421no—
236C22×C4—1424no—
237C22×C4(C2×D71)⋊C412yes—
238C22×C4(C2×D71)⋊C412yes—
239C22×C4#1176585412yes—
240C22×C4Dic28412yes—
241C22×C4C2×C4×D7111yescharacteristic
242C22×C4D4×D7112yes—
243C22×C4C2×C22⋊D7111yescharacteristic
244C22×C4#1176586412yes—
245C2×C8—44no—
246C2×C8—42no—
247C4⋊C4—2842no—
248C4⋊C4—2844no—
249C4⋊C4—2848no—
250C4⋊C4—2842no—
251C4⋊C4—1422no—
252C22⋊C4—2841no—
253C22⋊C4—2841no—
254C22⋊C4—2841no—
255C22⋊C4—2841no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×(C2×D71)⋊C411yescharacteristic
261C23#1455609811yescharacteristic
262C23#1455611211yescharacteristic
263C23C2×#1176585411yescharacteristic
264C23C2×Dic28411yescharacteristic
265C23#1455615611yescharacteristic
266C23#1455616111yescharacteristic
267Q8—5682no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—2844no—
284C2×C4—2844no—
285C2×C4—2841no—
286C2×C4—2841no—
287C2×C4—2844no—
288C2×C4—2842no—
289C2×C4—2842no—
290C2×C4—2842no—
291C2×C4—2842no—
292C2×C4—2842no—
293C2×C4—5684no—
294C2×C4—5684no—
295C2×C4#1455605314yes—
296C2×C4(C22⋊D71)⋊C412yes—
297C2×C4#1455610012yes—
298C8—82no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22#1798587611yescharacteristic
308C22#1798597212yes—
309C22C2×#1455605311yescharacteristic
310C22#1798600111yescharacteristic
311C22#1798601512yes—
312C22#1798615211yescharacteristic
313C22#1798615711yescharacteristic
314C22#1798633011yescharacteristic
315C22#1798633511yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—5684no—
321C4—5681no—
322C4—5681no—
323C4—5682no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2258687912yes—
329C2#2258690512yes—
330C2#2258864511yescharacteristic
331C2#2258878611yescharacteristic
332C2#2258879611yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1817C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#1817D71
hypocenterC71#9152
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #17985905C4semidirect
9. #17985897C4semidirect
172. C71#9152semidirect

Maximal subgroups

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

subgroupindex
#225966792
C2×#179858652
C2×#179857372
#225862202
C71×#18172
#915271