Finite Group Databasebeta

← #31888866

Subgroups — #31888866

Order 18176.

710 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 151 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#22589277C211yesmaximal, characteristic, complemented
2#22586727C212yesmaximal, complemented
3#22586722C212yesmaximal, complemented
4#22586664C211yesmaximal, characteristic, complemented
5C71×#3461C211yesmaximal, characteristic
6#17986052C2212yes—
7C2×C4⋊(C71⋊C8)C2211yescharacteristic
8C2×C4×C71⋊C8C2211yescharacteristic
9#17985754—22no—
10Q8⋊(C71⋊C8)—22no—
11Q8⋊(C71⋊C8)—22no—
12D4⋊(C71⋊C8)—22no—
13D4⋊(C71⋊C8)—22no—
14#17985725—22no—
15#544×C71C411yescharacteristic
16#542×C71—22no—
17#541×C71—22no—
18#539×C71—22no—
19#537×C71C411yescharacteristic
20#536×C71C412yes—
21#533×C71—22no—
22#529×C71—22no—
23#529×C71—22no—
24C4×#41×C71C2212yes—
25C2×C4×Q8×C71C2211yescharacteristic
26C22×C71⋊C8—21no—
27C22×C71⋊C8—21no—
28C4⋊(C71⋊C8)—22no—
29C4⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C4×C71⋊C8—22no—
32C4×C71⋊C8—22no—
33C2×#41×C71C2×C412yes—
34C22×Q8×C71C2×C411yescharacteristic
35C4⋊Q8×C71—22no—
36C4⋊Q8×C71—22no—
37#126×C71—24no—
38#126×C71—24no—
39#125×C71—22no—
40#125×C71—24no—
41#125×C71—22no—
42#124×C71—22no—
43#124×C71—22no—
44#123×C71—24no—
45#123×C71—24no—
46C22⋊Q8×C71—24no—
47C22⋊Q8×C71—22no—
48C22⋊Q8×C71—22no—
49C22⋊Q8×C71—22no—
50C22⋊Q8×C71—22no—
51#121×C71—22no—
52#121×C71—22no—
53C4×Q8×C71—22no—
54C4×Q8×C71—22no—
55C4×Q8×C71—22no—
56C4×Q8×C71D412yes—
57C4×Q8×C71D412yes—
58C4×D4×C71—22no—
59C4×D4×C71—22no—
60C4×D4×C71—22no—
61C4×D4×C71D412yes—
62C4×D4×C71D412yes—
63#117×C71—24no—
64#117×C71—22no—
65#117×C71C2×C412yes—
66C2×C4⋊C4×C71C2×C411yescharacteristic
67C2×C4⋊C4×C71D412yes—
68C2×C42×C71—22no—
69C2×C42×C71D412yes—
70C2×C42×C71C2311yescharacteristic
71C2×C71⋊C8—22no—
72C2×C71⋊C8—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8—42no—
75C2×C71⋊C8—42no—
76C2×C71⋊C8—41no—
77#41×C71—28no—
78C2×Q8×C71—22no—
79C2×Q8×C71—22no—
80C2×Q8×C71—24no—
81C2×Q8×C71C22⋊C412yes—
82C2×D4×C71—22no—
83C2×D4×C71—22no—
84C2×D4×C71C22⋊C412yes—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—22no—
88C22×C4×C71C22⋊C412yes—
89C22×C4×C71C22⋊C412yes—
90C22×C4×C71C22×C411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C4⋊C4×C71—24no—
94C4⋊C4×C71—24no—
95C4⋊C4×C71—22no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—24no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71—24no—
100C4⋊C4×C71—22no—
101C4⋊C4×C71—22no—
102C4⋊C4×C71C22⋊C412yes—
103C4⋊C4×C71C22⋊C412yes—
104C22⋊C4×C71—24no—
105C22⋊C4×C71—24no—
106C22⋊C4×C71—24no—
107C22⋊C4×C71—22no—
108C22⋊C4×C71—22no—
109C22⋊C4×C71—22no—
110C42×C71—22no—
111C42×C71—22no—
112C42×C71—22no—
113C42×C71—22no—
114C42×C71—22no—
115C42×C71C2×D412yes—
116C42×C71C2×D412yes—
117C23×C71—22no—
118C23×C71C2×C22⋊C411yescharacteristic
119Q8×C71—24no—
120Q8×C71—44no—
121Q8×C71—82no—
122D4×C71—24no—
123D4×C71—44no—
124D4×C71—82no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—22no—
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—24no—
133C2×C4×C71—41no—
134C2×C4×C71—44no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—82no—
139C2×C4×C71C2×C22⋊C412yes—
140C2×C4×C71#12011yescharacteristic
141C2×C4×C71#12012yes—
142C2×C4×C71#12011yescharacteristic
143C2×C4×C71#13114yes—
144C71⋊C8—42no—
145C71⋊C8—82no—
146C71⋊C8—81no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—42no—
150C22×C71—42no—
151C22×C71—42no—
152C22×C71#37911yescharacteristic
153C22×C71#40812yes—
154C4×C71—21no—
155C4×C71—22no—
156C4×C71—42no—
157C4×C71—42no—
