Finite Group Databasebeta

← #31890604

Subgroups — #31890604

Order 18176.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986077C211yesmaximal, characteristic
2#22588477C211yesmaximal, characteristic, complemented
3#22587086C211yesmaximal, characteristic, complemented
4#22586765C211yesmaximal, characteristic
5#22586236C211yesmaximal, characteristic, complemented
6#22586235C211yesmaximal, characteristic, complemented
7C71×#1486C211yesmaximal, characteristic
8C22×#11765849—21no—
9#17986077—21no—
10#17986077—21no—
11#17986077C412yes—
12#17986077C412yes—
13#17986052C2211yescharacteristic
14#17986023C2211yescharacteristic
15C2×C4⋊(C71⋊C8)C2211yescharacteristic
16C2×C22⋊(C71⋊C8)—21no—
17C2×C4×C71⋊C8C2211yescharacteristic
18#17985763—21no—
19#17985763—21no—
20#17985751—21no—
21#17985749—21no—
22D4⋊(C71⋊C8)—22no—
23C2×C4×D4×C71C2211yescharacteristic
24#431×C71C2211yescharacteristic
25#423×C71C2211yescharacteristic
26D4⋊C8×C71—22no—
27C2×#11765849—22no—
28C2×#11765849—22no—
29C2×#11765849—21no—
30C2×#11765849—42no—
31C2×#11765849—42no—
32C22×C71⋊C8—21no—
33C22×C71⋊C8D411yescharacteristic
34C22×C71⋊C8D411yescharacteristic
35C4⋊(C71⋊C8)—21no—
36C4⋊(C71⋊C8)—21no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)C2×C412yes—
40C22⋊(C71⋊C8)—22no—
41C22⋊(C71⋊C8)—22no—
42C22⋊(C71⋊C8)—21no—
43C4×C71⋊C8—21no—
44C4×C71⋊C8C2×C411yescharacteristic
45C4×C71⋊C8C2×C411yescharacteristic
46C4×C71⋊C8D411yescharacteristic
47C4×C71⋊C8D411yescharacteristic
48C22×D4×C71Q811yescharacteristic
49C23×C4×C71—21no—
50C2×M16×C71—21no—
51C4×D4×C71—21no—
52C4×D4×C71—21no—
53C4×D4×C71C2×C412yes—
54C4×D4×C71D412yes—
55C2×C4⋊C4×C71Q811yescharacteristic
56C2×C22⋊C4×C71—21no—
57C2×C42×C71C2311yescharacteristic
58C4⋊C8×C71—21no—
59C4⋊C8×C71—21no—
60C4⋊C8×C71—21no—
61C22⋊C8×C71—21no—
62C4×C8×C71—21no—
63#11765849—44no—
64#11765849—44no—
65#11765849—41no—
66#11765849—41no—
67C2×C71⋊C8—21no—
68C2×C71⋊C8—21no—
69C2×C71⋊C8—21no—
70C2×C71⋊C8—21no—
71C2×C71⋊C8—22no—
72C2×C71⋊C8—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8—41no—
75C2×C71⋊C8—41no—
76C2×C71⋊C8C22⋊C411yescharacteristic
77C2×C71⋊C8C22⋊C411yescharacteristic
78C2×C71⋊C8C22⋊C411yescharacteristic
79C2×C71⋊C8C22⋊C411yescharacteristic
80C24×C71—21no—
81C2×D4×C71—21no—
82C2×D4×C71—42no—
83C2×D4×C71C4⋊C412yes—
84C22×C4×C71—21no—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71—41no—
90C22×C4×C71—41no—
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×Q811yescharacteristic
93C22×C4×C71#4111yescharacteristic
94M16×C71—41no—
95M16×C71—41no—
96C2×C8×C71—41no—
97C2×C8×C71—41no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71C4⋊C412yes—
100C22⋊C4×C71—22no—
101C22⋊C4×C71—22no—
102C42×C71C22×C411yescharacteristic
103C42×C71C2×D411yescharacteristic
104C42×C71C2×D411yescharacteristic
105C42×C71#4111yescharacteristic
106C23×C71—22no—
107C23×C71—42no—
108C23×C71—42no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71C22⋊Q811yescharacteristic
112D4×C71—24no—
113D4×C71—42no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71C2×C22⋊C411yescharacteristic
124C2×C4×C71C2×C4⋊C411yescharacteristic
125C2×C4×C71#11711yescharacteristic
126C2×C4×C71#12111yescharacteristic
127C2×C4×C71#12111yescharacteristic
128C2×C4×C71C22⋊Q811yescharacteristic
129C2×C4×C71#13511yescharacteristic
130C2×C4×C71#13511yescharacteristic
131C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
133C8×C71—81no—
134C8×C71—81no—
135C71⋊C8—21no—
136C71⋊C8—21no—
137C71⋊C8—21no—
138C71⋊C8—21no—
139C71⋊C8—42no—
140C71⋊C8—81no—
141C71⋊C8—81no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—42no—
145C22×C71—42no—
146C22×C71—42no—
147C22×C71—82no—
148C22×C71—82no—
149C22×C71#38011yescharacteristic
150C22×C71#47811yescharacteristic
151C22×C71#47811yescharacteristic
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—42no—
156C4×C71#41611yescharacteristic
157C4×C71#41811yescharacteristic
158C4×C71#43012yes—
159C4×C71#46311yescharacteristic
