Finite Group Databasebeta

← #31906587

Subgroups — #31906587

Order 18176.

392 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#17986024C211yesmaximal, characteristic
2#22589949C211yesmaximal, characteristic
3C22⋊C16×D71C211yesmaximal, characteristic
4#22586280C211yesmaximal, characteristic, complemented
5#22586279C211yesmaximal, characteristic, complemented
6#22586279C211yesmaximal, characteristic, complemented
7C71×#1322C211yesmaximal, characteristic, complemented
8C22×#11765848C411yescharacteristic
9C22×C8×D71C2211yescharacteristic
10C2×#14556085C411yescharacteristic
11C2×#14556067—21no—
12C2×C16×D71—21no—
13#17986024—21no—
14#17986024—21no—
15#17986024—21no—
16#17986024—21no—
17C2×C4⋊(C71⋊C8)C411yescharacteristic
18C2×Dic71⋊C8C411yescharacteristic
19C2×Dic71⋊C8C2211yescharacteristic
20C22⋊(C71⋊C16)C2211yescharacteristic
21C22⋊(C71⋊C16)C2211yescharacteristic
22(C2×D71)⋊C16—21no—
23(C2×D71)⋊C16—21no—
24C2×C4⋊C8×C71C2211yescharacteristic
25C22⋊C16×C71C2211yescharacteristic
26C22⋊C16×C71C2211yescharacteristic
27C22×C4×D71C2×C411yescharacteristic
28C2×#11765848—21no—
29C2×#11765848—21no—
30C2×#11765848—42no—
31C22×C71⋊C8C2×C411yescharacteristic
32C22×C71⋊C8C2×C411yescharacteristic
33C2×C8×D71—42no—
34C2×C8×D71D411yescharacteristic
35C2×C8×D71D411yescharacteristic
36C2×C8×D71D411yescharacteristic
37C2×C8×D71D411yescharacteristic
38#14556085—21no—
39#14556085—21no—
40#14556085—21no—
41#14556085—21no—
42C2×Dic71⋊C4C811yescharacteristic
43C2×Dic71⋊C4C811yescharacteristic
44C2×(C2×D71)⋊C4C811yescharacteristic
45C2×(C2×D71)⋊C4C811yescharacteristic
46C2×C4×Dic71C2×C411yescharacteristic
47#14556067—42no—
48C2×C71⋊C16—21no—
49C2×C71⋊C16—21no—
50C16×D71—42no—
51C4⋊(C71⋊C8)—21no—
52C4⋊(C71⋊C8)—21no—
53Dic71⋊C8—21no—
54Dic71⋊C8—21no—
55Dic71⋊C8—21no—
56Dic71⋊C8—21no—
57C22×C8×C71C2×C411yescharacteristic
58C22×C8×C71C2311yescharacteristic
59C2×C42×C71C2×C411yescharacteristic
60C2×C16×C71—21no—
61C2×C16×C71—21no—
62C4⋊C8×C71—21no—
63C4⋊C8×C71—21no—
64C23×D71C2×C811yescharacteristic
65C22×Dic71C2×C811yescharacteristic
66C22×Dic71C2×C811yescharacteristic
67C22×Dic71C2×C811yescharacteristic
68C2×C4×D71—22no—
69C2×C4×D71—22no—
70C2×C4×D71C22⋊C411yescharacteristic
71C2×C4×D71C22⋊C411yescharacteristic
72C2×C4×D71M1611yescharacteristic
73C2×C4×D71M1611yescharacteristic
74#11765848—42no—
75#11765848—42no—
76C2×C71⋊C8—21no—
77C2×C71⋊C8—41no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8C22⋊C411yescharacteristic
80C2×C71⋊C8C22⋊C411yescharacteristic
81C8×D71—22no—
82C8×D71—22no—
83C8×D71—41no—
84C8×D71—41no—
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4—21no—
93C4×Dic71—21no—
94C4×Dic71—21no—
95C71⋊C16—41no—
96C71⋊C16—41no—
97C22×C4×C71C2×C811yescharacteristic