158C4×C71—42no—
159C4×C71—42no—
160C4×C71—42no—
161C4×C71#40814yes—
162C4×C71#45411yescharacteristic
163C4×C71#45611yescharacteristic
164#8321—711nomaximal, Sylow
165C2×C71—22no—
166C2×C71—82no—
167C2×C71#178011yescharacteristic
168C2×C71#179311yescharacteristic
169C2×C71#179411yescharacteristic
170#3461D7111yescharacteristic
171#1984—711no—
172#1477—712no—
173#1476—712no—
174#1461—711no—
175C71#832111yescharacteristic, complemented, Sylow
176#544Dic7111yescharacteristic
177#542—22no—
178#541—22no—
179#539—22no—
180#537Dic7111yescharacteristic
181#536Dic7112yes—
182#533—22no—
183#529—22no—
184#529—22no—
185C4×#41C2×D7112yes—
186C2×C4×Q8C2×D7111yescharacteristic
187#431—712no—
188C2×C4⋊C8—711no—
189C2×C4×C8—711no—
190#336—1422no—
191Q8⋊C8—1422no—
192Q8⋊C8—1422no—
193D4⋊C8—1422no—
194D4⋊C8—1422no—
195#324—1422no—
196C2×#41C2×Dic7112yes—
197C22×Q8C2×Dic7111yescharacteristic
198C22×C8—1421no—
199C22×C8—1421no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202#126—24no—
203#126—24no—
204#125—22no—
205#125—24no—
206#125—22no—
207#124—22no—
208#124—22no—
209#123—24no—
210#123—24no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216#121—22no—
217#121—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8C22⋊D7112yes—
222C4×Q8C22⋊D7112yes—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4C22⋊D7112yes—
227C4×D4C22⋊D7112yes—
228#117—24no—
229#117—22no—
230#117C2×Dic7112yes—
231C2×C4⋊C4C2×Dic7111yescharacteristic
232C2×C4⋊C4C22⋊D7112yes—
233C2×C42—22no—
234C2×C42C22⋊D7112yes—
235C2×C42C22×D7111yescharacteristic
236C4⋊C8—1422no—
237C4⋊C8—1422no—
238C22⋊C8—1422no—
239C4×C8—1422no—
240C4×C8—1422no—
241#41—28no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8C22⋊Dic7112yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4C22⋊Dic7112yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4C22⋊Dic7112yes—
253C22×C4C22⋊Dic7112yes—
254C22×C4C22×Dic7111yescharacteristic
255C22×C4C2×C22⋊D7111yescharacteristic
256C22×C4C2×C22⋊D7111yescharacteristic
257C2×C8—2841no—
258C2×C8—2841no—
259C2×C8—2842no—
260C2×C8—2842no—
261C2×C8—2841no—
262C2×C8—1422no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4C22⋊Dic7112yes—
273C4⋊C4C22⋊Dic7112yes—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42C2×C22⋊D7112yes—
286C42C2×C22⋊D7112yes—
287C23—22no—
288C23C2×C22⋊Dic7111yescharacteristic
289Q8—24no—
290Q8—44no—
291Q8—82no—
292D4—24no—
293D4—44no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4C2×C22⋊Dic7112yes—
310C2×C4#1455609711yescharacteristic
311C2×C4#1455609712yes—
312C2×C4#1455609711yescharacteristic
313C2×C4#1455613514yes—
314C8—2842no—
315C8—5682no—
316C8—5681no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22#1798585511yescharacteristic
323C22#1798596612yes—
324C4—21no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4#1798596614yes—
332C4#1798617211yescharacteristic
333C4#1798618211yescharacteristic
334C2—22no—
335C2—82no—
336C2#2258825211yescharacteristic
337C2#2258829911yescharacteristic
338C2#2258830411yescharacteristic
subgroupG/H
centerC2×C4#14556097
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3461C2
socleC22×C71#379
prime radical#537×C71C4
cosocleC2×C42×C71C23
hypercenter#3461D71
hypocenterC71#8321
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. #22589277C2semidirect
2. #22586727C2semidirect
3. #22586722C2semidirect
4. #22586664C2semidirect
175. C71#8321semidirect

Maximal subgroups

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

subgroupindex
#225892772
#225867272
#225867222
#225866642
C71×#34612
#832171