Finite Group Databasebeta

← #31891063

Subgroups — #31891063

Order 18176.

446 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4×#14556045C211yesmaximal, characteristic
2#22586681C211yesmaximal, characteristic
3#22586233C212yesmaximal, complemented
4#22586204C212yesmaximal, complemented
5C71×#2353C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×#14556045—21no—
8C4×#11765849C2212yes—
9#17985768—21no—
10#17985723—21no—
11#17985723—21no—
12C2×C4×D4×C71C2211yescharacteristic
13#388×C71—21no—
14#387×C71—21no—
15#386×C71—21no—
16#383×C71C2211yescharacteristic
17#381×C71C411yescharacteristic
18#380×C71C411yescharacteristic
19C4×C22⋊C4×C71—21no—
20C4×C22⋊C4×C71C411yescharacteristic
21#376×C71C2211yescharacteristic
22C2×#95×C71C411yescharacteristic
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—22no—
25#14556045—24no—
26C22⋊(C71⋊C8)—22no—
27#14556039—22no—
28#14556039—22no—
29C4×C71⋊C8—22no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71—21no—
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71C2×C411yescharacteristic
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71D411yescharacteristic
43C2×C42×C71—21no—
44C2×C42×C71—21no—
45C2×C42×C71C2311yescharacteristic
46#95×C71—21no—
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71C2×C411yescharacteristic
52#95×C71C2×C411yescharacteristic
53#95×C71C2×C411yescharacteristic
54#95×C71C2×C411yescharacteristic
55#11765849—24no—
56#11765849—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C24×C71C22⋊C411yescharacteristic
60C24×C71C22⋊C411yescharacteristic
61C2×D4×C71—22no—
62C2×D4×C71—22no—
63C2×D4×C71—41no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—42no—
73C22×C4×C71—42no—
74C22×C4×C71—81no—
75C22×C4×C71—81no—
76C22×C4×C71—81no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—22no—
83C4⋊C4×C71—81no—
84C22⋊C4×C71—22no—
85C22⋊C4×C71—22no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—81no—
89C42×C71—42no—
90C42×C71—81no—
91C42×C71#4112yes—
92C42×C71#4112yes—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71#10412yes—
103C23×C71C2×C22⋊C411yescharacteristic
104D4×C71—44no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—81no—
126C2×C4×C71—81no—
127C2×C4×C71—81no—
128C2×C4×C71—81no—
129C2×C4×C71—84no—
130C2×C4×C71#11711yescharacteristic
131C2×C4×C71#11711yescharacteristic
132C2×C4×C71#12312yes—
133C2×C4×C71#12312yes—
134C71⋊C8—44no—
135C71⋊C8—82no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—22no—
139C22×C71—42no—
140C22×C71—42no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71—81no—
147C22×C71—81no—
148C22×C71#38111yescharacteristic
149C22×C71C2×#10411yescharacteristic
150C22×C71#42111yescharacteristic
151C4×C71—22no—
152C4×C71—22no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—42no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71—81no—
159C4×C71—81no—
160C4×C71—81no—
161#8773—711nomaximal, Sylow
162C2×C71—21no—
163C2×C71—21no—
164C2×C71—42no—
165C2×C71—81no—
166C2×C71#182611yescharacteristic
167C2×C71#182711yescharacteristic
168C2×C71#182911yescharacteristic
169#2353D7111yescharacteristic
170C4×#100—711no—
171#1466—711no—
172#1300—712no—
173#1288—712no—
174C71#877311yescharacteristic, complemented, Sylow
175C2×C4×D4C2×D7111yescharacteristic
176#432—712no—
177C2×#100—1421no—
178C4×M16—712no—
179#388—21no—
180#387—21no—
181#386—21no—
182#383C2×D7111yescharacteristic
183#381Dic7111yescharacteristic
184#380Dic7111yescharacteristic
185C4×C22⋊C4—21no—
186C4×C22⋊C4Dic7111yescharacteristic
187#376C2×D7111yescharacteristic
188C2×#95Dic7111yescharacteristic
189#343—1421no—
190#323—1421no—
191#323—1421no—
192C22×D4C22⋊D7111yescharacteristic
193C23×C4—21no—
194C23×C4C2×Dic7111yescharacteristic
195C23×C4C2×Dic7111yescharacteristic
196C2×M16—1422no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22⋊D7111yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C22⋊D7111yescharacteristic
205C2×C22⋊C4C22⋊D7111yescharacteristic
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22×D7111yescharacteristic
209C4⋊C8—1422no—
210#100—1424no—
211C22⋊C8—1422no—
212#97—1422no—
213#97—1422no—
214C4×C8—1422no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C2×Dic7111yescharacteristic
221#95C2×Dic7111yescharacteristic
222#95C2×Dic7111yescharacteristic
223#95C2×Dic7111yescharacteristic
224C24C22⋊Dic7111yescharacteristic
225C24C22⋊Dic7111yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C22⋊Dic7111yescharacteristic
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22×Dic7111yescharacteristic
245C22×C4C2×C22⋊D7111yescharacteristic
246C22×C4C2×C22⋊D7111yescharacteristic
247M16—2842no—
248M16—1424no—
249C2×C8—2842no—
250C2×C8—2842no—
251C4⋊C4—22no—
252C4⋊C4—81no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—81no—
258C42—42no—
259C42—81no—
260C42#1176586512yes—
261C42#1176586512yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1455605512yes—
272C23C2×C22⋊Dic7111yescharacteristic
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—84no—
299C2×C4#1455608711yescharacteristic
300C2×C4#1455608711yescharacteristic
301C2×C4#1455611112yes—
302C2×C4#1455611112yes—
303C8—2844no—
304C8—5682no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22#1798586211yescharacteristic
318C22C2×#1455605511yescharacteristic
319C22#1798601611yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—81no—
334C2#2258843911yescharacteristic
335C2#2258844411yescharacteristic
336C2#2258845111yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2353C2
socleC22×C71#381
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2353D71
hypocenterC71#8773
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #22586233C2semidirect
4. #22586204C2semidirect
174. C71#8773semidirect

Maximal subgroups

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

subgroupindex
C4×#145560452
#225866812
#225862332
#225862042
C71×#23532
#877371