98C22×C4×C71C2×C811yescharacteristic
99C22×C4×C71C22×C411yescharacteristic
100C2×C8×C71—21no—
101C2×C8×C71—41no—
102C2×C8×C71—41no—
103C2×C8×C71C2×D411yescharacteristic
104C2×C8×C71C2×D411yescharacteristic
105C42×C71—21no—
106C42×C71—21no—
107C16×C71—41no—
108C16×C71—41no—
109C23×C71C22×C811yescharacteristic
110C22×D71—21no—
111C22×D71—21no—
112C22×D71—22no—
113C22×D71—41no—
114C22×D71C22⋊C811yescharacteristic
115C22×D71C22⋊C811yescharacteristic
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—41no—
123C2×C4×C71C2×C22⋊C411yescharacteristic
124C2×C4×C71C2×M1611yescharacteristic
125C2×Dic71—22no—
126C2×Dic71—21no—
127C2×Dic71—21no—
128C2×Dic71—21no—
129C2×Dic71—21no—
130C2×Dic71—41no—
131C2×Dic71C22⋊C811yescharacteristic
132C2×Dic71C22⋊C811yescharacteristic
133C4×D71—22no—
134C4×D71—21no—
135C4×D71—21no—
136C4×D71—22no—
137C4×D71#9912yes—
138C4×D71#10112yes—
139C8×C71—21no—
140C8×C71—21no—
141C8×C71—41no—
142C71⋊C8—21no—
143C71⋊C8—21no—
144C71⋊C8—41no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71C2×C22⋊C811yescharacteristic
148C22×C71#50411yescharacteristic
149C22×C71#50411yescharacteristic
150C2×D71—21no—
151C2×D71—21no—
152C2×D71—21no—
153C2×D71—21no—
154C2×D71—21no—
155C2×D71—42no—
156C2×D71—42no—
157C2×D71#32412yes—
158C4×C71—21no—
159C4×C71—41no—
160C4×C71—41no—
161C4×C71C2×#9911yescharacteristic
162C4×C71C2×#10111yescharacteristic
163Dic71—21no—
164Dic71—41no—
165Dic71—41no—
166Dic71#32412yes—
167#11441—711nomaximal, Sylow
168C2×C71—21no—
169C2×C71—21no—
170C2×C71C2×#32411yescharacteristic
171C2×C71#211311yescharacteristic
172C2×C71#211511yescharacteristic
173D71—22no—
174D71—41no—
175C2×#424—711no—
176#2112—711no—
177C2×C22⋊C16—711no—
178#1322—711no—
179#1322—711no—
180#1322—711no—
181#1322D7111yescharacteristic, complemented
182C71#1144111yescharacteristic, complemented, Sylow
183C22×M16—711no—
184C23×C8—711no—
185C2×#117—711no—
186C2×M32—1421no—
187C22×C16—1421no—
188#424—1421no—
189#424—1421no—
190#424—1421no—
191#424—1421no—
192C2×C4⋊C8—711no—
193C2×C4⋊C8—711no—
194C2×C4⋊C8—711no—
195C2×C4⋊C8C2×D7111yescharacteristic
196C22⋊C16—711no—
197C22⋊C16—711no—
198C22⋊C16—1421no—
199C22⋊C16—1421no—
200C22⋊C16C2×D7111yescharacteristic
201C22⋊C16C2×D7111yescharacteristic
202C23×C4—711no—
203C2×M16—2842no—
204C2×M16—1421no—
205C2×M16—1421no—
206C22×C8—2842no—
207C22×C8—711no—
208C22×C8—711no—
209C22×C8—711no—
210C22×C8—711no—
211C22×C8—711no—
212C22×C8—711no—
213C22×C8C4×D7111yescharacteristic
214C22×C8C22×D7111yescharacteristic
215#117—1421no—
216#117—1421no—
217#117—1421no—
218#117—1421no—
219C2×C4⋊C4—711no—
220C2×C4⋊C4—711no—
221C2×C22⋊C4—711no—
222C2×C22⋊C4—711no—
223C2×C42—711no—
224C2×C42C4×D7111yescharacteristic
225M32—2842no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—2842no—
