Finite Group Databasebeta

← #31965904

Subgroups — #31965904

Order 18176.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597451C211yesmaximal, characteristic, complemented
2#22596908C211yesmaximal, characteristic, complemented
3#22587017C213yesmaximal, complemented
4#22587012C213yesmaximal, complemented
5#22586989C213yesmaximal, complemented
6C2×#17985745C213yesmaximal, complemented
7C71×#3499C211yesmaximal, characteristic
8C2×#14556135—21no—
9Q8×C71⋊C8—21no—
10D4×C71⋊C8—23no—
11#17986052C2213yes—
12#17986026—23no—
13#17986023C2213yes—
14C2×C4⋊(C71⋊C8)C2213yes—
15C2×C4⋊(C71⋊C8)C2213yes—
16#17985745C2213yes—
17#17985745C2216yes—
18#17985745C2213yes—
19#17985743C2211yescharacteristic
20#17985743C2213yes—
21C2×2−1+4×C71—21no—
22D4×Q8×C71—21no—
23D4×Q8×C71C412yes—
24#547×C71—23no—
25#547×C71C416yes—
26#536×C71—23no—
27#532×C71C2213yes—
28C2×C4⋊Q8×C71C2213yes—
29C4×#41×C71C2211yescharacteristic
30#14556135—44no—
31C2×#11765849—23no—
32C22×C71⋊C8—23no—
33C22×C71⋊C8—23no—
34C4⋊(C71⋊C8)—23no—
35C4⋊(C71⋊C8)—23no—
36C4⋊(C71⋊C8)—23no—
37C4⋊(C71⋊C8)C2311yescharacteristic
38C4⋊(C71⋊C8)C2313yes—
39C4⋊(C71⋊C8)C2311yescharacteristic
40C4⋊(C71⋊C8)C2313yes—
41C22⋊(C71⋊C8)—23no—
42C4×C71⋊C8—23no—
432−1+4×C71—44no—
442−1+4×C71—44no—
45C2×#41×C71—26no—
46C2×#41×C71—23no—
47C2×#41×C71D411yescharacteristic
48C22×Q8×C71—22no—
49C22×Q8×C71—23no—
50C4⋊Q8×C71—23no—
51C4⋊Q8×C71—23no—
52C4⋊Q8×C71C2×C416yes—
53C4⋊Q8×C71C2313yes—
54#127×C71C2311yescharacteristic
55#124×C71—23no—
56#123×C71—212no—
57C22⋊Q8×C71—26no—
58C22⋊Q8×C71—26no—
59#121×C71—23no—
60#121×C71C2×C416yes—
61C4×Q8×C71C2×C411yescharacteristic
62C4×Q8×C71D411yescharacteristic
63C4×D4×C71—23no—
64C4×D4×C71C2×C416yes—
65C4×D4×C71C2×C413yes—
66C4×D4×C71D413yes—
67#117×C71D413yes—
68C2×C4⋊C4×C71—23no—
69C2×C4⋊C4×C71C2×C416yes—
70C2×C42×C71C2313yes—
71#11765849—46no—
72C2×C71⋊C8—21no—
73C2×C71⋊C8—23no—
74C2×C71⋊C8—21no—
75C2×C71⋊C8—23no—
76C2×C71⋊C8—46no—
77C2×C71⋊C8—83no—
78#41×C71—424no—
79#41×C71—46no—
80#41×C71—46no—
81#41×C71D814yes—
82#41×C71SD1614yes—
83C2×Q8×C71—26no—
84C2×Q8×C71—23no—
85C2×Q8×C71—48no—
86C2×Q8×C71—46no—
87C2×Q8×C71—46no—
88C2×Q8×C71C22⋊C411yescharacteristic
89C2×D4×C71—26no—
90C2×D4×C71—21no—
91C2×D4×C71—21no—
92C2×D4×C71C22⋊C413yes—
93C2×D4×C71C22×C412yes—
94C22×C4×C71—26no—
95C22×C4×C71—26no—
96C22×C4×C71C22×C413yes—
97C22×C4×C71C2×D413yes—
98C22×C4×C71C2×D413yes—
99C4⋊C4×C71—26no—
100C4⋊C4×C71—26no—
101C4⋊C4×C71—23no—
102C4⋊C4×C71C22⋊C413yes—
103C4⋊C4×C71C22⋊C413yes—
104C4⋊C4×C71C22×C416yes—
105C22⋊C4×C71—26no—
106C22⋊C4×C71—26no—
107C22⋊C4×C71C22⋊C416yes—
