Finite Group Databasebeta

← #31885959

Subgroups — #31885959

Order 18176.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#22596707C211yesmaximal, characteristic
2#22588086C211yesmaximal, characteristic, complemented
3#22586162C212yesmaximal, complemented
4#22586160C212yesmaximal
5C71×#1745C211yesmaximal, characteristic, complemented
6C2×#14556122—21no—
7C2×C4×C71⋊C8—21no—
8#17985901C411yescharacteristic
9#17985901C411yescharacteristic
10#17985901C2211yescharacteristic
11#17985897C411yescharacteristic
12#17985897C411yescharacteristic
13#17985897C2211yescharacteristic
14#17985890—21no—
15#17985890—21no—
16#17985890—21no—
17#17985754—21no—
18#17985754C2212yes—
19C2×#97×C71—21no—
20C43×C71C2211yescharacteristic
21#336×C71—21no—
22#336×C71C2212yes—
23C22×D284C2×C411yescharacteristic
24C22×C71⋊C8—22no—
25#14556122—44no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×Dic71⋊C4—21no—
30C2×Dic71⋊C4—21no—
31C2×Dic71⋊C4—21no—
32C2×(C2×D71)⋊C4—21no—
33C2×(C2×D71)⋊C4—21no—
34C2×(C2×D71)⋊C4—21no—
35C2×(C2×D71)⋊C4—21no—
36C2×(C2×D71)⋊C4—21no—
37C2×(C2×D71)⋊C4—21no—
38C4×C71⋊C8—24no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C22×C8×C71—22no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71C2311yescharacteristic
50#97×C71—24no—
51C23×D71—21no—
52C2×D284—22no—
53C2×D284—41no—
54C2×D284—41no—
55C22×Dic71—21no—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C2×C71⋊C8—24no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—22no—
62C4⋊Dic71—41no—
63C4⋊Dic71—41no—
64Dic71⋊C4—42no—
65Dic71⋊C4—42no—
66Dic71⋊C4—42no—
67(C2×D71)⋊C4—42no—
68(C2×D71)⋊C4—42no—
69(C2×D71)⋊C4—42no—
70(C2×D71)⋊C4—42no—
71(C2×D71)⋊C4—42no—
72(C2×D71)⋊C4—42no—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C2×C8×C71—24no—
81C2×C8×C71—42no—
82C2×C8×C71—42no—
83C42×C71—22no—
84C42×C71—22no—
85C42×C71—22no—
86C42×C71—22no—
87C42×C71—22no—
88C42×C71—41no—
89C42×C71—41no—
90C23×C71C2×C22⋊C411yescharacteristic
91C22×D71—41no—
92C22×D71—41no—
93C22×D71—41no—
94C22×D71—41no—
95C22×D71—41no—
96C22×D71—41no—
97C22×D71—41no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
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—21no—
114C2×C4×C71#10412yes—
115C2×C4×C71#10412yes—
116C2×C4×C71#13112yes—
117C2×C4×C71#13512yes—
118C2×C4×C71AGL(1,ℤ/8)12yes—
119C2×Dic71—41no—
120C2×Dic71—41no—
121C2×Dic71—41no—
122C2×Dic71—41no—
123C2×Dic71—41no—
124C2×Dic71—41no—
125C2×Dic71—41no—
126C2×Dic71—41no—
127C2×Dic71—41no—
128D284—82no—
129C8×C71—44no—
130C71⋊C8—44no—
131C22×C71#40811yescharacteristic
132C22×C71#41011yescharacteristic
133C22×C71#41311yescharacteristic
134C22×C71#41611yescharacteristic
135C22×C71#41811yescharacteristic
136C22×C71C2×#10411yescharacteristic
137C22×C71C2×#10411yescharacteristic
138C2×D71—81no—
139C2×D71—81no—
140C2×D71—81no—
141C2×D71—81no—
142C2×D71—81no—
143C2×D71—81no—
144C2×D71—81no—
145C4×C71—22no—
146C4×C71—22no—
147C4×C71—22no—
148C4×C71—22no—
149C4×C71—22no—
150C4×C71—41no—
151C4×C71—41no—
152Dic71—81no—
153Dic71—81no—
154Dic71—81no—
155#7692—711nomaximal, Sylow
156C2×C71#147311yescharacteristic
157C2×C71#148411yescharacteristic
158C2×C71#149411yescharacteristic
159C2×C71#150011yescharacteristic
160C2×C71#150511yescharacteristic
161C2×C71#152411yescharacteristic
162C2×C71#155311yescharacteristic
163D71—161no—
164#2860—711no—
165#1745D7111yescharacteristic, complemented
166#1744—711no—
167#1271—712no—
