Finite Group Databasebeta

← #31891013

Subgroups — #31891013

Order 18176.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#22588328C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586219C212yesmaximal, complemented
4#22586207C212yesmaximal, complemented
5C71×#2353C211yesmaximal, characteristic
6C2×#14556043—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985859C2212yes—
9#17985766—21no—
10#17985725—21no—
11#17985723—21no—
12C2×C4×D4×C71C2211yescharacteristic
13#388×C71—21no—
14#387×C71—21no—
15#386×C71—21no—
16#383×C71C411yescharacteristic, complemented
17#381×C71C411yescharacteristic, complemented
18#380×C71C2211yescharacteristic
19C4×C22⋊C4×C71—21no—
20C4×C22⋊C4×C71C411yescharacteristic, complemented
21#376×C71C411yescharacteristic, complemented
22C2×#95×C71C2211yescharacteristic
23C22×C71⋊C8—22no—
24C2×C4⋊Dic71—22no—
25C2×C22⋊Dic71—22no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C71⋊#95—22no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71—21no—
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71C2311yescharacteristic
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71D411yescharacteristic
43C2×C42×C71—21no—
44C2×C42×C71—21no—
45C2×C42×C71C2×C411yescharacteristic
46#95×C71—21no—
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71C2×C411yescharacteristic
52#95×C71C2×C411yescharacteristic
53#95×C71C2×C411yescharacteristic
54#95×C71C2×C411yescharacteristic
55C22×Dic71—42no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—82no—
58C4⋊Dic71—82no—
59C22⋊Dic71—42no—
60C22⋊Dic71—42no—
61C24×C71C22⋊C411yescharacteristic
62C24×C71C2×D411yescharacteristic
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—21no—
66C2×D4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71—81no—
76C22×C4×C71—81no—
77C22×C4×C71—81no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71#4112yes—
84C22×C4×C71#4112yes—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—81no—
87C22⋊C4×C71—22no—
88C22⋊C4×C71—22no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—81no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—42no—
96C42×C71—81no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71C2×C22⋊C411yescharacteristic
105C23×C71#12312yes—
106C23×C71#13512yes—
107C23×C71AGL(1,ℤ/8)11yescharacteristic
108C23×C71#13711yescharacteristic
109D4×C71—42no—
110D4×C71—42no—
111C2×C4×C71—22no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
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×C71—41no—
134C2×C4×C71—81no—
135C2×C4×C71—81no—
136C2×C4×C71—84no—
137C2×C4×C71#11711yescharacteristic
138C2×C4×C71#11711yescharacteristic
139C2×C4×C71#12312yes—
140C2×Dic71—82no—
141C2×Dic71—82no—
142C2×Dic71—82no—
143C71⋊C8—82no—
144C22×C71—22no—
145C22×C71—21no—
146C22×C71—22no—
147C22×C71—21no—
148C22×C71—42no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—41no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71#38111yescharacteristic
157C22×C71#41611yescharacteristic
158C22×C71#41711yescharacteristic
159C22×C71#48212yes—
160C22×C71#48512yes—
161C4×C71—22no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—42no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—41no—
168C4×C71—81no—
169C4×C71—81no—
170C4×C71—81no—
171Dic71—162no—
172#8763—711nomaximal, Sylow
173C2×C71—22no—
174C2×C71—21no—
175C2×C71—21no—
176C2×C71—22no—
177C2×C71—81no—
178C2×C71#182511yescharacteristic
179C2×C71#182711yescharacteristic
180C2×C71#182811yescharacteristic
181#2353D7111yescharacteristic
182#1799—711no—
183#1459—711no—
184#1294—712no—
185#1289—712no—
186C71#876311yescharacteristic, complemented, Sylow
187C2×C4×D4C2×D7111yescharacteristic
188C2×#99—1421no—
189C2×C22⋊C8—712no—
190#388—21no—
191#387—21no—
192#386—21no—
193#383Dic7111yescharacteristic, complemented
194#381Dic7111yescharacteristic, complemented
195#380—712no—
196#380C2×D7111yescharacteristic
197C4×C22⋊C4—21no—
198C4×C22⋊C4Dic7111yescharacteristic, complemented
199#376Dic7111yescharacteristic, complemented
200C2×#95C2×D7111yescharacteristic
201#342—1421no—
202#324—1421no—
203#323—1421no—
204C22×D4C22⋊D7111yescharacteristic
205C23×C4—21no—
206C23×C4C2×Dic7111yescharacteristic
207C23×C4C22×D7111yescharacteristic
208C22×C8—1422no—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1422no—
213C2×C4⋊C4C22⋊D7111yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1422no—
218C2×C22⋊C4C22⋊D7111yescharacteristic
219C2×C22⋊C4C22⋊D7111yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×Dic7111yescharacteristic
223#99—2842no—
224C22⋊C8—1422no—
225C22⋊C8—1422no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—1422no—
232#95C2×Dic7111yescharacteristic
233#95C2×Dic7111yescharacteristic
234#95C2×Dic7111yescharacteristic
235#95C2×Dic7111yescharacteristic
236C24C22⋊Dic7111yescharacteristic
237C24C2×C22⋊D7111yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—2842no—
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C22×Dic7111yescharacteristic
258C22×C4C2×C22⋊D7111yescharacteristic
259C22×C4#1176586512yes—
260C22×C4#1176586512yes—
261C2×C8—2842no—
262C2×C8—5682no—
263C4⋊C4—22no—
264C4⋊C4—81no—
265C4⋊C4—5682no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—2842no—
273C22⋊C4—2842no—
274C42—21no—
275C42—21no—
276C42—42no—
277C42—81no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊Dic7111yescharacteristic
286C23#1455611112yes—
287C23#1455614812yes—
288C23#1455615311yescharacteristic
289C23#1455615811yescharacteristic
290D4—42no—
291D4—42no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—84no—
318C2×C4—5682no—
319C2×C4—5682no—
320C2×C4—5682no—
321C2×C4#1455608711yescharacteristic
322C2×C4#1455608711yescharacteristic
323C2×C4#1455611112yes—
324C8—5682no—
325C22—21no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1798586211yescharacteristic
338C22#1798599511yescharacteristic
339C22#1798600011yescharacteristic
340C22#1798632312yes—
341C22#1798634012yes—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—11362no—
353C2—21no—
354C2—21no—
355C2—22no—
356C2—22no—
357C2—81no—
358C2#2258843511yescharacteristic
359C2#2258844411yescharacteristic
360C2#2258844811yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2353C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2353D71
hypocenterC71#8763
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #22586219C2semidirect
4. #22586207C2semidirect
16. #383×C71C4semidirect
17. #381×C71C4semidirect
20. C4×C22⋊C4×C71C4semidirect
21. #376×C71C4semidirect
186. C71#8763semidirect
193. #383Dic71semidirect
194. #381Dic71semidirect
198. C4×C22⋊C4Dic71semidirect
199. #376Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866582
#225862192
#225862072
C71×#23532
#876371