Finite Group Databasebeta

← #31916106

Subgroups — #31916106

Order 18176.

484 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588328C211yesmaximal, characteristic
2C4×#14556045C211yesmaximal, characteristic
3#22586458C212yesmaximal
4#22586458C212yesmaximal
5C71×#2579C211yesmaximal, characteristic
6C2×#14556045C2212yes—
7C2×#14556043C2212yes—
8C4×#11765849—21no—
9#17985859—21no—
10#17985768—21no—
11#17985766—21no—
12C2×C4×D4×C71C2211yescharacteristic
13C22×C4⋊C4×C71C411yescharacteristic, complemented
14#398×C71—21no—
15#396×C71—21no—
16#387×C71—21no—
17#386×C71C2211yescharacteristic
18#386×C71C2211yescharacteristic
19#385×C71—21no—
20#385×C71C411yescharacteristic, complemented
21#385×C71C411yescharacteristic, complemented
22#380×C71C411yescharacteristic, complemented
23C2×#11765849—22no—
24C2×C4⋊Dic71—41no—
25C2×C22⋊Dic71—22no—
26#14556045—22no—
27#14556045Q814yes—
28#14556043—24no—
29#14556039—41no—
30C4×C71⋊C8—41no—
31C71⋊#95—41no—
32C22×D4×C71C2311yescharacteristic
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—41no—
45C2×C4⋊C4×C71—41no—
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71C2×C411yescharacteristic
50C2×C22⋊C4×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71C2×C411yescharacteristic
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59C22×Dic71—42no—
60#11765849—24no—
61#11765849—42no—
62C2×C71⋊C8—42no—
63C4⋊Dic71—82no—
64C22⋊Dic71—42no—
65C22⋊Dic71—42no—
66C24×C71C22×C411yescharacteristic
67C24×C71C2×D411yescharacteristic
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71C2×Q812yes—
71C2×D4×C71#4112yes—
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×C71—42no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—81no—
84C22×C4×C71—81no—
85C22×C4×C71—81no—
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C4⋊C4×C71—41no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—81no—
97C4⋊C4×C71—81no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—81no—
104C42×C71—21no—
105C42×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71C2×C22⋊C411yescharacteristic
115C23×C71C4×Q812yes—
116C23×C71#12411yescharacteristic
117C23×C71C4⋊Q811yescharacteristic
118D4×C71—42no—
119D4×C71—42no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—22no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—41no—
127C2×C4×C71—42no—
128C2×C4×C71—41no—
129C2×C4×C71—42no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—42no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—81no—
145C2×C4×C71—81no—
146C2×C4×C71—81no—
147C2×C4×C71—82no—
148C2×C4×C71#9911yescharacteristic
149C2×C4×C71#9911yescharacteristic
150C2×C4×C71C22⋊Q812yes—
151C2×Dic71—82no—
152C2×Dic71—82no—
153C2×Dic71—82no—
154C71⋊C8—44no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—22no—
159C22×C71—21no—
160C22×C71—42no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—42no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—81no—
169C22×C71#39111yescharacteristic
170C22×C71C2×#9911yescharacteristic
171C22×C71#41011yescharacteristic
172C4×C71—21no—
173C4×C71—21no—
174C4×C71—22no—
175C4×C71—41no—
176C4×C71—42no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71—81no—
180C4×C71—81no—
181C4×C71—81no—
182Dic71—162no—
183#13294—711nomaximal, Sylow
184C2×C71—21no—
185C2×C71—21no—
186C2×C71—41no—
187C2×C71—42no—
188C2×C71—41no—
189C2×C71#199611yescharacteristic
190C2×C71#212011yescharacteristic
191C2×C71#212111yescharacteristic
192#2579D7111yescharacteristic
193#1799—711no—
194C4×#100—711no—
195#1390—712no—
196#1390—712no—
197C71#1329411yescharacteristic, complemented, Sylow
198C2×C4×D4C2×D7111yescharacteristic
199C22×C4⋊C4Dic7111yescharacteristic, complemented
200C2×#100—712no—
201C2×#99—712no—
202C4×M16—1421no—
203#398—21no—
204#396—21no—
205#387—21no—
206#386C2×D7111yescharacteristic
207#386C2×D7111yescharacteristic
208#385—21no—
209#385Dic7111yescharacteristic, complemented
210#385Dic7111yescharacteristic, complemented
211#380—1421no—
212#380Dic7111yescharacteristic, complemented
213#343—1421no—
214#342—1421no—
215C22×D4C22×D7111yescharacteristic
216C23×C4C2×Dic7111yescharacteristic
217C23×C4C2×Dic7111yescharacteristic
218C23×C4C2×Dic7111yescharacteristic
219C23×C4C22⋊D7111yescharacteristic
220C2×M16—1422no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—2841no—
231C2×C4⋊C4C22⋊D7111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—1422no—
235C2×C22⋊C4C2×Dic7111yescharacteristic
236C2×C22⋊C4C2×Dic7111yescharacteristic
237C2×C22⋊C4C2×Dic7111yescharacteristic
238C2×C22⋊C4C22⋊D7111yescharacteristic
239C2×C42C22⋊D7111yescharacteristic
240#100—714no—
241#100—1422no—
242#99—1424no—
243#97—2841no—
244C4×C8—2841no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—2841no—
251C24C22×Dic7111yescharacteristic
252C24C2×C22⋊D7111yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4Q8×D7112yes—
256C2×D4#1176586612yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—2842no—
272C22×C4C22⋊Dic7111yescharacteristic
273C22×C4C22⋊Dic7111yescharacteristic
274C22×C4C22⋊Dic7111yescharacteristic
275C22×C4C22⋊Dic7111yescharacteristic
276C22×C4C2×C22⋊D7111yescharacteristic
277M16—2842no—
278M16—1424no—
279C2×C8—2842no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—5682no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—81no—
294C22⋊C4—2842no—
295C22⋊C4—2842no—
296C42—21no—
297C42—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23C2×C22⋊Dic7111yescharacteristic
307C23Q8×Dic7112yes—
308C23#1455611711yescharacteristic
309C23#1455612611yescharacteristic
310D4—42no—
311D4—42no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—5682no—
341C2×C4—5682no—
342C2×C4—5682no—
343C2×C4#1455604311yescharacteristic
344C2×C4#1455604311yescharacteristic
345C2×C4#1455610412yes—
346C8—2844no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22#1798590611yescharacteristic
362C22C2×#1455604311yescharacteristic
363C22#1798597311yescharacteristic
364C4—21no—
365C4—21no—
366C4—22no—
367C4—42no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—11362no—
375C2—21no—
376C2—21no—
377C2—41no—
378C2—42no—
379C2—41no—
380C2#2258932611yescharacteristic
381C2#2258998111yescharacteristic
382C2#2258998611yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2579C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenter#2579D71
hypocenterC71#13294
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
13. C22×C4⋊C4×C71C4semidirect
20. #385×C71C4semidirect
21. #385×C71C4semidirect
22. #380×C71C4semidirect
197. C71#13294semidirect
199. C22×C4⋊C4Dic71semidirect
209. #385Dic71semidirect
210. #385Dic71semidirect
212. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
C4×#145560452
#225864582
C71×#25792
#1329471