Finite Group Databasebeta

← #31916623

Subgroups — #31916623

Order 18176.

446 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986241C211yesmaximal, characteristic
2C2×#17985804C211yesmaximal, characteristic, complemented
3C2×#17985802C211yesmaximal, characteristic, complemented
4C2×(C2×D71)⋊C16C211yesmaximal, characteristic
5#22588366C211yesmaximal, characteristic
6#22586431C211yesmaximal, characteristic, complemented
7C2×#367×C71C211yesmaximal, characteristic, complemented
8C22×Dic284C2211yescharacteristic
9C22×C8×D71C2211yescharacteristic
10C2×#14556108—21no—
11C22×C71⋊C16C2211yescharacteristic
12#17986241—22no—
13#17986241—22no—
14C2×#14556062C2211yescharacteristic
15C2×#14556051—21no—
16#17985856—21no—
17#17985804—22no—
18#17985802—22no—
19(C2×D71)⋊C16—22no—
20#17985761—21no—
21C22×C16×C71C2211yescharacteristic
22C2×#107×C71C2211yescharacteristic
23C2×#107×C71C2211yescharacteristic
24#367×C71—22no—
25C22×Dic142—21no—
26C22×C4×D71D411yescharacteristic
27C2×Dic284—21no—
28C2×Dic284—21no—
29C2×Dic284—21no—
30C2×Dic284—21no—
31C2×Dic284—21no—
32C2×Dic284D412yes—
33C22×C71⋊C8D411yescharacteristic
34C2×C8×D71—21no—
35C2×C8×D71—21no—
36C2×C8×D71—21no—
37C2×C8×D71—21no—
38C2×C8×D71—21no—
39C2×C8×D71—21no—
40#14556108—24no—
41#14556108—42no—
42C2×C4⋊Dic71—21no—
43C2×C4⋊Dic71—41no—
44C2×(C2×D71)⋊C4—41no—
45C2×(C2×D71)⋊C4—41no—
46C2×C71⋊C16—21no—
47C2×C71⋊C16—21no—
48C2×C71⋊C16—21no—
49#14556062—21no—
50#14556062—21no—
51#14556051—24no—
52C71⋊#95—41no—
53C22×C8×C71C2311yescharacteristic
54C2×C4⋊C4×C71—21no—
55C2×C4⋊C4×C71—21no—
56C2×C16×C71—21no—
57C2×C16×C71—21no—
58C2×C16×C71D412yes—
59#107×C71—21no—
60#107×C71—21no—
61#107×C71—21no—
62#107×C71D412yes—
63C23×D71D811yescharacteristic
64C2×Dic142—22no—
65C2×Dic142—41no—
66C2×Dic142—41no—
67C2×Dic142—41no—
68C2×Dic142—41no—
69C2×Dic142—41no—
70C22×Dic71—41no—
71C22×Dic71—41no—
72C22×Dic71D811yescharacteristic
73C2×C4×D71—21no—
74C2×C4×D71—21no—
75C2×C4×D71—21no—
76C2×C4×D71—21no—
77C2×C4×D71—21no—
78C2×C4×D71—21no—
79Dic284—22no—
80Dic284—22no—
81Dic284—41no—
82Dic284—41no—
83C2×C71⋊C8—21no—
84C2×C71⋊C8—21no—
85C2×C71⋊C8—21no—
86C8×D71—41no—
87C8×D71—41no—
88C8×D71—41no—
89C8×D71—41no—
90C4⋊Dic71—44no—
91C4⋊Dic71—41no—
92C4⋊Dic71—41no—
93(C2×D71)⋊C4—44no—
94(C2×D71)⋊C4—82no—
95C71⋊C16—41no—
96C22×C4×C71—41no—
97C22×C4×C71—41no—
98C22×C4×C71C2×D411yescharacteristic
99C2×C8×C71C2×D411yescharacteristic
100C2×C8×C71C2×D411yescharacteristic
101C2×C8×C71C2×D411yescharacteristic
102C2×C8×C71#4111yescharacteristic
103C2×C8×C71#4111yescharacteristic
104C2×C8×C71#4111yescharacteristic
105C4⋊C4×C71—22no—
106C4⋊C4×C71—41no—
107C4⋊C4×C71—41no—
108C4⋊C4×C71—41no—
109C16×C71—22no—
110C23×C71C2×D811yescharacteristic
111C22×D71—21no—
112C22×D71—21no—
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—21no—
116C22×D71—21no—
117C22×D71SD3212yes—
118C2×C4×C71—42no—
119C2×C4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71—81no—
124C2×C4×C71#12011yescharacteristic
