Finite Group Databasebeta

← #31908178

Subgroups — #31908178

Order 18176.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588417C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3#22586361C214yesmaximal
4C71×#2628C211yesmaximal, characteristic
5#17986056—21no—
6#17985862—21no—
7#17985766—21no—
8#17985766C2212yes—
9#17985725—21no—
10#17985723C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic, complemented
12C2×#117×C71C2211yescharacteristic
13#400×C71—21no—
14#399×C71—21no—
15#395×C71C412yescomplemented
16#393×C71C2212yes—
17#389×C71—21no—
18#382×C71C411yescharacteristic, complemented
19#380×C71—21no—
20C2×C22⋊Dic71—22no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23#14556039—41no—
24C71⋊#95—41no—
25C71⋊#95—41no—
26C22×Q8×C71C2×C412yes—
27C23×C4×C71D411yescharacteristic
28C4⋊Q8×C71—41no—
29C4⋊Q8×C71—41no—
30#117×C71—41no—
31#117×C71—41no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71C2×C411yescharacteristic
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71D411yescharacteristic
40C2×C22⋊C4×C71C2311yescharacteristic
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71D411yescharacteristic
43#95×C71—22no—
44#95×C71—21no—
45#95×C71—22no—
46#95×C71C2×C412yes—
47C22×Dic71—42no—
48C2×C71⋊C8—42no—
49C22⋊Dic71—42no—
50C22⋊Dic71—42no—
51C24×C71C2×D411yescharacteristic
52C2×Q8×C71—42no—
53C2×Q8×C71—82no—
54C22×C4×C71—22no—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—41no—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22×C411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—82no—
73C22⋊C4×C71—41no—
74C22⋊C4×C71—41no—
75C22⋊C4×C71—82no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C23×C71—21no—
81C23×C71—21no—
82C23×C71—41no—
83C23×C71—41no—
84C23×C71C2×C22⋊C411yescharacteristic
85C23×C71#13712yes—
86Q8×C71—84no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71#9911yescharacteristic
109C2×C4×C71#9911yescharacteristic
110C2×Dic71—82no—
111C2×Dic71—82no—
112C2×Dic71—82no—
113C71⋊C8—82no—
114C22×C71—21no—
115C22×C71—21no—
116C22×C71—22no—
117C22×C71—42no—
118C22×C71—41no—
119C22×C71—41no—
120C22×C71—81no—
121C22×C71C2×#9911yescharacteristic
122C22×C71#41911yescharacteristic
123C22×C71#42111yescharacteristic
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—41no—
127C4×C71—41no—
128C4×C71—41no—
129C4×C71—41no—
130C4×C71—41no—
131C4×C71—82no—
132C4×C71—82no—
133Dic71—162no—
134#11812—711nomaximal, Sylow
135C2×C71—21no—
136C2×C71—21no—
137C2×C71—42no—
138C2×C71#151111yescharacteristic
139C2×C71#212211yescharacteristic
140C2×C71#212911yescharacteristic
141#2628D7111yescharacteristic
142#1820—711no—
143#1464—711no—
144#1351—714no—
145C71#1181211yescharacteristic, complemented, Sylow
146C2×C4⋊Q8Dic7111yescharacteristic, complemented
147C2×#117C2×D7111yescharacteristic
148#432—1421no—
149#400—21no—
150#399—21no—
151#395Dic7112yescomplemented
152#393C2×D7112yes—
153#389—21no—
154#382Dic7111yescharacteristic, complemented
155#381—1421no—
156#380—21no—
157#342—712no—
158#342—1421no—
159#324—1421no—
160#323—712no—
161C22×Q8C2×Dic7112yes—
162C23×C4C22⋊D7111yescharacteristic
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C2×Dic7111yescharacteristic
172C2×C4⋊C4C22⋊D7111yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1422no—
175C2×C22⋊C4C22⋊D7111yescharacteristic
176C2×C22⋊C4C22×D7111yescharacteristic
177C2×C42C2×Dic7111yescharacteristic
178C2×C42C22⋊D7111yescharacteristic
179C4⋊C8—2841no—
180C22⋊C8—1422no—
181#97—2841no—
182#95—22no—
183#95—21no—
184#95—22no—
185#95—2841no—
186#95—2841no—
187#95C2×Dic7112yes—
188C24C2×C22⋊D7111yescharacteristic
189C2×Q8—42no—
190C2×Q8—82no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—2842no—
198C22×C4C22⋊Dic7111yescharacteristic
199C22×C4C22⋊Dic7111yescharacteristic
200C22×C4C22⋊Dic7111yescharacteristic
201C22×C4C22⋊Dic7111yescharacteristic
202C22×C4C22×Dic7111yescharacteristic
203C22×C4C2×C22⋊D7111yescharacteristic
204C2×C8—2842no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—82no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—82no—
215C22⋊C4—2842no—
216C22⋊C4—2842no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊Dic7111yescharacteristic
226C23#1455615912yes—
227Q8—84no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—5682no—
250C2×C4—5682no—
251C2×C4—5682no—
252C2×C4#1455604311yescharacteristic
253C2×C4#1455604311yescharacteristic
254C8—5682no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×#1455604311yescharacteristic
263C22#1798600811yescharacteristic
264C22#1798601611yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—11362no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2#2258687311yescharacteristic
279C2#2258999011yescharacteristic
280C2#2259001511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2628C2
socleC22×C71C2×#99
prime radicalC24×C71C2×D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2628D71
hypocenterC71#11812
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
11. C2×C4⋊Q8×C71C4semidirect
15. #395×C71C4semidirect
18. #382×C71C4semidirect
145. C71#11812semidirect
146. C2×C4⋊Q8Dic71semidirect
151. #395Dic71semidirect
154. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884172
#225866742
#225863612
C71×#26282
#1181271