Finite Group Databasebeta

← #31913435

Subgroups — #31913435

Order 18176.

680 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 123 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×#17985792C212yesmaximal
2#22588328C212yesmaximal, complemented
3#22586465C212yesmaximal, complemented
4C4×#127×C71C211yesmaximal, characteristic
5C2×#14556043C2214yes—
6#17985859—22no—
7#17985792—24no—
8#17985766—22no—
9C2×#127×C71C2211yescharacteristic
10C2×C4×D4×C71—22no—
11C2×C4×D4×C71C411yescharacteristic, complemented
12C2×C4×D4×C71C2211yescharacteristic
13#390×C71—22no—
14#390×C71C411yescharacteristic, complemented
15#390×C71C2211yescharacteristic
16#384×C71C411yescharacteristic, complemented
17C43×C71C411yescharacteristic, complemented
18C2×C4⋊Dic71—42no—
19C2×C22⋊Dic71—24no—
20#14556043—24no—
21#14556043—24no—
22C71⋊#95—42no—
23C22×D4×C71—22no—
24C22×D4×C71C2×C411yescharacteristic
25C22×D4×C71C2311yescharacteristic
26C23×C4×C71—21no—
27C23×C4×C71D412yes—
28#127×C71—22no—
29#127×C71D412yes—
30#127×C71Q812yes—
31C4×D4×C71—41no—
32C4×D4×C71—41no—
33C4×D4×C71—48no—
34C4×D4×C71—41no—
35C4×D4×C71—41no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C22⋊C4×C71—22no—
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71D412yes—
43C2×C42×C71—21no—
44C2×C42×C71—21no—
45C2×C42×C71C2×C411yescharacteristic
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71C2×C411yescharacteristic
48C22×Dic71—44no—
49C4⋊Dic71—84no—
50C22⋊Dic71—44no—
51C22⋊Dic71—44no—
52C24×C71—21no—
53C24×C71C2×D412yes—
54C2×D4×C71—24no—
55C2×D4×C71—24no—
56C2×D4×C71—21no—
57C2×D4×C71—41no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61C2×D4×C71—44no—
62C2×D4×C71—44no—
63C2×D4×C71C4⋊C412yescomplemented
64C2×D4×C71C2×D411yescharacteristic
65C2×D4×C71C2×Q811yescharacteristic
66C2×D4×C71#4112yes—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—44no—
73C22×C4×C71—44no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—44no—
83C22⋊C4×C71—42no—
84C22⋊C4×C71—44no—
85C22⋊C4×C71—44no—
86C22⋊C4×C71—42no—
87C42×C71—24no—
88C42×C71—21no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—41no—
92C42×C71—42no—
93C42×C71C4⋊C412yescomplemented
94C23×C71—22no—
95C23×C71—22no—
96C23×C71—22no—
97C23×C71—42no—
98C23×C71—42no—
99C23×C71—42no—
100C23×C71—42no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71C2×C22⋊C411yescharacteristic
104C23×C71#12112yes—
105C23×C71C22⋊Q812yes—
106D4×C71—41no—
107D4×C71—41no—
108D4×C71—42no—
109D4×C71—42no—
110D4×C71—48no—
111D4×C71—48no—
112D4×C71—42no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—22no—
119C2×C4×C71—22no—
120C2×C4×C71—21no—
121C2×C4×C71—22no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—44no—
125C2×C4×C71—41no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—41no—
129C2×C4×C71—82no—
130C2×C4×C71—82no—
131C2×C4×C71—82no—
132C2×C4×C71#9911yescharacteristic
133C2×C4×C71#9911yescharacteristic
134C2×C4×C71C2×C4⋊C411yescharacteristic
135C2×C4×C71#11711yescharacteristic
136C2×Dic71—84no—
137C2×Dic71—84no—
138C2×Dic71—84no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—22no—
142C22×C71—22no—
143C22×C71—42no—
144C22×C71—42no—
145C22×C71—44no—
146C22×C71—42no—
147C22×C71—42no—
148C22×C71—42no—
149C22×C71—82no—
150C22×C71—82no—
151C22×C71—81no—
152C22×C71—81no—
153C22×C71#38011yescharacteristic
154C22×C71C2×#9911yescharacteristic
155C22×C71#41011yescharacteristic
156C4×C71—24no—
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—42no—
161C4×C71—42no—
162C4×C71—41no—
163C4×C71—81no—
164C4×C71#35312yescomplemented
165Dic71—164no—
166#12797—711nomaximal, Sylow
167C2×C71—21no—
168C2×C71—21no—
169C2×C71—42no—
170C2×C71—42no—