108C42×C71C22×C413yes—
109C42×C71C2×D413yes—
110C42×C71C2×D413yes—
111C42×C71C2411yescharacteristic
112C23×C71—22no—
113C23×C71C2×C22⋊C413yes—
114Q8×C71—43no—
115Q8×C71—43no—
116Q8×C71—412no—
117Q8×C71#10214yes—
118D4×C71—42no—
119D4×C71—412no—
120D4×C71—41no—
121D4×C71—41no—
122D4×C71—41no—
123D4×C71—41no—
124D4×C71#102112yes—
125C2×C4×C71—23no—
126C2×C4×C71—23no—
127C2×C4×C71—26no—
128C2×C4×C71—43no—
129C2×C4×C71—412no—
130C2×C4×C71—412no—
131C2×C4×C71C2×C22⋊C413yes—
132C2×C4×C71C2×C22⋊C413yes—
133C2×C4×C71C2×C22⋊C413yes—
134C2×C4×C71C2×D816yes—
135C2×C4×C71C2×SD1616yes—
136C2×C4×C71C23×C411yescharacteristic
137C2×C4×C71C22×D411yescharacteristic
138C2×C4×C71C22×D411yescharacteristic
139C71⋊C8—41no—
140C71⋊C8—43no—
141C71⋊C8—43no—
142C71⋊C8—41no—
143C22×C71—23no—
144C22×C71—23no—
145C22×C71—42no—
146C22×C71—42no—
147C22×C71—42no—
148C22×C71C2×#10216yes—
149C22×C71C22×C22⋊C411yescharacteristic
150C4×C71—21no—
151C4×C71—23no—
152C4×C71—23no—
153C4×C71—46no—
154C4×C71C2×#10216yes—
155C4×C71C22×D811yescharacteristic
156C4×C71C22×SD1611yescharacteristic
157C4×C71#57711yescharacteristic
158C4×C71#57811yescharacteristic
159#17721—711nomaximal, Sylow
160C2×C71—26no—
161C2×C71—82no—
162C2×C71#288911yescharacteristic
163C2×C71C22×#10211yescharacteristic
164C2×C71#289811yescharacteristic
165#3499D7111yescharacteristic
166#2988—711no—
167#2906—711no—
168#1547—713no—
169#1546—713no—
170#1540—713no—
171C2×#332—713no—
172C71#1772111yescharacteristic, complemented, Sylow
173C2×2−1+4—21no—
174C2×#131—1421no—
175D4×Q8—21no—
176D4×Q8Dic7112yes—
177#547—23no—
178#547Dic7116yes—
179#536—23no—
180#532C2×D7113yes—
181C2×C4⋊Q8C2×D7113yes—
182C4×#41C2×D7111yescharacteristic
183C8×Q8—1421no—
184C8×D4—1423no—
185#431—713no—
186#424—1423no—
187#423—713no—
188C2×C4⋊C8—713no—
189C2×C4⋊C8—713no—
190#332—713no—
191#332—716no—
192#332—713no—
193#331—711no—
194#331—713no—
1952−1+4—44no—
1962−1+4—44no—
197C2×#41—26no—
198C2×#41—23no—
199C2×#41C22⋊D7111yescharacteristic
200C22×Q8—22no—
201C22×Q8—23no—
202#131—2844no—
203C2×M16—1423no—
204C22×C8—1423no—
205C22×C8—1423no—
206C4⋊Q8—23no—
207C4⋊Q8—23no—
208C4⋊Q8C2×Dic7116yes—
209C4⋊Q8C22×D7113yes—
210#127C22×D7111yescharacteristic
211#124—23no—
212#123—212no—
213C22⋊Q8—26no—
214C22⋊Q8—26no—
215#121—23no—
216#121C2×Dic7116yes—
217C4×Q8C2×Dic7111yescharacteristic
218C4×Q8C22⋊D7111yescharacteristic
219C4×D4—23no—
220C4×D4C2×Dic7116yes—
221C4×D4C2×Dic7113yes—
222C4×D4C22⋊D7113yes—
223#117C22⋊D7113yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4C2×Dic7116yes—
226C2×C42C22×D7113yes—
227C4⋊C8—711no—
228C4⋊C8—713no—