168#1270—712no—
169C71#769211yescharacteristic, complemented, Sylow
170C2×#126—1421no—
171C2×#97—21no—
172C2×C4×C8—1421no—
173#390—711no—
174#390—711no—
175#390—711no—
176#389—711no—
177#389—711no—
178#389—711no—
179#388—1421no—
180#388—1421no—
181#388—1421no—
182C43C2×D7111yescharacteristic
183#336—21no—
184#336—712no—
185#336—1421no—
186#336C2×D7112yes—
187C22×D4—711no—
188C22×C8—22no—
189C22×C8—1422no—
190#126—2844no—
191C2×C4⋊C4—711no—
192C2×C4⋊C4—711no—
193C2×C4⋊C4—711no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4—1421no—
196C2×C4⋊C4—1421no—
197C2×C22⋊C4—1421no—
198C2×C22⋊C4—1421no—
199C2×C22⋊C4—1421no—
200C2×C22⋊C4—1421no—
201C2×C22⋊C4—1421no—
202C2×C22⋊C4—1421no—
203C2×C42C4×D7111yescharacteristic
204C2×C42C4×D7111yescharacteristic
205C2×C42D28411yescharacteristic
206C2×C42D28411yescharacteristic
207C2×C42C22⋊D7111yescharacteristic
208C2×C42C22⋊D7111yescharacteristic
209C2×C42C22×D7111yescharacteristic
210#97—24no—
211C4×C8—1424no—
212#95—1421no—
213#95—1421no—
214#95—1421no—
215C24—1421no—
216C2×D4—2841no—
217C2×D4—2841no—
218C2×D4—1422no—
219C22×C4—1421no—
220C22×C4—1421no—
221C22×C4—1421no—
222C22×C4(C2×D71)⋊C411yescharacteristic
223C22×C4(C2×D71)⋊C411yescharacteristic
224C22×C4(C2×D71)⋊C411yescharacteristic
225C22×C4(C2×D71)⋊C411yescharacteristic
226C22×C4C2×C4×D7111yescharacteristic
227C22×C4C2×C22⋊D7111yescharacteristic
228C22×C4C2×D28411yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—2842no—
233C2×C8—2842no—
234C2×C8—1424no—
235C4⋊C4—2841no—
236C4⋊C4—2841no—
237C4⋊C4—2842no—
238C4⋊C4—2842no—
239C4⋊C4—2842no—
240C4⋊C4—1422no—
241C22⋊C4—2842no—
242C22⋊C4—2842no—
243C22⋊C4—2842no—
244C22⋊C4—2842no—
245C22⋊C4—2842no—
246C22⋊C4—2842no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C23—2841no—
255C23—2841no—
256C23—2841no—
257C23—2841no—
258C23—2841no—
259C23—2841no—
260C23—2841no—
261C23C2×(C2×D71)⋊C411yescharacteristic
262D4—5682no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—2841no—
280C2×C4—2841no—
281C2×C4—2841no—
282C2×C4—2841no—
283C2×C4—2841no—
284C2×C4—2841no—
285C2×C4—2841no—
286C2×C4—2841no—
287C2×C4—2841no—
288C2×C4#1455605412yes—
289C2×C4#1455605612yes—
290C2×C4#1455613612yes—
291C2×C4#1455614812yes—
292C2×C4#1455615612yes—
293C8—44no—
294C8—2844no—
295C22—5681no—
296C22—5681no—
297C22—5681no—
298C22—5681no—
299C22—5681no—
300C22—5681no—
301C22—5681no—
302C22#1798596711yescharacteristic
303C22#1798597211yescharacteristic
304C22#1798598111yescharacteristic
305C22#1798599311yescharacteristic
306C22#1798600511yescharacteristic
307C22C2×#1455605411yescharacteristic
308C22C2×#1455605611yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—5681no—
317C4—5681no—
318C4—5681no—
319C2—11361no—
320C2#2258671011yescharacteristic
321C2#2258675811yescharacteristic
322C2#2258679811yescharacteristic
323C2#2258682411yescharacteristic
324C2#2258684911yescharacteristic
325C2#2258692911yescharacteristic
326C2#2258704611yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1745C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1745D71
hypocenterC71#7692
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #22588086C2semidirect
3. #22586162C2semidirect
5. C71×#1745C2semidirect
165. #1745D71semidirect
169. C71#7692semidirect

Maximal subgroups

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

subgroupindex
#225967072
#225880862
#225861622
#225861602
C71×#17452
#769271