Finite Group Databasebeta

← #31892522

Subgroups — #31892522

Order 18176.

534 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 111 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#22597369C211yesmaximal, characteristic
2C4×#14556045C211yesmaximal, characteristic
3#22586787C212yesmaximal, complemented
4#22586233C212yesmaximal, complemented
5C71×#2398C211yesmaximal, characteristic
6#17986071—24no—
7#17986056C2212yes—
8C2×#14556045—21no—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)—21no—
11C4×#11765849C2212yes—
12#17985768—21no—
13Q8⋊(C71⋊C8)—24no—
14C2×C4×Q8×C71C2211yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16#400×C71—21no—
17#397×C71C411yescharacteristic
18#397×C71C411yescharacteristic
19#394×C71—21no—
20#393×C71C411yescharacteristic
21#393×C71C411yescharacteristic
22C4×C22⋊C4×C71—21no—
23C4×C22⋊C4×C71—21no—
24#376×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×#11765849—22no—
27C22×C71⋊C8—22no—
28C4⋊(C71⋊C8)D414yes—
29#14556045—24no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33#14556039—22no—
34#14556039—22no—
35C4×C71⋊C8—22no—
36C22×Q8×C71C2×C411yescharacteristic
37C22×D4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C4×Q8×C71—22no—
41C4×Q8×C71—22no—
42C4×D4×C71—22no—
43C4×D4×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
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×C71C2×C411yescharacteristic
53C2×C42×C71—21no—
54C2×C42×C71—21no—
55C2×C42×C71C2311yescharacteristic
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71C2×C411yescharacteristic
62#95×C71C2×C411yescharacteristic
63#11765849—24no—
64#11765849—42no—
65#11765849—82no—
66C2×C71⋊C8—24no—
67C2×C71⋊C8—42no—
68C2×C71⋊C8—82no—
69C24×C71C22⋊C411yescharacteristic
70C24×C71C22⋊C411yescharacteristic
71C2×Q8×C71—22no—
72C2×Q8×C71—42no—
73C2×D4×C71—22no—
74C2×D4×C71—22no—
75C2×D4×C71—41no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—42no—
85C22×C4×C71—42no—
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—42no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—42no—
99C22⋊C4×C71—42no—
100C42×C71—42no—
101C42×C71—41no—
102C42×C71—41no—
103C42×C71C2×D412yes—
104C42×C71#4112yes—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71#10412yes—
115C23×C71C2×C22⋊C411yescharacteristic
116Q8×C71—24no—
117Q8×C71—42no—
118D4×C71—44no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—22no—
122C2×C4×C71—22no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—81no—
142C2×C4×C71—81no—
143C2×C4×C71C4×D412yes—
144C2×C4×C71#12011yescharacteristic
145C2×C4×C71#12112yes—
146C2×C4×C71#12311yescharacteristic
147C71⋊C8—44no—
148C71⋊C8—44no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—22no—
152C22×C71—42no—
153C22×C71—42no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—81no—
160C22×C71—81no—
161C22×C71#38611yescharacteristic
162C22×C71C2×#10411yescharacteristic
163C22×C71#42111yescharacteristic
164C4×C71—22no—
165C4×C71—21no—
166C4×C71—22no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—42no—
170C4×C71—81no—
171C4×C71—81no—
172C4×C71—81no—
173C4×C71#45512yes—
174#9029—711nomaximal, Sylow
175C2×C71—21no—
176C2×C71—21no—
177C2×C71—42no—
178C2×C71—81no—
179C2×C71#187111yescharacteristic
180C2×C71#188511yescharacteristic
181C2×C71#189111yescharacteristic
182#2973—711no—
183#2398D7111yescharacteristic
184C4×#100—711no—
185#1491—712no—
186#1300—712no—
187C71#902911yescharacteristic, complemented, Sylow
188C2×C4×Q8C2×D7111yescharacteristic
189C2×C4×D4C2×D7111yescharacteristic
190#435—1424no—
191#432—712no—
192C2×#100—1421no—
193#407—1421no—
194C2×C22⋊C8—1421no—
195C4×M16—712no—
196#400—21no—
197#397Dic7111yescharacteristic
198#397Dic7111yescharacteristic
199#394—21no—
200#393Dic7111yescharacteristic
201#393Dic7111yescharacteristic
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204#376C2×D7111yescharacteristic
205#343—1421no—
206Q8⋊C8—1424no—
207C22×Q8C2×Dic7111yescharacteristic
208C22×D4C22⋊D7111yescharacteristic
209C23×C4C22⋊D7111yescharacteristic
210C23×C4C22⋊D7111yescharacteristic
211C2×M16—1422no—
212C2×M16—1422no—
213C22×C8—1422no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×D4—22no—
217C4×D4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C2×Dic7111yescharacteristic
220C2×C4⋊C4C22⋊D7111yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic7111yescharacteristic
226C2×C22⋊C4C2×Dic7111yescharacteristic
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22×D7111yescharacteristic
230C4⋊C8—714no—
231#100—1424no—
232C22⋊C8—1422no—
233C22⋊C8—1422no—
234C22⋊C8—1422no—
235#97—1422no—
236#97—1422no—
237C4×C8—1422no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C2×Dic7111yescharacteristic
244#95C2×Dic7111yescharacteristic
245C24C22⋊Dic7111yescharacteristic
246C24C22⋊Dic7111yescharacteristic
247C2×Q8—22no—
248C2×Q8—42no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C22⋊Dic7111yescharacteristic
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22×Dic7111yescharacteristic
265C22×C4C2×C22⋊D7111yescharacteristic
266C22×C4C2×C22⋊D7111yescharacteristic
267M16—2842no—
268M16—5682no—
269M16—1424no—
270C2×C8—2842no—
271C2×C8—5682no—
272C2×C8—1424no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—42no—
283C42—41no—
284C42—41no—
285C42D4×D7112yes—
286C42#1176586412yes—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#1455605512yes—
297C23C2×C22⋊Dic7111yescharacteristic
298Q8—24no—
299Q8—42no—
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4D4×Dic7112yes—
326C2×C4#1455609611yescharacteristic
327C2×C4#1455610012yes—
328C2×C4#1455610911yescharacteristic
329C8—2844no—
330C8—2844no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—81no—
343C22#1798587911yescharacteristic
344C22C2×#1455605511yescharacteristic
345C22#1798601611yescharacteristic
346C4—21no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4#1798617612yes—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—81no—
360C2#2258864311yescharacteristic
361C2#2258872911yescharacteristic
362C2#2258875911yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2398C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2398D71
hypocenterC71#9029
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #22586787C2semidirect
4. #22586233C2semidirect
187. C71#9029semidirect

Maximal subgroups

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

subgroupindex
#225973692
C4×#145560452
#225867872
#225862332
C71×#23982
#902971