Finite Group Databasebeta

← #31886481

Subgroups — #31886481

Order 18176.

430 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 91 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#22589898C211yesmaximal, characteristic, complemented
2#22589748C211yesmaximal, characteristic, complemented
3#22589743C211yesmaximal, characteristic
4#22586166C211yesmaximal, characteristic, complemented
5#22586163C211yesmaximal, characteristic, complemented
6#22586161C211yesmaximal, characteristic
7C71×#2732C211yesmaximal, characteristic
8C2×#14556052—21no—
9C2×#14556049—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18C2×C22⋊Q8×C71—21no—
19#399×C71—21no—
20#398×C71—21no—
21#396×C71—21no—
22#391×C71C2211yescharacteristic
23#390×C71C2211yescharacteristic
24#389×C71C2211yescharacteristic
25#386×C71—21no—
26#385×C71—21no—
27C22×C71⋊C8—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33#14556052—42no—
34#14556049—42no—
35C22×Q8×C71D411yescharacteristic
36C22×D4×C71D411yescharacteristic
37C23×C4×C71—21no—
38C22⋊Q8×C71—42no—
39C22⋊Q8×C71—42no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C42×C71C2311yescharacteristic
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58C22×Dic71—41no—
59C2×C71⋊C8—41no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C4⋊Dic71—42no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C24×C71—21no—
74C2×Q8×C71—21no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71D812yes—
78C2×D4×C71—21no—
79C2×D4×C71—41no—
80C2×D4×C71—41no—
81C2×D4×C71SD1612yes—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—42no—
88C22×C4×C71—81no—
89C22×C4×C71—81no—
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—42no—
99C4⋊C4×C71—42no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—41no—
102C4⋊C4×C71—81no—
103C22⋊C4×C71—42no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—41no—
106C22⋊C4×C71—41no—
107C22⋊C4×C71—41no—
108C22⋊C4×C71—81no—
109C22⋊C4×C71—81no—
110C42×C71—41no—
111C23×C71—22no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71#12011yescharacteristic
119Q8×C71—42no—
120Q8×C71—81no—
121D4×C71—42no—
122D4×C71—81no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—22no—
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—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—84no—
145C2×C4×C71C2×D811yescharacteristic
146C2×C4×C71C2×SD1611yescharacteristic
147C2×C4×C71#13511yescharacteristic
148C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C71#13711yescharacteristic
150C2×C4×C71#13711yescharacteristic
151C2×Dic71—81no—
152C2×Dic71—81no—
153C2×Dic71—81no—
154C71⋊C8—81no—
155C71⋊C8—81no—
156C71⋊C8—81no—
157C22×C71—42no—
158C22×C71—42no—
159C22×C71—42no—
160C22×C71—81no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71#44711yescharacteristic
165C22×C71#44811yescharacteristic
166C22×C71#45211yescharacteristic
167C22×C71#45411yescharacteristic
168C22×C71#45511yescharacteristic
169C22×C71#45511yescharacteristic
170C22×C71#45811yescharacteristic
171C4×C71—21no—
172C4×C71—21no—
173C4×C71—21no—
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—42no—
178C4×C71—81no—
179C4×C71—81no—
180C4×C71—81no—
181C4×C71—81no—
182Dic71—161no—
183#7787—711nomaximal, Sylow
184C2×C71—82no—
185C2×C71#160711yescharacteristic
186C2×C71#160811yescharacteristic
187C2×C71#161511yescharacteristic
188C2×C71#162711yescharacteristic
189C2×C71#164111yescharacteristic
190C2×C71#164311yescharacteristic
191C2×C71#166211yescharacteristic
192#2732D7111yescharacteristic
193#2099—711no—
194#2072—711no—
195#2071—711no—
196#1272—711no—
197#1271—711no—
198#1270—711no—
199C71#778711yescharacteristic, complemented, Sylow
200C2×C22⋊Q8—21no—
201C2×#103—1421no—
202C2×#102—1421no—
203#399—21no—
204#398—21no—
205#396—21no—
206#391C2×D7111yescharacteristic
207#390C2×D7111yescharacteristic
208#389C2×D7111yescharacteristic
209#386—21no—
210#385—21no—
211#384—711no—
212#340—1421no—
213#340—1421no—
214#340—1421no—
215#340—1421no—
216#336—711no—
217#336—711no—
218#336—711no—
219C22×Q8C22⋊D7111yescharacteristic
220C22×D4C22⋊D7111yescharacteristic
221C23×C4—21no—
222C22×C8—1421no—
223C22×C8—1421no—
224C22×C8—1421no—
225C22⋊Q8—42no—
226C22⋊Q8—42no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—1421no—
232C2×C4⋊C4—1421no—
233C2×C4⋊C4—1421no—
234C2×C4⋊C4C22⋊D7111yescharacteristic
235C2×C4⋊C4C22⋊D7111yescharacteristic
236C2×C4⋊C4C22⋊D7111yescharacteristic
237C2×C4⋊C4C22⋊D7111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42C22×D7111yescharacteristic
244#103—2842no—
245#102—2842no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C24—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8#1176585012yes—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4#1176585212yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—2841no—
268C22×C4C2×C22⋊D7111yescharacteristic
269C22×C4C2×C22⋊D7111yescharacteristic
270C22×C4C2×C22⋊D7111yescharacteristic
271C2×C8—2841no—
272C2×C8—2841no—
273C2×C8—2841no—
274C2×C8—2841no—
275C2×C8—2841no—
276C2×C8—2841no—
277C2×C8—2841no—
278C2×C8—2841no—
279C2×C8—2841no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—81no—
290C4⋊C4—2842no—
291C4⋊C4—2841no—
292C4⋊C4—2841no—
293C4⋊C4—2841no—
294C4⋊C4—2841no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—41no—
303C23—22no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#1455609711yescharacteristic
311Q8—42no—
312Q8—81no—
313D4—42no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
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—84no—
337C2×C4—5681no—
338C2×C4—5681no—
339C2×C4—5681no—
340C2×C4C2×#1176585011yescharacteristic
341C2×C4C2×#1176585211yescharacteristic
342C2×C4#1455614811yescharacteristic
343C2×C4#1455615411yescharacteristic
344C2×C4#1455615811yescharacteristic
345C2×C4#1455615911yescharacteristic
346C8—5681no—
347C8—5681no—
348C8—5681no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1798613311yescharacteristic
357C22#1798613811yescharacteristic
358C22#1798616111yescharacteristic
359C22#1798617211yescharacteristic
360C22#1798617711yescharacteristic
361C22#1798617711yescharacteristic
362C22#1798618911yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—11361no—
375C2—82no—
376C2#2258732611yescharacteristic
377C2#2258733111yescharacteristic
378C2#2258737111yescharacteristic
379C2#2258744311yescharacteristic
380C2#2258753311yescharacteristic
381C2#2258754711yescharacteristic
382C2#2258766311yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2732C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2732D71
hypocenterC71#7787
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
1. #22589898C2semidirect
2. #22589748C2semidirect
4. #22586166C2semidirect
5. #22586163C2semidirect
199. C71#7787semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225897482
#225897432
#225861662
#225861632
#225861612
C71×#27322
#778771