Finite Group Databasebeta

← #31885590

Subgroups — #31885590

Order 18176.

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

normal subgroups only

This group has 484 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#22590804C211yesmaximal, characteristic
2#22588483C211yesmaximal, characteristic, complemented
3#22586162C212yesmaximal, complemented
4#22586160C212yesmaximal
5C71×#1745C211yesmaximal, characteristic, complemented
6C2×C4×D284C411yescharacteristic
7C2×#14556085—21no—
8#17985901C411yescharacteristic
9#17985901C2211yescharacteristic
10#17985897C411yescharacteristic
11#17985897C2211yescharacteristic
12#17985890—21no—
13#17985890—21no—
14C4×C4⋊Dic71C411yescharacteristic
15C4×(C2×D71)⋊C4—21no—
16#17985754—21no—
17#17985754—21no—
18#17985754C2212yes—
19C2×#97×C71—21no—
20C43×C71C2211yescharacteristic
21#336×C71—21no—
22#336×C71C2212yes—
23C22×D284C2×C411yescharacteristic
24C22×C4×D71—21no—
25C22×C71⋊C8—22no—
26C4×D284—22no—
27C4×D284—22no—
28#14556085—44no—
29C2×C4⋊Dic71C2×C411yescharacteristic
30C2×C4⋊Dic71C2×C411yescharacteristic
31C2×C4⋊Dic71C2×C411yescharacteristic
32C2×Dic71⋊C4—21no—
33C2×Dic71⋊C4—21no—
34C2×(C2×D71)⋊C4—21no—
35C2×(C2×D71)⋊C4—21no—
36C2×(C2×D71)⋊C4—21no—
37C2×(C2×D71)⋊C4—21no—
38C2×(C2×D71)⋊C4—21no—
39C2×C4×Dic71—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C71⋊#95—21no—
43C22×C8×C71—22no—
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71C2×C411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71C2311yescharacteristic
51#97×C71—24no—
52C23×D71—21no—
53C2×D284—21no—
54C2×D284—41no—
55C2×D284—41no—
56C2×D284M1612yes—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C2×C4×D71—24no—
61C2×C4×D71—42no—
62C2×C4×D71—42no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C4⋊Dic71—21no—
67C4⋊Dic71—41no—
68C4⋊Dic71—41no—
69C4⋊Dic71M1612yes—
70Dic71⋊C4—42no—
71Dic71⋊C4—42no—
72(C2×D71)⋊C4—24no—
73(C2×D71)⋊C4—42no—
74(C2×D71)⋊C4—42no—
75(C2×D71)⋊C4—42no—
76(C2×D71)⋊C4—42no—
77C4×Dic71—42no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C2×C8×C71—24no—
86C2×C8×C71—42no—
87C2×C8×C71—42no—
88C42×C71—22no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—22no—
92C42×C71—22no—
93C42×C71—41no—
94C42×C71—41no—
95C42×C71—41no—
96C23×C71C2×C22⋊C411yescharacteristic
97C22×D71—22no—
98C22×D71—41no—
99C22×D71—41no—
100C22×D71—41no—
101C22×D71—41no—
102C22×D71—41no—
103C22×D71—41no—
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—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71#10412yes—
122C2×C4×C71C2×M1611yescharacteristic
123C2×C4×C71C2×M1611yescharacteristic
124C2×C4×C71#13512yes—
125C2×C4×C71AGL(1,ℤ/8)12yes—
126C2×Dic71—22no—
127C2×Dic71—41no—
128C2×Dic71—41no—
129C2×Dic71—41no—
130C2×Dic71—41no—
131C2×Dic71—41no—
132C2×Dic71—41no—
133C2×Dic71—41no—
134C2×Dic71—41no—
135D284—42no—
136D284—42no—
137C4×D71—48no—
138C8×C71—44no—
139C71⋊C8—82no—
140C22×C71#40711yescharacteristic
141C22×C71#41011yescharacteristic
142C22×C71#41311yescharacteristic
143C22×C71#41611yescharacteristic
144C22×C71#41811yescharacteristic
145C22×C71C2×#10411yescharacteristic
146C22×C71#42111yescharacteristic
147C2×D71—42no—
148C2×D71—42no—
149C2×D71—42no—
150C2×D71—81no—
151C2×D71—81no—
152C2×D71—81no—
153C2×D71—81no—
154C4×C71—21no—
155C4×C71—21no—
156C4×C71—22no—
157C4×C71—22no—
158C4×C71—22no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71—41no—
162Dic71—42no—
163Dic71—81no—
164Dic71—81no—
165#7608—711nomaximal, Sylow
166C2×C71#146411yescharacteristic
167C2×C71#148811yescharacteristic
168C2×C71#149011yescharacteristic
169C2×C71#150011yescharacteristic
170C2×C71#151411yescharacteristic
