Finite Group Databasebeta

← #31895620

Subgroups — #31895620

Order 18176.

356 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589059C211yesmaximal, characteristic
2#22588209C211yesmaximal, characteristic
3#22586181C211yesmaximal, characteristic
4#22586181C211yesmaximal, characteristic
5#22586175C211yesmaximal, characteristic
6C71⋊#1273C211yesmaximal, characteristic
7C4×#107×C71C211yesmaximal, characteristic
8C2×C4⋊(C71⋊C8)—21no—
9C2×C4⋊(C71⋊C8)C2211yescharacteristic
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×#14556039—21no—
12C2×C4×C71⋊C8C2211yescharacteristic
13C2×C4×C71⋊C8C2211yescharacteristic
14#17985754—21no—
15#17985754—21no—
16C2×#107×C71C411yescharacteristic
17C2×#107×C71C411yescharacteristic
18C2×C4×C8×C71C2211yescharacteristic
19C4×C4⋊C4×C71C2211yescharacteristic
20C4×C4⋊C4×C71C2211yescharacteristic
21#340×C71C411yescharacteristic
22#340×C71C411yescharacteristic
23C22×C71⋊C8—21no—
24C22×C71⋊C8—21no—
25C22×C71⋊C8—21no—
26C22×C71⋊C8D411yescharacteristic
27C22×C71⋊C8D411yescharacteristic
28C4⋊(C71⋊C8)—21no—
29C4⋊(C71⋊C8)—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)—42no—
32C4⋊(C71⋊C8)Q812yes—
33#14556039—42no—
34C4×C71⋊C8—21no—
35C4×C71⋊C8—21no—
36C4×C71⋊C8—21no—
37C4×C71⋊C8—21no—
38C22×C8×C71C2×C411yescharacteristic
39C22×C8×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C42×C71—21no—
45C2×C42×C71—21no—
46C2×C42×C71C2311yescharacteristic
47#107×C71—22no—
48#107×C71—22no—
49C4×C8×C71—21no—
50C4×C8×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53C2×C71⋊C8—21no—
54C2×C71⋊C8—21no—
55C2×C71⋊C8—22no—
56C2×C71⋊C8—22no—
57C2×C71⋊C8—21no—
58C2×C71⋊C8—21no—
59C2×C71⋊C8—21no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8Q1612yes—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71#4111yescharacteristic
75C2×C8×C71—21no—
76C2×C8×C71—21no—
77C2×C8×C71—21no—
78C2×C8×C71—21no—
79C2×C8×C71—21no—
80C2×C8×C71M1612yes—
81C4⋊C4×C71—21no—
82C4⋊C4×C71—21no—
83C4⋊C4×C71—21no—
84C4⋊C4×C71—21no—
85C4⋊C4×C71—21no—
86C4⋊C4×C71—21no—
87C4⋊C4×C71—21no—
88C4⋊C4×C71M1612yes—
89C42×C71—41no—
90C42×C71—41no—
91C42×C71—41no—
92C42×C71—41no—
93C42×C71C2×Q811yescharacteristic
94C42×C71#4111yescharacteristic
95C42×C71#4111yescharacteristic
96C42×C71#4111yescharacteristic
97C23×C71C4×D411yescharacteristic
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—22no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71#11711yescharacteristic
113C2×C4×C71C4×Q811yescharacteristic
114C2×C4×C71C22⋊Q811yescharacteristic
115C2×C4×C71#12311yescharacteristic
116C2×C4×C71#12511yescharacteristic
117C2×C4×C71#12611yescharacteristic
118C2×C4×C71C2×M1611yescharacteristic
119C2×C4×C71C2×M1611yescharacteristic
120C2×C4×C71#13111yescharacteristic
121C2×C4×C71#13111yescharacteristic
122C2×C4×C71C2×Q1611yescharacteristic
123C2×C4×C71#13511yescharacteristic
124C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
126C8×C71—22no—
127C8×C71—41no—
128C71⋊C8—22no—
129C71⋊C8—42no—
130C71⋊C8—42no—
131C71⋊C8—41no—
132C71⋊C8—81no—
133C22×C71#38711yescharacteristic
134C22×C71#43511yescharacteristic
135C22×C71#43611yescharacteristic
136C22×C71C4×Q1611yescharacteristic
137C22×C71#44211yescharacteristic
138C22×C71#44311yescharacteristic
139C22×C71#44411yescharacteristic
140C4×C71—21no—
141C4×C71—21no—
142C4×C71—42no—
143C4×C71—81no—
144C4×C71—81no—
145C4×C71—81no—
146C4×C71#43211yescharacteristic
147C4×C71#43311yescharacteristic
148C4×C71#44611yescharacteristic
149C4×C71#44611yescharacteristic
150C4×C71#47711yescharacteristic
151C4×C71#47811yescharacteristic
152C4×C71#48211yescharacteristic
153C4×C71#48411yescharacteristic
154#9540—711nomaximal, Sylow
155C2×C71#158411yescharacteristic
156C2×C71#158611yescharacteristic
157C2×C71#159111yescharacteristic
158C2×C71#159411yescharacteristic
159C2×C71#191711yescharacteristic
160C2×C71#192111yescharacteristic
161C2×C71#192211yescharacteristic
162#1939—711no—
