Finite Group Databasebeta

← #31890850

Subgroups — #31890850

Order 18176.

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

normal subgroups only

This group has 554 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#382×D71C211yesmaximal, characteristic
2#22588103C212yesmaximal
3#22586222C212yesmaximal, complemented
4#22586221C211yesmaximal, characteristic, complemented
5C71×#1295C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D71C411yescharacteristic
7C2×C4⋊C4×D71C411yescharacteristic
8C2×C42×D71C2211yescharacteristic
9C2×(C2×D71)⋊C8—22no—
10C2×#14556039C2212yes—
11#17985865C2211yescharacteristic
12#17985864C411yescharacteristic
13#17985864C411yescharacteristic
14#17985863—22no—
15#95×D71—22no—
16#17985753—22no—
17C2×#97×C71C2212yes—
18#382×C71C2211yescharacteristic
19C22×C4×D71—21no—
20C22×C4×D71—21no—
21C22×C4×D71C2×C411yescharacteristic
22C22×C4×D71D412yes—
23C22×C71⋊C8—22no—
24C2×C4⋊Dic71C2×C411yescharacteristic
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×Dic71⋊C4—21no—
27C2×Dic71⋊C4—21no—
28C4⋊C4×D71—42no—
29C4⋊C4×D71—42no—
30C2×C4×Dic71C2×C411yescharacteristic
31C2×C4×Dic71D412yes—
32C42×D71—21no—
33C42×D71—22no—
34C42×D71—21no—
35(C2×D71)⋊C8—28no—
36#14556039—22no—
37#14556039—22no—
38C71⋊#95—22no—
39C71⋊#95—22no—
40C71⋊#95—22no—
41C22×C8×C71—22no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C42×C71C2311yescharacteristic
45#97×C71—22no—
46#97×C71—22no—
47#95×C71—22no—
48C23×D71C22⋊C411yescharacteristic
49C22×Dic71—21no—
50C22×Dic71—21no—
51C22×Dic71C22⋊C412yes—
52C22×Dic71C22⋊C411yescharacteristic
53C2×C4×D71—22no—
54C2×C4×D71—21no—
55C2×C4×D71—21no—
56C2×C4×D71—22no—
57C2×C4×D71—24no—
58C2×C4×D71—22no—
59C2×C4×D71—22no—
60C2×C4×D71—42no—
61C2×C4×D71—44no—
62C2×C4×D71—42no—
63C2×C4×D71—42no—
64C2×C4×D71—44no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8—42no—
68C4⋊Dic71—41no—
69C4⋊Dic71—41no—
70Dic71⋊C4—42no—
71Dic71⋊C4—42no—
72C4×Dic71—21no—
73C4×Dic71—21no—
74C4×Dic71—22no—
75C4×Dic71—22no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D412yes—
80C2×C8×C71—24no—
81C2×C8×C71—42no—
82C2×C8×C71—42no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C42×C71#4111yescharacteristic
86C42×C71#4112yes—
87C42×C71#4111yescharacteristic
88C23×C71C2×C22⋊C411yescharacteristic
89C22×D71—21no—
90C22×D71—21no—
91C22×D71—22no—
92C22×D71#10012yes—
93C22×D71#10414yes—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71#11711yescharacteristic
103C2×C4×C71#11711yescharacteristic
104C2×C4×C71#12312yes—
105C2×C4×C71#12312yes—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—41no—
112C2×Dic71—41no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71#10112yes—
116C4×D71—24no—
117C4×D71—24no—
118C4×D71—44no—
119C4×D71—44no—
120C4×D71—84no—
121C4×D71—84no—
122C8×C71—44no—
123C71⋊C8—44no—
124C22×C71#38111yescharacteristic
125C22×C71C2×#10011yescharacteristic
126C22×C71C2×#10111yescharacteristic
127C22×C71C2×#10412yes—
128C22×C71#42112yes—
129C2×D71—22no—
130C2×D71—24no—
131C2×D71—24no—
132C2×D71—22no—
133C2×D71#32914yes—
134C4×C71—21no—
135C4×C71—21no—
136C4×C71—22no—
137C4×C71—22no—
138C4×C71—81no—
139C4×C71—81no—
140Dic71—22no—
141Dic71—42no—
142Dic71—81no—
143Dic71—81no—
144#8718—711nomaximal, Sylow
145C2×C71C2×#32911yescharacteristic
146C2×C71#152812yes—
