Finite Group Databasebeta

← #31908043

Subgroups — #31908043

Order 18176.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14556047C211yesmaximal, characteristic, complemented
2#22586684C211yesmaximal, characteristic, complemented
3#22586343C214yesmaximal, complemented
4C71×#2548C211yesmaximal, characteristic
5#17986056—21no—
6C2×#14556047—21no—
7C4×#11765849—21no—
8#17985768C2212yes—
9#17985725—21no—
10#17985725C2212yes—
11C2×#127×C71C411yescharacteristic
12C2×#124×C71—21no—
13C2×C4×Q8×C71C2211yescharacteristic
14#394×C71C412yes—
15#394×C71C2212yes—
16#390×C71—21no—
17#390×C71—21no—
18#390×C71—21no—
19C4×C4⋊C4×C71C411yescharacteristic
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—41no—
22#14556047—24no—
23C22⋊(C71⋊C8)—22no—
24#14556039—41no—
25#14556039—41no—
26C4×C71⋊C8—41no—
27C22×Q8×C71D411yescharacteristic
28C22×D4×C71—21no—
29C22×D4×C71—21no—
30C22×D4×C71C2×C412yes—
31#127×C71—41no—
32#127×C71—41no—
33#124×C71—44no—
34C4×Q8×C71—41no—
35C4×Q8×C71—41no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71C2311yescharacteristic
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—22no—
44C2×C42×C71—21no—
45C2×C42×C71C2×C411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48#95×C71C2×C412yes—
49#11765849—24no—
50#11765849—42no—
51C2×C71⋊C8—42no—
52C2×C71⋊C8—42no—
53C24×C71—22no—
54C2×Q8×C71—22no—
55C2×Q8×C71—22no—
56C2×Q8×C71—41no—
57C2×D4×C71—42no—
58C2×D4×C71—42no—
59C2×D4×C71—42no—
60C2×D4×C71—42no—
61C2×D4×C71—42no—
62C2×D4×C71—41no—
63C2×D4×C71—41no—
64C2×D4×C71—42no—
65C2×D4×C71—82no—
66C22×C4×C71—22no—
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C4⋊C4×C71—41no—
77C4⋊C4×C71—42no—
78C22⋊C4×C71—44no—
79C22⋊C4×C71—44no—
80C22⋊C4×C71—82no—
81C22⋊C4×C71—82no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71—21no—
87C42×C71—21no—
88C42×C71—42no—
89C23×C71—42no—
90C23×C71—42no—
91C23×C71—42no—
92C23×C71—42no—
93C23×C71—42no—
94C23×C71—42no—
95C23×C71—42no—
96C23×C71C2×C22⋊C411yescharacteristic
97Q8×C71—44no—
98D4×C71—82no—
99D4×C71—84no—
100D4×C71—84no—
101D4×C71—82no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—22no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71#9911yescharacteristic
120C2×C4×C71#9911yescharacteristic
121C2×C4×C71#10412yes—
122C2×C4×C71AGL(1,ℤ/8)12yes—
123C71⋊C8—44no—
124C71⋊C8—82no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—82no—
128C22×C71—82no—
129C22×C71—82no—
130C22×C71—82no—
131C22×C71—82no—
132C22×C71—82no—
133C22×C71—82no—
134C22×C71C2×#9911yescharacteristic
135C22×C71#41811yescharacteristic
136C22×C71C2×#10411yescharacteristic
137C4×C71—21no—
138C4×C71—22no—
139C4×C71—22no—
140C4×C71—41no—
141C4×C71—41no—
142C4×C71—41no—
143C4×C71—41no—
144C4×C71—82no—
145C4×C71#35512yes—
146#11781—711nomaximal, Sylow
147C2×C71—21no—
148C2×C71—21no—
149C2×C71—162no—
150C2×C71#150711yescharacteristic
151C2×C71#212211yescharacteristic
152C2×C71C2×#35511yescharacteristic
153#2548D7111yescharacteristic
154C4×#101—711no—
155#1467—711no—
156#1345—714no—
157C71#1178111yescharacteristic, complemented, Sylow
158C2×#127Dic7111yescharacteristic
159C2×#124—21no—
160C2×C4×Q8C2×D7111yescharacteristic
161#432—1421no—
162C2×#101—1421no—
163C4×M16—1421no—
164#394Dic7112yes—
165#394C2×D7112yes—
166#390—21no—
167#390—21no—
168#390—21no—
169C4×C4⋊C4Dic7111yescharacteristic
170#343—712no—
171#324—712no—
172#324—1421no—
173C22×Q8C22⋊D7111yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×Dic7112yes—
177C2×M16—1422no—
178#127—41no—
179#127—41no—
180#124—44no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×Dic7111yescharacteristic
185C2×C4⋊C4C22⋊D7111yescharacteristic
186C2×C4⋊C4C22×D7111yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—21no—
192C2×C42C2×Dic7111yescharacteristic
193C2×C42C22⋊D7111yescharacteristic
194C2×C42C22⋊D7111yescharacteristic
195C4⋊C8—2841no—
196#101—1424no—
197C22⋊C8—1422no—
198#97—2841no—
199#97—2841no—
200C4×C8—2841no—
201#95C2×Dic7112yes—
202C24—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C22×C4—22no—
216C22×C4C22⋊Dic7111yescharacteristic
217C22×C4C22⋊Dic7111yescharacteristic
218C22×C4C22⋊Dic7111yescharacteristic
219C22×C4C22⋊Dic7111yescharacteristic
220C22×C4C22×Dic7111yescharacteristic
221C22×C4C2×C22⋊D7111yescharacteristic
222C22×C4C2×C22⋊D7111yescharacteristic
223M16—2842no—
224M16—1424no—
225C2×C8—2842no—
226C2×C8—2842no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊Dic7111yescharacteristic
250Q8—44no—
251D4—82no—
252D4—84no—
253D4—84no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#1455604311yescharacteristic
273C2×C4#1455604311yescharacteristic
274C2×C4#1455605512yes—
275C2×C4#1455615412yes—
276C8—2844no—
277C8—5682no—
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#1455604311yescharacteristic
288C22#1798600411yescharacteristic
289C22C2×#1455605511yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4#1798579612yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2#2258685611yescharacteristic
303C2#2258999011yescharacteristic
304C2C2×#1798579611yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2548C2
socleC22×C71C2×#99
prime radicalC2×#127×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2548D71
hypocenterC71#11781
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C4×#14556047C2semidirect
2. #22586684C2semidirect
3. #22586343C2semidirect
157. C71#11781semidirect

Maximal subgroups

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

subgroupindex
C4×#145560472
#225866842
#225863432
C71×#25482
#1178171