Finite Group Databasebeta

← #31907654

Subgroups — #31907654

Order 18176.

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

normal subgroups only

This group has 536 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#22588079C211yesmaximal, characteristic, complemented
2#22586654C211yesmaximal, characteristic, complemented
3#22586346C212yesmaximal, complemented
4#22586343C212yesmaximal, complemented
5C71×#2392C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17986052—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C4×D4×C71—21no—
13C2×C4×D4×C71C411yescharacteristic
14C2×#117×C71C2211yescharacteristic
15#398×C71C2211yescharacteristic
16#397×C71C411yescharacteristic
17#396×C71—21no—
18#394×C71—21no—
19#394×C71C2211yescharacteristic
20#392×C71C411yescharacteristic
21C4×C4⋊C4×C71C411yescharacteristic
22C4×C22⋊C4×C71—21no—
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C22×C71⋊C8—41no—
28C4⋊(C71⋊C8)—41no—
29C22⋊(C71⋊C8)—22no—
30C4×C71⋊C8—41no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71D411yescharacteristic
35C4×D4×C71—41no—
36C4×D4×C71—41no—
37C4×D4×C71—42no—
38C4×D4×C71—42no—
39#117×C71—41no—
40#117×C71—41no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71C2311yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71C2×C411yescharacteristic
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C42×C71—21no—
54C2×C42×C71C2×C411yescharacteristic
55C2×C42×C71D411yescharacteristic
56#95×C71—21no—
57#95×C71—21no—
58#95×C71C2×C411yescharacteristic
59#95×C71C2×C411yescharacteristic
60#11765849—24no—
61#11765849—44no—
62#11765849—42no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—44no—
65C2×C71⋊C8—42no—
66C24×C71—21no—
67C24×C71C22⋊C411yescharacteristic
68C2×D4×C71—41no—
69C2×D4×C71—42no—
70C2×D4×C71—42no—
71C2×D4×C71—42no—
72C2×D4×C71—42no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—22no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—42no—
82C22×C4×C71—42no—
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71D812yes—
87C22×C4×C71SD1612yes—
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—42no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—42no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—41no—
105C42×C71—21no—
106C42×C71—21no—
107C42×C71—21no—
108C42×C71—21no—
109C42×C71—42no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—22no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71#10212yes—
122C23×C71C2×C22⋊C411yescharacteristic
123D4×C71—42no—
124D4×C71—48no—
125D4×C71—42no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—22no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—82no—
150C2×C4×C71—82no—
151C2×C4×C71#9911yescharacteristic
152C2×C4×C71#9911yescharacteristic
153C2×C4×C71#10212yes—
154C2×C4×C71#10412yes—
155C2×C4×C71C2×D811yescharacteristic
156C2×C4×C71C2×SD1611yescharacteristic
157C71⋊C8—44no—
158C71⋊C8—82no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—42no—
163C22×C71—42no—
164C22×C71—42no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—42no—
168C22×C71—81no—
169C22×C71—81no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71—81no—
173C22×C71C2×#9911yescharacteristic
174C22×C71C2×#10211yescharacteristic
175C22×C71C2×#10411yescharacteristic
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—42no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—81no—
184C4×C71—81no—
185C4×C71#32712yes—
186C4×C71#32712yes—
187#11697—711nomaximal, Sylow
188C2×C71—21no—
189C2×C71—21no—
190C2×C71—42no—
191C2×C71—82no—
192C2×C71C2×#32711yescharacteristic
193C2×C71#212011yescharacteristic
194C2×C71#213211yescharacteristic
195#2392D7111yescharacteristic
196#1742—711no—
197#1458—711no—
198#1346—712no—
199#1345—712no—
200C71#1169711yescharacteristic, complemented, Sylow
201C22×M16—1421no—
202C2×C4×D4—21no—
203C2×C4×D4Dic7111yescharacteristic
204C2×#117C2×D7111yescharacteristic
205#431—1421no—
206#398C2×D7111yescharacteristic
207#397Dic7111yescharacteristic
208#396—21no—
209#394—21no—
210#394C2×D7111yescharacteristic
211#392Dic7111yescharacteristic
212C4×C4⋊C4Dic7111yescharacteristic
213C4×C22⋊C4—21no—
214#343—712no—
215#343—1421no—
216#324—712no—
217#323—1421no—
218C22×D4—21no—
219C22×D4C2×Dic7111yescharacteristic
220C23×C4—21no—
221C23×C4C22⋊D7111yescharacteristic
222C2×M16—2841no—
223C2×M16—2842no—
224C2×M16—1424no—
225C2×M16—1422no—
226C22×C8—2841no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—42no—
230C4×D4—42no—
231#117—41no—
232#117—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C2×Dic7111yescharacteristic
236C2×C4⋊C4C22⋊D7111yescharacteristic
237C2×C4⋊C4C22×D7111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4C2×Dic7111yescharacteristic
244C2×C22⋊C4C22⋊D7111yescharacteristic
245C2×C42—21no—
246C2×C42C2×Dic7111yescharacteristic
247C2×C42C22⋊D7111yescharacteristic
248C4⋊C8—2841no—
249C22⋊C8—1422no—
250C4×C8—2841no—
251#95—21no—
252#95—21no—
253#95C2×Dic7111yescharacteristic
254#95C2×Dic7111yescharacteristic
255C24—21no—
256C24C22⋊Dic7111yescharacteristic
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C22⋊Dic7111yescharacteristic
273C22×C4C22⋊Dic7111yescharacteristic
274C22×C4C22⋊Dic7111yescharacteristic
275C22×C4#1176585012yes—
276C22×C4#1176585312yes—
277C22×C4C22×Dic7111yescharacteristic
278C22×C4C2×C22⋊D7111yescharacteristic
279C22×C4C2×C22⋊D7111yescharacteristic
280M16—2844no—
281M16—2842no—
282M16—1424no—
283C2×C8—2842no—
284C2×C8—2844no—
285C2×C8—2842no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—42no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23#1455604912yes—
317C23C2×C22⋊Dic7111yescharacteristic
318D4—42no—
319D4—48no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#1455604311yescharacteristic
347C2×C4#1455604311yescharacteristic
348C2×C4#1455604912yes—
349C2×C4#1455605512yes—
350C2×C4C2×#1176585011yescharacteristic
351C2×C4C2×#1176585311yescharacteristic
352C8—2844no—
353C8—5682no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#1455604311yescharacteristic
369C22C2×#1455604911yescharacteristic
370C22C2×#1455605511yescharacteristic
371C4—21no—
372C4—21no—
373C4—42no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4#1798573312yes—
381C4#1798573312yes—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—82no—
386C2C2×#1798573311yescharacteristic
387C2#2258998111yescharacteristic
388C2#2259002511yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2392C2
socleC22×C71C2×#99
prime radical#392×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2392D71
hypocenterC71#11697
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #22588079C2semidirect
2. #22586654C2semidirect
3. #22586346C2semidirect
4. #22586343C2semidirect
200. C71#11697semidirect

Maximal subgroups

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

subgroupindex
#225880792
#225866542
#225863462
#225863432
C71×#23922
#1169771