147C2×C71#153311yescharacteristic
148C2×C71#182111yescharacteristic
149C2×C71#182612yes—
150D71—24no—
151C2×#382—711no—
152#1749—712no—
153#1295—712no—
154#1295—711no—
155#1295D7111yescharacteristic, complemented
156C71#871811yescharacteristic, complemented, Sylow
157C22×C4⋊C4—711no—
158C22×C4⋊C4—711no—
159C22×C42—711no—
160C2×C22⋊C8—1422no—
161C2×#97—712no—
162C2×#97C2×D7112yes—
163#382—711no—
164#382—711no—
165#382—711no—
166#382—1422no—
167#382C2×D7111yescharacteristic
168C2×#95—1422no—
169#336—1422no—
170C23×C4—712no—
171C23×C4—711no—
172C23×C4—1421no—
173C23×C4—1421no—
174C22×C8—22no—
175C22×C8—1422no—
176C2×C4⋊C4—2842no—
177C2×C4⋊C4—2842no—
178C2×C4⋊C4—711no—
179C2×C4⋊C4—711no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4—1421no—
182C2×C4⋊C4C4×D7111yescharacteristic
183C2×C4⋊C4C4×D7111yescharacteristic
184C2×C42—711no—
185C2×C42—712no—
186C2×C42—1421no—
187C2×C42—1422no—
188C2×C42—1421no—
189C2×C42C22×D7111yescharacteristic
190C22⋊C8—1428no—
191#97—22no—
192#97—22no—
193#97—1422no—
194#97—1422no—
195#95—22no—
196#95—1422no—
197#95—1422no—
198#95—1422no—
199C24—711no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—2842no—
203C22×C4—2842no—
204C22×C4—2842no—
205C22×C4—2844no—
206C22×C4—2844no—
207C22×C4—711no—
208C22×C4—712no—
209C22×C4—1424no—
210C22×C4—1422no—
211C22×C4—1422no—
212C22×C4—1421no—
213C22×C4—1421no—
214C22×C4—1422no—
215C22×C4—1422no—
216C22×C4—1421no—
217C22×C4—1421no—
218C22×C4C2×C4×D7111yescharacteristic
219C22×C4D4×D7112yes—
220C2×C8—24no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—2842no—
224C2×C8—2842no—
225C2×C8—1424no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—2841no—
229C4⋊C4—2841no—
230C4⋊C4—2842no—
231C4⋊C4—2842no—
232C42—1421no—
233C42—1421no—
234C42—1422no—
235C42—1422no—
236C42#1176586411yescharacteristic
237C42#1176586512yes—
238C42#1176586611yescharacteristic
239C23—712no—
240C23—714no—
241C23—1421no—
242C23—1421no—
243C23—1422no—
244C23C22⋊C4×D7111yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—2844no—
254C2×C4—2844no—
255C2×C4—2841no—
256C2×C4—2841no—
257C2×C4—2842no—
258C2×C4—2842no—
259C2×C4—5684no—
260C2×C4—5684no—
261C2×C4—712no—
262C2×C4—1424no—
263C2×C4—1424no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4—1421no—
268C2×C4—1421no—
269C2×C4#1455608511yescharacteristic
270C2×C4#1455608611yescharacteristic
271C2×C4#1455611012yes—
272C2×C4#1455611312yes—
273C8—44no—
274C8—2844no—
275C22—714no—
276C22—1422no—
277C22—1424no—
278C22—1424no—
279C22—1422no—
280C22#1798586111yescharacteristic
281C22#100×D7111yescharacteristic
282C22#101×D7111yescharacteristic
283C22#104×D7112yes—
284C22#1798601412yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—2842no—
292C4—5681no—
293C4—5681no—
294C4—1422no—
295C2—1424no—
296C2#329×D7111yescharacteristic
297C2#2258694012yes—
298C2#2258696211yescharacteristic
299C2#2258841911yescharacteristic
300C2#2258843812yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1295D71
hypocenterC71#8718
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #22586222C2semidirect
4. #22586221C2semidirect
5. C71×#1295C2semidirect
155. #1295D71semidirect
156. C71#8718semidirect

Maximal subgroups

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

subgroupindex
#382×D712
#225881032
#225862222
#225862212
C71×#12952
#871871