125C2×C4×C71#12111yescharacteristic
126C2×C4×C71#12311yescharacteristic
127C2×C4×C71#13511yescharacteristic
128C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C71#13711yescharacteristic
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71—21no—
133C2×Dic71—42no—
134C2×Dic71—81no—
135C2×Dic71—81no—
136C2×Dic71—81no—
137C2×Dic71—81no—
138C2×Dic71—81no—
139C4×D71—41no—
140C4×D71—41no—
141C4×D71—41no—
142C4×D71—41no—
143Dic142—42no—
144Dic142—42no—
145Dic142—81no—
146Dic142—81no—
147C8×C71#12111yescharacteristic
148C8×C71#12111yescharacteristic
149C8×C71#12311yescharacteristic
150C8×C71#12411yescharacteristic
151C71⋊C8—41no—
152C22×C71#44711yescharacteristic
153C22×C71#46611yescharacteristic
154C22×C71#48011yescharacteristic
155C22×C71C2×SD3211yescharacteristic
156C22×C71#50811yescharacteristic
157C22×C71#51011yescharacteristic
158C22×C71#51011yescharacteristic
159C2×D71—22no—
160C2×D71—22no—
161C2×D71—22no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—41no—
166C4×C71—82no—
167C4×C71—161no—
168C4×C71#39411yescharacteristic
169C4×C71#45211yescharacteristic
170C4×C71#46811yescharacteristic
171C4×C71#48211yescharacteristic
172Dic71—41no—
173Dic71—82no—
174Dic71—161no—
175#13382—711nomaximal, Sylow
176C2×C71#203311yescharacteristic
177C2×C71#218511yescharacteristic
178C2×C71#218911yescharacteristic
179C2×C71#221711yescharacteristic
180C2×C71#221911yescharacteristic
181C2×C71#223511yescharacteristic
182C2×C71#223711yescharacteristic
183D71—42no—
184C2×#467—711no—
185C2×#367D7111yescharacteristic, complemented
186C2×#358—711no—
187C2×#358—711no—
188C2×C22⋊C16—711no—
189#1808—711no—
190#1380—711no—
191C71#1338211yescharacteristic, complemented, Sylow
192C22×Q16—711no—
193C23×C8—711no—
194C2×C22⋊Q8—1421no—
195C22×C16—711no—
196C22×C16C2×D7111yescharacteristic
197#467—1422no—
198#467—1422no—
199C2×#107—711no—
200C2×#107C2×D7111yescharacteristic
201C2×#107C2×D7111yescharacteristic
202C2×#103—1421no—
203#380—1421no—
204#367—22no—
205#358—1422no—
206#358—1422no—
207C22⋊C16—1422no—
208#340—1421no—
209C22×Q8—1421no—
210C23×C4—711no—
211C2×Q16—712no—
212C2×Q16—1421no—
213C2×Q16—1421no—
214C2×Q16—1421no—
215C2×Q16—1421no—
216C2×Q16—1421no—
217C22×C8—711no—
218C22×C8—1421no—
219C22×C8—1421no—
220C22×C8—1421no—
221C22×C8—1421no—
222C22×C8—1421no—
223C22×C8—1421no—
224C22×C8C22×D7111yescharacteristic
225C22⋊Q8—2842no—
226C22⋊Q8—1424no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—2841no—
230C2×C4⋊C4—1421no—
231C2×C22⋊C4—2841no—
232C2×C22⋊C4—2841no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—1421no—
236C2×C16—1421no—
237C2×C16—1421no—
238C2×C16D28412yes—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—1421no—
243#107—1421no—
244#107C22⋊D7112yes—
245#103—1424no—
246#95—2841no—
247C24—711no—
248C2×Q8—2841no—
249C2×Q8—2841no—
250C2×Q8—2841no—
251C2×Q8—2841no—
252C2×Q8—2841no—
253C2×Q8—1422no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—2841no—
257C22×C4—2841no—
