Finite Group Databasebeta

← #31901870

Subgroups — #31901870

Order 18176.

604 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 119 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#22594454C211yesmaximal, characteristic
2#22588533C212yesmaximal
3#22586219C212yesmaximal
4#22586183C211yesmaximal, characteristic
5C71×#1444C211yesmaximal, characteristic
6C2×#14556107—21no—
7C2×#14556059—22no—
8C2×C22⋊(C71⋊C8)C2212yes—
9#17985931—22no—
10#17985920—22no—
11#17985859—21no—
12#17985859C2212yes—
13#17985761—22no—
14C22×C4⋊C4×C71—21no—
15C22×C4⋊C4×C71C2211yescharacteristic
16C2×#95×C71—21no—
17C2×#95×C71C2211yescharacteristic
18C2×#95×C71C2211yescharacteristic
19C22×Dic142—21no—
20C22×C71⋊C8—22no—
21#14556107—42no—
22#14556107—42no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71D412yes—
27C2×C4⋊Dic71D412yes—
28C2×Dic71⋊C4—22no—
29C2×C22⋊Dic71—22no—
30#14556059—44no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)Q814yescomplemented
33C71⋊#95—22no—
34C71⋊#95—22no—
35C23×C4×C71—21no—
36C23×C4×C71—21no—
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—42no—
42C2×C4⋊C4×C71—42no—
43C2×C4⋊C4×C71—42no—
44C2×C4⋊C4×C71—42no—
45#95×C71—22no—
46#95×C71—21no—
47#95×C71—42no—
48#95×C71Q812yescomplemented
49C2×Dic142—42no—
50C2×Dic142—81no—
51C2×Dic142—81no—
52C22×Dic71—22no—
53C22×Dic71—22no—
54C2×C71⋊C8—24no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—42no—
59C4⋊Dic71—42no—
60C4⋊Dic71—81no—
61C4⋊Dic71—81no—
62Dic71⋊C4—44no—
63C22⋊Dic71—42no—
64C22⋊Dic71—42no—
65C24×C71C2×D411yescharacteristic
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—22no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71—41no—
90C22×C4×C71SD1612yes—
91C22×C4×C71SD1612yes—
92C22×C4×C71C2×D412yes—
93C22×C4×C71C2×Q812yes—
94C22×C4×C71C2×Q811yescharacteristic
95C22×C4×C71#4111yescharacteristic
96C4⋊C4×C71—24no—
97C4⋊C4×C71—42no—
98C4⋊C4×C71—48no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71#12011yescharacteristic
104C23×C71C22⋊Q811yescharacteristic
105C23×C71C4⋊Q811yescharacteristic
106C23×C71C2×SD1611yescharacteristic
107C23×C71C2×SD1611yescharacteristic
108C23×C71AGL(1,ℤ/8)11yescharacteristic
109C23×C71#13711yescharacteristic
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—24no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—42no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—81no—
134C2×C4×C71—81no—
135C2×C4×C71—81no—
136C2×C4×C71—81no—
137C2×C4×C71—81no—
138C2×C4×C71—81no—
139C2×C4×C71C22⋊Q812yes—
140C2×C4×C71C4⋊Q812yes—
141C2×Dic71—42no—
142C2×Dic71—42no—
143C2×Dic71—42no—
144C2×Dic71—42no—
145C2×Dic71—42no—
146C2×Dic71—42no—
147Dic142—84no—
148C71⋊C8—44no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71#39511yescharacteristic
162C22×C71#44811yescharacteristic
163C22×C71#45011yescharacteristic
164C22×C71#45311yescharacteristic
165C22×C71#45611yescharacteristic
166C22×C71B(4,𝔽2)11yescharacteristic
167C22×C71#45811yescharacteristic
168C22×C71#47511yescharacteristic
169C22×C71#47611yescharacteristic
170C22×C71#49811yescharacteristic
171C22×C71#50111yescharacteristic
172C4×C71—22no—
173C4×C71—22no—
174C4×C71—44no—
175C4×C71—42no—
176C4×C71—42no—
177C4×C71—82no—
178Dic71—82no—
179Dic71—82no—
180#10570—711nomaximal, Sylow
181C2×C71—21no—
182C2×C71—21no—
183C2×C71—21no—
184C2×C71—21no—
185C2×C71#159611yescharacteristic
186C2×C71#160411yescharacteristic
187C2×C71#161111yescharacteristic
188C2×C71#161311yescharacteristic
189C2×C71#205811yescharacteristic
190C2×C71#206011yescharacteristic
191C2×C71#206211yescharacteristic