163C4×#107D7111yescharacteristic
164#1770—711no—
165#1279—711no—
166#1279—711no—
167#1277—711no—
168#1273—711no—
169C71#954011yescharacteristic, complemented, Sylow
170C2×#107Dic7111yescharacteristic
171C2×#107Dic7111yescharacteristic
172C2×C4⋊C8—711no—
173C2×C4⋊C8—711no—
174C2×C4⋊C8—1421no—
175C2×#97—1421no—
176C2×C4×C8—711no—
177C2×C4×C8—711no—
178C2×C4×C8C2×D7111yescharacteristic
179C4×C4⋊C4C2×D7111yescharacteristic
180C4×C4⋊C4C2×D7111yescharacteristic
181#340Dic7111yescharacteristic
182#340Dic7111yescharacteristic
183#336—1421no—
184#336—1421no—
185C22×C8—711no—
186C22×C8—711no—
187C22×C8—1421no—
188C22×C8—1421no—
189C22×C8—1421no—
190C22×C8C2×Dic7111yescharacteristic
191C22×C8C2×Dic7111yescharacteristic
192C2×C4⋊C4C2×Dic7111yescharacteristic
193C2×C4⋊C4C2×Dic7111yescharacteristic
194C2×C4⋊C4C2×Dic7111yescharacteristic
195C2×C4⋊C4C2×Dic7111yescharacteristic
196C2×C42—21no—
197C2×C42—21no—
198C2×C42C22×D7111yescharacteristic
199#107—22no—
200#107—22no—
201C4⋊C8—2842no—
202C4⋊C8—712no—
203C4⋊C8—1421no—
204C4⋊C8—1421no—
205C4⋊C8—1421no—
206#97—2842no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—1421no—
210C4×C8—1421no—
211C4×C8—1421no—
212C4×C8—1421no—
213#95—21no—
214#95—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4C22×Dic7111yescharacteristic
220C22×C4D4×D7111yescharacteristic
221C22×C4#1176586411yescharacteristic
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—2841no—
228C2×C8—2841no—
229C2×C8—2841no—
230C2×C8—2841no—
231C2×C8—2841no—
232C2×C8—2841no—
233C2×C8—2841no—
234C2×C8—712no—
235C2×C8—1422no—
236C2×C8—1422no—
237C2×C8—1421no—
238C2×C8—1421no—
239C2×C8—1421no—
240C2×C8—1421no—
241C2×C8—1421no—
242C2×C8#1176584912yes—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4#1176584912yes—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42Q8×D7111yescharacteristic
256C42#1176586511yescharacteristic
257C42#1176586511yescharacteristic
258C42#1176586611yescharacteristic
259C23D4×Dic7111yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4#1455608711yescharacteristic
275C2×C4Q8×Dic7111yescharacteristic
276C2×C4#1455610611yescharacteristic
277C2×C4#1455611311yescharacteristic
278C2×C4#1455611811yescharacteristic
279C2×C4#1455612311yescharacteristic
280C2×C4C2×#1176584911yescharacteristic
281C2×C4C2×#1176584911yescharacteristic
282C2×C4#1455613511yescharacteristic
283C2×C4#1455613511yescharacteristic
284C2×C4Q16×D7111yescharacteristic
285C2×C4#1455615111yescharacteristic
286C2×C4#1455615211yescharacteristic
287C2×C4#1455615211yescharacteristic
288C8—22no—
289C8—41no—
290C8—2842no—
291C8—2841no—
292C8—2842no—
293C8—5681no—
294C8—1422no—
295C22#1798588511yescharacteristic
296C22#1798607111yescharacteristic
297C22#1798607711yescharacteristic
298C22Q16×Dic7111yescharacteristic
299C22#1798611111yescharacteristic
300C22#1798611611yescharacteristic
301C22#1798612111yescharacteristic
302C4—21no—
303C4—21no—
304C4—42no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4#1798605611yescharacteristic
309C4#1798606011yescharacteristic
310C4#1798613011yescharacteristic
311C4#1798613011yescharacteristic
312C4#1798629311yescharacteristic
313C4#1798630211yescharacteristic
314C4#1798632611yescharacteristic
315C4#1798633611yescharacteristic
316C2#2258719111yescharacteristic
317C2#2258720511yescharacteristic
318C2#2258723711yescharacteristic
319C2#2258725411yescharacteristic
320C2#2258890611yescharacteristic
321C2#2258893411yescharacteristic
322C2#2258894111yescharacteristic
subgroupG/H
centerC22×C4D4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×#107×C71C2
socle, prime radicalC23×C71C4×D4
cosocleC2×C42×C71C23
hypercenterC4×#107D71
hypocenterC71#9540
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
169. C71#9540semidirect

Maximal subgroups

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

subgroupindex
#225890592
#225882092
#225861812
#225861752
C71⋊#12732
C4×#107×C712
#954071