229C2×C16—1421no—
230C2×C16—1421no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—1421no—
234C4⋊C8—1421no—
235C4⋊C8—1421no—
236C4⋊C8—1421no—
237C4⋊C8—1421no—
238C4⋊C8—1421no—
239C24—711no—
240C22×C4—711no—
241C22×C4—711no—
242C22×C4—711no—
243C22×C4—711no—
244C22×C4—711no—
245C22×C4—711no—
246C22×C4—711no—
247C22×C4—1422no—
248C22×C4—1422no—
249C22×C4C8×D7111yescharacteristic
250C22×C4C8×D7111yescharacteristic
251C22×C4C2×C4×D7111yescharacteristic
252M16—2842no—
253M16—2842no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—2841no—
258C2×C8—2841no—
259C2×C8—2841no—
260C2×C8—2841no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8—1422no—
264C2×C8—1422no—
265C2×C8—1421no—
266C2×C8D4×D7111yescharacteristic
267C2×C8D4×D7111yescharacteristic
268C4⋊C4—1421no—
269C4⋊C4—1421no—
270C4⋊C4—1421no—
271C4⋊C4—1421no—
272C22⋊C4—1421no—
273C22⋊C4—1421no—
274C22⋊C4—1421no—
275C22⋊C4—1421no—
276C42—21no—
277C42—21no—
278C42—1421no—
279C42—1421no—
280C16—41no—
281C16—41no—
282C16—2841no—
283C16—2841no—
284C23—2841no—
285C23—711no—
286C23—711no—
287C23—1422no—
288C23—1421no—
289C23—1421no—
290C23C2×C8×D7111yescharacteristic
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—2841no—
299C2×C4—711no—
300C2×C4—712no—
301C2×C4—712no—
302C2×C4—711no—
303C2×C4—1422no—
304C2×C4—1422no—
305C2×C4—1421no—
306C2×C4—1421no—
307C2×C4—1421no—
308C2×C4—1421no—
309C2×C4—1422no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4C22⋊C4×D7111yescharacteristic
313C2×C4M16×D7111yescharacteristic
314C8—21no—
315C8—21no—
316C8—41no—
317C8—2841no—
318C8—1421no—
319C8—1421no—
320C22—21no—
321C22—21no—
322C22—2842no—
323C22—2842no—
324C22—712no—
325C22—1421no—
326C22—1421no—
327C22—1421no—
328C22—1421no—
329C22—1421no—
330C22C22⋊C8×D7111yescharacteristic
331C22#1798641511yescharacteristic
332C22#1798641711yescharacteristic
333C4—21no—
334C4—41no—
335C4—41no—
336C4—2841no—
337C4—2841no—
338C4—712no—
339C4—1421no—
340C4#99×D7111yescharacteristic
341C4#101×D7111yescharacteristic
342C2—21no—
343C2—21no—
344C2—2841no—
345C2—1422no—
346C2#324×D7111yescharacteristic
347C2#2258995211yescharacteristic
348C2#2258995911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC23×C71C22×C8
FrattiniC22×C8C22×D71
FittingC71×#1322C2
socleC22×C71C2×C22⋊C8
prime radicalC23×D71C2×C8
cosocleC22×C8×C71C23
hypercenter#1322D71
hypocenterC71#11441
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
4. #22586280C2semidirect
5. #22586279C2semidirect
6. #22586279C2semidirect
7. C71×#1322C2semidirect
181. #1322D71semidirect
182. C71#11441semidirect

Maximal subgroups

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

subgroupindex
C2×#179860242
#225899492
C22⋊C16×D712
#225862802
#225862792
C71×#13222
#1144171