192#2659—711no—
193#1853—712no—
194#1444D7111yescharacteristic
195#1294—712no—
196#1280—711no—
197C71#1057011yescharacteristic, complemented, Sylow
198C2×C22⋊Q8—1421no—
199C22×C4⋊C4—21no—
200C22×C4⋊C4C2×D7111yescharacteristic
201C2×#106—1422no—
202C2×C22⋊C8—712no—
203#398—1422no—
204#395—1422no—
205#380—712no—
206#380—1421no—
207C2×#95—21no—
208C2×#95C2×D7111yescharacteristic
209C2×#95C2×D7111yescharacteristic
210#340—1422no—
211C22×Q8—1421no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4C22⋊D7111yescharacteristic
215C23×C4C22⋊D7111yescharacteristic
216C23×C4C22×D7111yescharacteristic
217C22×C8—1422no—
218C22⋊Q8—2842no—
219C22⋊Q8—2842no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—712no—
226C2×C4⋊C4—712no—
227C2×C4⋊C4—1422no—
228C2×C4⋊C4—1421no—
229C2×C4⋊C4—1421no—
230C2×C4⋊C4—1421no—
231C2×C22⋊C4—1422no—
232#106—2844no—
233C22⋊C8—714no—
234C22⋊C8—1422no—
235#95—22no—
236#95—21no—
237#95—42no—
238#95—1422no—
239#95—1422no—
240#95Dic14212yescomplemented
241C24C2×C22⋊D7111yescharacteristic
242C2×Q8—2842no—
243C2×Q8—5681no—
244C2×Q8—5681no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—1422no—
270C22×C4—1422no—
271C22×C4#1176585212yes—
272C22×C4#1176585312yes—
273C22×C4D4×D7112yes—
274C22×C4Q8×D7112yes—
275C22×C4C2×Dic14211yescharacteristic
276C22×C4#1176586511yescharacteristic
277C2×C8—2842no—
278C2×C8—2842no—
279C2×C8—1424no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—48no—
283C4⋊C4—2842no—
284C4⋊C4—2842no—
285C4⋊C4—2844no—
286C4⋊C4—2842no—
287C4⋊C4—5681no—
288C4⋊C4—5681no—
289C22⋊C4—2842no—
290C22⋊C4—2842no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1455609611yescharacteristic
296C23#1455610711yescharacteristic
297C23#1455612611yescharacteristic
298C23C2×#1176585211yescharacteristic
299C23C2×#1176585311yescharacteristic
300C23#1455615311yescharacteristic
301C23#1455615811yescharacteristic
302Q8—5684no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—2842no—
333C2×C4—2842no—
334C2×C4—2842no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—2842no—
338C2×C4#1455610612yes—
339C2×C4#1455612712yes—
340C8—2844no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1798591911yescharacteristic
354C22#1798613711yescharacteristic
355C22#1798614911yescharacteristic
356C22#1798616611yescharacteristic
357C22#1798618111yescharacteristic
358C22#1798618511yescharacteristic
359C22#1798618811yescharacteristic
360C22#1798628411yescharacteristic
361C22#1798628911yescharacteristic
362C22#1798639511yescharacteristic
363C22#1798640911yescharacteristic
364C4—22no—
365C4—22no—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—82no—
370C4—5682no—
371C4—5682no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2258726211yescharacteristic
377C2#2258730811yescharacteristic
378C2#2258734711yescharacteristic
379C2#2258735911yescharacteristic
380C2#2258967311yescharacteristic
381C2#2258968511yescharacteristic
382C2#2258969311yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC23×C4×C71C23
FrattiniC23×C4C22×D71
FittingC71×#1444C2
socleC23×C71#120
prime radicalC24×C71C2×D4
hypercenter#1444D71
hypocenterC71#10570
LCSG ▷ C23×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1444

Complements

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

NH ≅ G/Ntype
32. C22⋊(C71⋊C8)Q8semidirect
48. #95×C71Q8semidirect
197. C71#10570semidirect
240. #95Dic142semidirect

Maximal subgroups

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

subgroupindex
#225944542
#225885332
#225862192
#225861832
C71×#14442
#1057071