160C4×C71#46311yescharacteristic
161#8664—711nomaximal, Sylow
162C2×C71—21no—
163C2×C71—21no—
164C2×C71—82no—
165C2×C71#180411yescharacteristic
166C2×C71#181111yescharacteristic
167C2×C71#181411yescharacteristic
168C2×#436—711no—
169#1837—711no—
170#1564—711no—
171#1486—711no—
172#1486D7111yescharacteristic
173#1301—711no—
174#1301—711no—
175C71#866411yescharacteristic, complemented, Sylow
176C22×M16—1421no—
177C2×C4×D4C2×D7111yescharacteristic
178#436—712no—
179#436—712no—
180#436—1421no—
181#436—1421no—
182#431—711no—
183#431C2×D7111yescharacteristic
184#423—711no—
185#423C2×D7111yescharacteristic
186C2×C4⋊C8—711no—
187C2×C22⋊C8—1421no—
188C2×C4×C8—711no—
189#341—1421no—
190#341—1421no—
191#335—1421no—
192#334—1421no—
193D4⋊C8—22no—
194D4⋊C8—1422no—
195C22×D4Dic14211yescharacteristic
196C23×C4—21no—
197C2×M16—21no—
198C2×M16—2842no—
199C2×M16—2842no—
200C2×M16—1421no—
201C2×M16—1422no—
202C2×M16—1422no—
203C22×C8—711no—
204C22×C8—711no—
205C22×C8—1421no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C4×D7112yes—
209C4×D4C22⋊D7112yes—
210C2×C4⋊C4Dic14211yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D7111yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—712no—
217C4⋊C8—1421no—
218C4⋊C8—1421no—
219C4⋊C8—1421no—
220C4⋊C8—1421no—
221C22⋊C8—21no—
222C22⋊C8—1421no—
223C22⋊C8—1422no—
224C22⋊C8—1422no—
225C4×C8—21no—
226C4×C8—711no—
227C4×C8—711no—
228C4×C8—711no—
229C4×C8—711no—
230C4×C8—1421no—
231C24—21no—
232C2×D4—21no—
233C2×D4—42no—
234C2×D4Dic71⋊C412yes—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4D4×D7111yescharacteristic
243C22×C4C2×Dic14211yescharacteristic
244C22×C4#1176586411yescharacteristic
245M16—41no—
246M16—41no—
247M16—2844no—
248M16—2844no—
249M16—2841no—
250M16—2841no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—2841no—
254C2×C8—2841no—
255C2×C8—2841no—
256C2×C8—2841no—
257C2×C8—711no—
258C2×C8—711no—
259C2×C8—711no—
260C2×C8—711no—
261C2×C8—1422no—
262C2×C8—1421no—
263C2×C8—1421no—
264C2×C8—1421no—
265C2×C8—1421no—
266C4⋊C4—21no—
267C4⋊C4Dic71⋊C412yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42C2×C4×D7111yescharacteristic
271C42D4×D7111yescharacteristic
272C42C2×C22⋊D7111yescharacteristic
273C42#1176586411yescharacteristic
274C23—22no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23#1455610511yescharacteristic
280D4—24no—
281D4—42no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4C22⋊C4×D7111yescharacteristic
292C2×C4C2×Dic71⋊C411yescharacteristic
293C2×C4#1455608411yescharacteristic
294C2×C4#1455610011yescharacteristic
295C2×C4#1455610011yescharacteristic
296C2×C4#1455610511yescharacteristic
297C2×C4#1455614711yescharacteristic
298C2×C4#1455614911yescharacteristic
299C2×C4#1455615211yescharacteristic
300C2×C4#1455615511yescharacteristic
301C8—81no—
302C8—81no—
303C8—2842no—
304C8—5681no—
305C8—5681no—
306C8—1421no—
307C8—1421no—
308C8—1421no—
309C8—1421no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—82no—
316C22—82no—
317C22#1798585711yescharacteristic
318C22#1798629711yescharacteristic
319C22#1798630111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4#1798599211yescharacteristic
325C4#1798600211yescharacteristic
326C4#1798604712yes—
327C4#1798621211yescharacteristic
328C4#1798621611yescharacteristic
329C2—21no—
330C2—21no—
331C2—82no—
332C2#2258834811yescharacteristic
333C2#2258838011yescharacteristic
334C2#2258839211yescharacteristic
subgroupG/H
centerC2×C4#14556100
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1486C2
socleC22×C71#380
prime radicalC22×D4×C71Q8
cosocleC2×C42×C71C23
hypercenter#1486D71
hypocenterC71#8664
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #22588477C2semidirect
3. #22587086C2semidirect
5. #22586236C2semidirect
6. #22586235C2semidirect
175. C71#8664semidirect

Maximal subgroups

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

subgroupindex
C2×#179860772
#225884772
#225870862
#225867652
#225862362
#225862352
C71×#14862
#866471