171C2×C71—82no—
172C2×C71#179911yescharacteristic
173C2×C71C2×#35311yescharacteristic
174C2×C71#212611yescharacteristic
175C4×#127D7111yescharacteristic
176C2×#353—712no—
177#1799—712no—
178#1393—712no—
179C71#1279711yescharacteristic, complemented, Sylow
180C2×#127C2×D7111yescharacteristic
181C2×C4×D4—22no—
182C2×C4×D4Dic7111yescharacteristic, complemented
183C2×C4×D4C2×D7111yescharacteristic
184C2×#99—714no—
185#390—22no—
186#390Dic7111yescharacteristic, complemented
187#390C2×D7111yescharacteristic
188#384Dic7111yescharacteristic, complemented
189#380—1422no—
190C43Dic7111yescharacteristic, complemented
191#353—1424no—
192#342—1422no—
193C22×D4—22no—
194C22×D4C2×Dic7111yescharacteristic
195C22×D4C22×D7111yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D7112yes—
198#127—22no—
199#127Dic14212yes—
200#127D28412yes—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—48no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—2842no—
208C2×C4⋊C4C2×Dic7111yescharacteristic
209C2×C4⋊C4C2×Dic7111yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1424no—
214C2×C22⋊C4C22⋊D7112yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C2×Dic7111yescharacteristic
218C2×C42C2×Dic7111yescharacteristic
219C2×C42C2×Dic7111yescharacteristic
220#99—1424no—
221#99—1424no—
222#95—2842no—
223C24—21no—
224C24C2×C22⋊D7112yes—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4C4⋊Dic7112yescomplemented
235C2×D4C2×D28411yescharacteristic
236C2×D4C2×Dic14211yescharacteristic
237C2×D4#1176586512yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—2844no—
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4C22⋊Dic7112yes—
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22×Dic7111yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—5684no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—2844no—
261C22⋊C4—2844no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—42no—
268C42C4⋊Dic7112yescomplemented
269C23—22no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊Dic7111yescharacteristic
279C23#1455610212yes—
280C23#1455610712yes—
281D4—41no—
282D4—41no—
283D4—42no—
284D4—42no—
285D4—48no—
286D4—48no—
287D4—42no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—5684no—
308C2×C4—5684no—
309C2×C4—5684no—
310C2×C4#1455604311yescharacteristic
311C2×C4#1455604311yescharacteristic
312C2×C4C2×C4⋊Dic7111yescharacteristic
313C2×C4#1455608711yescharacteristic
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1798585911yescharacteristic
329C22C2×#1455604311yescharacteristic
330C22#1798597311yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—81no—
339C4—11364no—
340C4#1798579212yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—42no—
345C2—82no—
346C2#2258832811yescharacteristic
347C2C2×#1798579211yescharacteristic
348C2#2259000511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×D4C22×D71
FittingC4×#127×C71C2
socleC22×C71C2×#99
prime radicalC2×#127×C71C22
cosocleC22×D4×C71C23
hypercenterC4×#127D71
hypocenterC71#12797
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. #22588328C2semidirect
3. #22586465C2semidirect
11. C2×C4×D4×C71C4semidirect
14. #390×C71C4semidirect
16. #384×C71C4semidirect
17. C43×C71C4semidirect
63. C2×D4×C71C4⋊C4semidirect
93. C42×C71C4⋊C4semidirect
164. C4×C71#353semidirect
179. C71#12797semidirect
182. C2×C4×D4Dic71semidirect
186. #390Dic71semidirect
188. #384Dic71semidirect
190. C43Dic71semidirect
234. C2×D4C4⋊Dic71semidirect
268. C42C4⋊Dic71semidirect
340. C4#17985792semidirect

Maximal subgroups

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

subgroupindex
C2×#179857922
#225883282
#225864652
C4×#127×C712
#1279771