258C22×C4—711no—
259C22×C4—1421no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4—1421no—
263C22×C4—1421no—
264C22×C4—1421no—
265C22×C4D4×D7111yescharacteristic
266Q16—2841no—
267Q16—2841no—
268Q16—1422no—
269Q16—1422no—
270C2×C8—2841no—
271C2×C8—2841no—
272C2×C8—2841no—
273C2×C8—2841no—
274C2×C8—1421no—
275C2×C8—1421no—
276C2×C8—1421no—
277C2×C8D4×D7111yescharacteristic
278C2×C8C2×C22⋊D7111yescharacteristic
279C2×C8C2×D28411yescharacteristic
280C2×C8#1176586511yescharacteristic
281C2×C8#1176586611yescharacteristic
282C2×C8#1176586611yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—2844no—
288C4⋊C4—2841no—
289C4⋊C4—2841no—
290C22⋊C4—2844no—
291C22⋊C4—5682no—
292C16—22no—
293C16—2841no—
294C23—712no—
295C23—1421no—
296C23—1421no—
297C23—1421no—
298C23—1421no—
299C23—1421no—
300C23—1421no—
301C23D8×D7111yescharacteristic
302Q8—2842no—
303Q8—2842no—
304Q8—5681no—
305Q8—5681no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—2841no—
313C2×C4—2841no—
314C2×C4—2841no—
315C2×C4—2841no—
316C2×C4—2842no—
317C2×C4—5681no—
318C2×C4—5681no—
319C2×C4—5681no—
320C2×C4—5681no—
321C2×C4—5681no—
322C2×C4—1421no—
323C2×C4—1421no—
324C2×C4—1421no—
325C2×C4#1455609611yescharacteristic
326C2×C4#1455610311yescharacteristic
327C2×C4#1455611311yescharacteristic
328C2×C4#1455615111yescharacteristic
329C2×C4#1455615211yescharacteristic
330C2×C4#1455616011yescharacteristic
331C8—2841no—
332C8#1455610211yescharacteristic
333C8#1455610311yescharacteristic
334C8#1455611311yescharacteristic
335C8#1455611711yescharacteristic
336C22—2841no—
337C22—2841no—
338C22—2841no—
339C22—2841no—
340C22—1422no—
341C22—1422no—
342C22—1422no—
343C22#1798613111yescharacteristic
344C22#1798623611yescharacteristic
345C22#1798631411yescharacteristic
346C22SD32×D7111yescharacteristic
347C22#1798643011yescharacteristic
348C22#1798643811yescharacteristic
349C22#1798643811yescharacteristic
350C4—82no—
351C4—161no—
352C4—2841no—
353C4—5682no—
354C4—11361no—
355C4#1798591611yescharacteristic
356C4#1798616011yescharacteristic
357C4#1798624611yescharacteristic
358C4#1798632611yescharacteristic
359C2—2842no—
360C2#2258952411yescharacteristic
361C2#2259025511yescharacteristic
362C2#2259027711yescharacteristic
363C2#2259043711yescharacteristic
364C2#2259044911yescharacteristic
365C2#2259053911yescharacteristic
366C2#2259055111yescharacteristic
subgroupG/H
centerC23D8×D71
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC2×#367×C71C2
socleC23×C71C2×D8
prime radicalC23×D71D8
hypercenterC2×#367D71
hypocenterC71#13382
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#367

Complements

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

NH ≅ G/Ntype
2. C2×#17985804C2semidirect
3. C2×#17985802C2semidirect
6. #22586431C2semidirect
7. C2×#367×C71C2semidirect
185. C2×#367D71semidirect
191. C71#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#179862412
C2×#179858042
C2×#179858022
C2×(C2×D71)⋊C162
#225883662
#225864312
C2×#367×C712
#1338271