Finite Group Databasebeta

← #31892542

Subgroups — #31892542

Order 18176.

528 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596904C211yesmaximal, characteristic
2#22586769C212yesmaximal, complemented
3#22586654C211yesmaximal, characteristic
4#22586243C212yesmaximal, complemented
5C71×#2396C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17986052C2212yes—
8#17986026—24no—
9#17986023—21no—
10#17986023C2212yes—
11Q8⋊(C71⋊C8)—24no—
12#17985725—21no—
13#17985723—21no—
14C2×C4×Q8×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#400×C71C411yescharacteristic
17#399×C71C411yescharacteristic
18#398×C71—21no—
19#396×C71—21no—
20#395×C71C411yescharacteristic
21#393×C71C411yescharacteristic
22C4×C4⋊C4×C71—21no—
23C4×C4⋊C4×C71C2211yescharacteristic
24C4×C22⋊C4×C71—21no—
25C2×#11765849—22no—
26C2×#11765849—22no—
27C2×#11765849—42no—
28C2×#11765849—42no—
29C22×C71⋊C8—22no—
30C4⋊(C71⋊C8)—24no—
31C4⋊(C71⋊C8)—22no—
32C4⋊(C71⋊C8)—22no—
33C4⋊(C71⋊C8)D414yes—
34C22⋊(C71⋊C8)—22no—
35C4×C71⋊C8—22no—
36C22×Q8×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C4×Q8×C71—22no—
39C4×Q8×C71—22no—
40#117×C71—22no—
41#117×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71C2×C411yescharacteristic
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C42×C71—21no—
54C2×C42×C71—21no—
55C2×C42×C71D411yescharacteristic
56C2×C42×C71C2311yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71C2×C411yescharacteristic
62#95×C71C2×C411yescharacteristic
63#11765849—44no—
64#11765849—42no—
65#11765849—42no—
66C2×C71⋊C8—24no—
67C2×C71⋊C8—44no—
68C2×C71⋊C8—42no—
69C24×C71C22⋊C411yescharacteristic
70C2×Q8×C71—22no—
71C2×Q8×C71—42no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71SD1612yes—
84C22×C4×C71Q1612yes—
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—22no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—22no—
91C4⋊C4×C71—42no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71—42no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—22no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—42no—
99C42×C71—21no—
100C42×C71—21no—
101C42×C71—41no—
102C42×C71—41no—
103C42×C71—41no—
104C42×C71—41no—
105C42×C71C2×D412yes—
106C42×C71#4112yes—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71#10312yes—
112C23×C71C2×C22⋊C411yescharacteristic
113Q8×C71—24no—
114Q8×C71—42no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—22no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71#10312yes—
140C2×C4×C71C4×D412yes—
141C2×C4×C71#12011yescharacteristic
142C2×C4×C71#12112yes—
143C2×C4×C71#12311yescharacteristic
144C2×C4×C71C2×SD1611yescharacteristic
145C2×C4×C71C2×Q1611yescharacteristic
146C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C71#13711yescharacteristic
148C71⋊C8—44no—
149C71⋊C8—82no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—22no—
153C22×C71—42no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—81no—
157C22×C71#38611yescharacteristic
158C22×C71C2×#10311yescharacteristic
159C22×C71#41711yescharacteristic
160C4×C71—22no—
161C4×C71—22no—
162C4×C71—22no—
163C4×C71—42no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—42no—
167C4×C71—81no—
168C4×C71—81no—
169C4×C71#45011yescharacteristic
170C4×C71#45111yescharacteristic
171C4×C71#48311yescharacteristic
172C4×C71#48411yescharacteristic
173#9033—711nomaximal, Sylow
174C2×C71—21no—
175C2×C71—21no—
176C2×C71—42no—
177C2×C71#187211yescharacteristic
178C2×C71#188411yescharacteristic
179C2×C71#189111yescharacteristic
180#2905—711no—
181#2396D7111yescharacteristic
182#1487—712no—
183#1458—711no—
184#1303—712no—
185C71#903311yescharacteristic, complemented, Sylow
186C22×M16—1421no—
187C2×C4×Q8C2×D7111yescharacteristic
188C2×#117C2×D7111yescharacteristic
189#431—712no—
190#424—1424no—
191#423—712no—
192#423—1421no—
193#400Dic7111yescharacteristic
194#399Dic7111yescharacteristic
195#398—21no—
196#396—21no—
197#395Dic7111yescharacteristic
198#393Dic7111yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2×D7111yescharacteristic
201C4×C22⋊C4—21no—
202Q8⋊C8—1424no—
203#324—1421no—
204#323—1421no—
205C22×Q8C2×Dic7111yescharacteristic
206C23×C4C22⋊D7111yescharacteristic
207C2×M16—2842no—
208C2×M16—2842no—
209C2×M16—1422no—
210C2×M16—1422no—
211C22×C8—1422no—
212C4×Q8—22no—
213C4×Q8—22no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic7111yescharacteristic
221C2×C4⋊C4C2×Dic7111yescharacteristic
222C2×C4⋊C4C22⋊D7111yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic7111yescharacteristic
226C2×C22⋊C4C22⋊D7111yescharacteristic
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22⋊D7111yescharacteristic
230C2×C42C22×D7111yescharacteristic
231C4⋊C8—714no—
232C4⋊C8—1424no—
233C4⋊C8—1422no—
234C4⋊C8—1422no—
235C22⋊C8—1422no—
236C4×C8—1422no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95C2×Dic7111yescharacteristic
242#95C2×Dic7111yescharacteristic
243C24C22⋊Dic7111yescharacteristic
244C2×Q8—22no—
245C2×Q8—42no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4#1176585212yes—
258C22×C4#1176585512yes—
259C22×C4C22×Dic7111yescharacteristic
260C22×C4C2×C22⋊D7111yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262M16—2844no—
263M16—2842no—
264M16—2842no—
265C2×C8—2844no—
266C2×C8—2842no—
267C2×C8—1424no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—21no—
280C42—21no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42D4×D7112yes—
286C42#1176586412yes—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1455605212yes—
292C23C2×C22⋊Dic7111yescharacteristic
293Q8—24no—
294Q8—42no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#1455605212yes—
320C2×C4D4×Dic7112yes—
321C2×C4#1455609611yescharacteristic
322C2×C4#1455610012yes—
323C2×C4#1455610911yescharacteristic
324C2×C4C2×#1176585211yescharacteristic
325C2×C4C2×#1176585511yescharacteristic
326C2×C4#1455615311yescharacteristic
327C2×C4#1455615811yescharacteristic
328C8—2844no—
329C8—5682no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#1798587911yescharacteristic
338C22C2×#1455605211yescharacteristic
339C22#1798600011yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—41no—
345C4—42no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4#1798614911yescharacteristic
350C4#1798615411yescharacteristic
351C4#1798632711yescharacteristic
352C4#1798633211yescharacteristic
353C2—21no—
354C2—21no—
355C2—42no—
356C2#2258864811yescharacteristic
357C2#2258872411yescharacteristic
358C2#2258875911yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2396C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2396D71
hypocenterC71#9033
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #22586769C2semidirect
4. #22586243C2semidirect
185. C71#9033semidirect

Maximal subgroups

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

subgroupindex
#225969042
#225867692
#225866542
#225862432
C71×#23962
#903371