171C2×C71#152911yescharacteristic
172C2×C71#155311yescharacteristic
173D71—82no—
174#2293—711no—
175#1839—711no—
176#1745D7111yescharacteristic, complemented
177#1271—712no—
178#1270—712no—
179C71#760811yescharacteristic, complemented, Sylow
180C2×C4×D4—711no—
181C2×#117—1421no—
182C2×#97—21no—
183#390—711no—
184#390—711no—
185#389—711no—
186#389—711no—
187#388—1421no—
188#388—1421no—
189C4×C4⋊C4—711no—
190C4×C22⋊C4—1421no—
191C43C2×D7111yescharacteristic
192#336—21no—
193#336—712no—
194#336—1421no—
195#336—1421no—
196#336C2×D7112yes—
197C22×D4—711no—
198C23×C4—1421no—
199C22×C8—22no—
200C22×C8—1422no—
201C4×D4—1422no—
202C4×D4—1422no—
203#117—2844no—
204C2×C4⋊C4—711no—
205C2×C4⋊C4—711no—
206C2×C4⋊C4—711no—
207C2×C4⋊C4—1421no—
208C2×C4⋊C4—1421no—
209C2×C22⋊C4—1421no—
210C2×C22⋊C4—1421no—
211C2×C22⋊C4—1421no—
212C2×C22⋊C4—1421no—
213C2×C22⋊C4—1421no—
214C2×C42—1421no—
215C2×C42C4×D7111yescharacteristic
216C2×C42C4×D7111yescharacteristic
217C2×C42D28411yescharacteristic
218C2×C42D28411yescharacteristic
219C2×C42C22⋊D7111yescharacteristic
220C2×C42C22⋊D7111yescharacteristic
221C2×C42C22×D7111yescharacteristic
222#97—24no—
223#95—1421no—
224#95—1421no—
225#95—1421no—
226C24—1421no—
227C2×D4—2841no—
228C2×D4—2841no—
229C2×D4—712no—
230C2×D4—1421no—
231C22×C4—2842no—
232C22×C4—2842no—
233C22×C4—1424no—
234C22×C4—1421no—
235C22×C4—1421no—
236C22×C4—1421no—
237C22×C4(C2×D71)⋊C411yescharacteristic
238C22×C4(C2×D71)⋊C411yescharacteristic
239C22×C4(C2×D71)⋊C411yescharacteristic
240C22×C4(C2×D71)⋊C411yescharacteristic
241C22×C4C2×C4×D7111yescharacteristic
242C22×C4C2×C22⋊D7111yescharacteristic
243C22×C4C2×D28411yescharacteristic
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—2842no—
248C2×C8—2842no—
249C2×C8—2842no—
250C4⋊C4—2841no—
251C4⋊C4—2841no—
252C4⋊C4—2842no—
253C4⋊C4—2842no—
254C4⋊C4—712no—
255C4⋊C4—1421no—
256C22⋊C4—2842no—
257C22⋊C4—2842no—
258C22⋊C4—2842no—
259C22⋊C4—2842no—
260C22⋊C4—1424no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—2842no—
270C23—2841no—
271C23—2841no—
272C23—2841no—
273C23—2841no—
274C23—2841no—
275C23—2841no—
276C23—1422no—
277C23C2×(C2×D71)⋊C411yescharacteristic
278D4—2842no—
279D4—2842no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—2848no—
298C2×C4—2841no—
299C2×C4—2841no—
300C2×C4—2841no—
301C2×C4—2841no—
302C2×C4—2841no—
303C2×C4—2841no—
304C2×C4—2841no—
305C2×C4—2841no—
306C2×C4—1422no—
307C2×C4#1455605612yes—
308C2×C4M16×D7111yescharacteristic
309C2×C4M16×D7111yescharacteristic
310C2×C4#1455614812yes—
311C2×C4#1455615612yes—
312C8—44no—
313C8—5682no—
314C22—2842no—
315C22—2842no—
316C22—2842no—
317C22—5681no—
318C22—5681no—
319C22—5681no—
320C22—5681no—
321C22#1798596211yescharacteristic
322C22#1798597211yescharacteristic
323C22#1798598111yescharacteristic
324C22#1798599311yescharacteristic
325C22#1798600511yescharacteristic
326C22C2×#1455605611yescharacteristic
327C22#1798601511yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—2842no—
337C4—5681no—
338C4—5681no—
339C2—5682no—
340C2#2258667211yescharacteristic
341C2#2258677211yescharacteristic
342C2#2258678411yescharacteristic
343C2#2258682411yescharacteristic
344C2#2258688811yescharacteristic
345C2#2258694511yescharacteristic
346C2#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#7608
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. #22588483C2semidirect
3. #22586162C2semidirect
5. C71×#1745C2semidirect
176. #1745D71semidirect
179. C71#7608semidirect

Maximal subgroups

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

subgroupindex
#225908042
#225884832
#225861622
#225861602
C71×#17452
#760871