229C4⋊C8—713no—
230C4⋊C8—711no—
231C4⋊C8—1423no—
232C4⋊C8—1423no—
233C4⋊C8—1423no—
234C22⋊C8—1423no—
235C4×C8—1423no—
236#41—424no—
237#41—46no—
238#41—46no—
239#41#1176585014yes—
240#41#1176585314yes—
241C2×Q8—23no—
242C2×Q8—26no—
243C2×Q8—48no—
244C2×Q8—46no—
245C2×Q8—46no—
246C2×Q8C22⋊Dic7111yescharacteristic
247C2×D4—26no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4C22⋊Dic7113yes—
251C2×D4C22×Dic7112yes—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4C22×Dic7113yes—
255C22×C4C2×C22⋊D7113yes—
256C22×C4C2×C22⋊D7113yes—
257M16—2846no—
258C2×C8—2846no—
259C2×C8—5683no—
260C2×C8—1421no—
261C2×C8—1423no—
262C2×C8—1421no—
263C2×C8—1423no—
264C4⋊C4—23no—
265C4⋊C4—26no—
266C4⋊C4—26no—
267C4⋊C4C22⋊Dic7113yes—
268C4⋊C4C22⋊Dic7113yes—
269C4⋊C4C22×Dic7116yes—
270C22⋊C4—26no—
271C22⋊C4—26no—
272C22⋊C4C22⋊Dic7116yes—
273C42C22×Dic7113yes—
274C42C2×C22⋊D7113yes—
275C42C2×C22⋊D7113yes—
276C42C23×D7111yescharacteristic
277C23—22no—
278C23C2×C22⋊Dic7113yes—
279Q8—412no—
280Q8—43no—
281Q8—43no—
282Q8#1455604914yes—
283D4—42no—
284D4—412no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4#14556049112yes—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—26no—
293C2×C4—43no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C2×C22⋊Dic7113yes—
297C2×C4C2×C22⋊Dic7113yes—
298C2×C4C2×C22⋊Dic7113yes—
299C2×C4C2×#1176585016yes—
300C2×C4C2×#1176585316yes—
301C2×C4C23×Dic7111yescharacteristic
302C2×C4C22×C22⋊D7111yescharacteristic
303C2×C4C22×C22⋊D7111yescharacteristic
304C8—2841no—
305C8—2843no—
306C8—2843no—
307C8—2841no—
308C22—23no—
309C22—23no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#1455604916yes—
314C22C22×C22⋊Dic7111yescharacteristic
315C4—23no—
316C4—21no—
317C4—23no—
318C4—46no—
319C4C2×#1455604916yes—
320C4C22×#1176585011yescharacteristic
321C4C22×#1176585311yescharacteristic
322C4#1798680711yescharacteristic
323C4#1798681411yescharacteristic
324C2—26no—
325C2—82no—
326C2#2259682311yescharacteristic
327C2C22×#1455604911yescharacteristic
328C2#2259687311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
Fitting, prime radicalC71×#3499C2
socleC22×C71C22×C22⋊C4
cosocleC42×C71C24
hypercenter#3499D71
hypocenterC71#17721
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #22597451C2semidirect
2. #22596908C2semidirect
3. #22587017C2semidirect
4. #22587012C2semidirect
5. #22586989C2semidirect
6. C2×#17985745C2semidirect
172. C71#17721semidirect

Maximal subgroups

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

subgroupindex
#225974512
#225969082
#225870172
#225870122
#225869892
C2×#179857452
C71×#34992
#1772171