Finite Group Databasebeta

← #31907771

Subgroups — #31907771

Order 18176.

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

normal subgroups only

This group has 432 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
2#22586654C211yesmaximal, characteristic
3#22586370C214yesmaximal, complemented
4C71×#2628C211yesmaximal, characteristic
5C22×#11765849—21no—
6#17986052—21no—
7#17985768—21no—
8#17985768C2212yes—
9#17985725C2212yes—
10#17985723—21no—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×#117×C71C2211yescharacteristic
13#400×C71—21no—
14#399×C71—21no—
15#395×C71C2212yes—
16#393×C71C412yes—
17#389×C71—21no—
18#382×C71C411yescharacteristic
19#380×C71—21no—
20C2×#11765849—24no—
21C2×#11765849—22no—
22C2×#11765849—41no—
23C2×#11765849—42no—
24C22×C71⋊C8—41no—
25C4⋊(C71⋊C8)—41no—
26C22⋊(C71⋊C8)—22no—
27C4×C71⋊C8—41no—
28C22×Q8×C71C2×C412yes—
29C23×C4×C71D411yescharacteristic
30C4⋊Q8×C71—41no—
31C4⋊Q8×C71—41no—
32#117×C71—41no—
33#117×C71—41no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71C2311yescharacteristic
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71D411yescharacteristic
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71D411yescharacteristic
45#95×C71—21no—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71C2×C412yes—
49#11765849—24no—
50#11765849—44no—
51#11765849—42no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—44no—
54C2×C71⋊C8—42no—
55C24×C71C22⋊C411yescharacteristic
56C2×Q8×C71—42no—
57C2×Q8×C71—82no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—22no—
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71SD1612yes—
67C22×C4×C71Q1612yes—
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—82no—
78C22⋊C4×C71—41no—
79C22⋊C4×C71—41no—
80C22⋊C4×C71—82no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—41no—
88C23×C71—41no—
89C23×C71#10312yes—
90C23×C71C2×C22⋊C411yescharacteristic
91Q8×C71—84no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71#9911yescharacteristic
113C2×C4×C71#9911yescharacteristic
114C2×C4×C71#10312yes—
115C2×C4×C71#10412yes—
116C2×C4×C71C2×SD1611yescharacteristic
117C2×C4×C71C2×Q1611yescharacteristic
118C71⋊C8—44no—
119C71⋊C8—82no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—22no—
123C22×C71—42no—
124C22×C71—41no—
125C22×C71—41no—
126C22×C71—81no—
127C22×C71C2×#9911yescharacteristic
128C22×C71C2×#10311yescharacteristic
129C22×C71C2×#10411yescharacteristic
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—41no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71—82no—
137C4×C71—82no—
138C4×C71#32812yes—
139C4×C71#32812yes—
140#11724—711nomaximal, Sylow
141C2×C71—21no—
142C2×C71—21no—
143C2×C71—42no—
144C2×C71C2×#32811yescharacteristic
145C2×C71#212211yescharacteristic
146C2×C71#213511yescharacteristic
147#2628D7111yescharacteristic
148#1742—711no—
149#1458—711no—
150#1354—714no—
151C71#1172411yescharacteristic, complemented, Sylow
152C22×M16—1421no—
153C2×C4⋊Q8Dic7111yescharacteristic
154C2×#117C2×D7111yescharacteristic
155#431—1421no—
156#400—21no—
157#399—21no—
158#395C2×D7112yes—
159#393Dic7112yes—
160#389—21no—
161#382Dic7111yescharacteristic
162#380—21no—
163#343—712no—
164#343—1421no—
165#324—712no—
166#323—1421no—
167C22×Q8C2×Dic7112yes—
168C23×C4C22⋊D7111yescharacteristic
169C2×M16—2841no—
170C2×M16—2842no—
171C2×M16—1424no—
172C2×M16—1422no—
173C22×C8—2841no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22⋊D7111yescharacteristic
183C2×C4⋊C4C22×D7111yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic7111yescharacteristic
186C2×C22⋊C4C22⋊D7111yescharacteristic
187C2×C42C2×Dic7111yescharacteristic
188C2×C42C22⋊D7111yescharacteristic
189C4⋊C8—2841no—
190C22⋊C8—1422no—
191C4×C8—2841no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95C2×Dic7112yes—
196C24C22⋊Dic7111yescharacteristic
197C2×Q8—42no—
198C2×Q8—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4C22⋊Dic7111yescharacteristic
205C22×C4C22⋊Dic7111yescharacteristic
206C22×C4C22⋊Dic7111yescharacteristic
207C22×C4#1176585212yes—
208C22×C4#1176585512yes—
209C22×C4C22×Dic7111yescharacteristic
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4C2×C22⋊D7111yescharacteristic
212M16—2844no—
213M16—2842no—
214M16—1424no—
215C2×C8—2842no—
216C2×C8—2844no—
217C2×C8—2842no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1455605212yes—
237C23C2×C22⋊Dic7111yescharacteristic
238Q8—84no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#1455604311yescharacteristic
260C2×C4#1455604311yescharacteristic
261C2×C4#1455605212yes—
262C2×C4#1455605512yes—
263C2×C4C2×#1176585211yescharacteristic
264C2×C4C2×#1176585511yescharacteristic
265C8—2844no—
266C8—5682no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1455604311yescharacteristic
275C22C2×#1455605211yescharacteristic
276C22C2×#1455605511yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1798573612yes—
286C4#1798573612yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1798573611yescharacteristic
291C2#2258999011yescharacteristic
292C2#2259003511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2628C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2628D71
hypocenterC71#11724
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #22586370C2semidirect
151. C71#11724semidirect

Maximal subgroups

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

subgroupindex
#225880792
#225866542
#225863702
C71×#26282
#1172471