Finite Group Databasebeta

← #31914363

Subgroups — #31914363

Order 18176.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22601443C211yesmaximal, characteristic, complemented
2#22589968C211yesmaximal, characteristic, complemented
3#22589968C211yesmaximal, characteristic, complemented
4#22589961C211yesmaximal, characteristic, complemented
5#22586424C211yesmaximal, characteristic, complemented
6#22586420C211yesmaximal, characteristic, complemented
7C2×#444×C71C211yesmaximal, characteristic
8#17986416—22no—
9#17986416—22no—
10C2×#14556068—22no—
11C2×#14556068C2211yescharacteristic
12C2×#14556068C2211yescharacteristic
13C2×#14556068C2211yescharacteristic
14C2×#14556068C2211yescharacteristic
15#17985783—22no—
16#17985783—22no—
17#17985781—22no—
18C2×#135×C71C411yescharacteristic
19C2×#131×C71C2211yescharacteristic
20C2×#131×C71C2211yescharacteristic
21#444×C71—22no—
22#444×C71—22no—
23C2×#108×C71C411yescharacteristic
24C2×#104×C71C411yescharacteristic
25C2×#104×C71C411yescharacteristic
26C2×#97×C71C2211yescharacteristic
27#14556068—24no—
28#14556068—21no—
29#14556068—21no—
30#14556068—21no—
31#14556068—21no—
32#14556068—21no—
33#14556068—21no—
34#14556068—21no—
35#14556068D412yes—
36C2×C71⋊C16—24no—
37C2×C71⋊C16—21no—
38C2×C71⋊C16—21no—
39C2×C71⋊C16—21no—
40C2×C71⋊C16D412yes—
41C2×#41×C71C2×C411yescharacteristic
42C2×#41×C71C2×C411yescharacteristic
43#135×C71—22no—
44#135×C71—22no—
45C2×Q16×C71C811yescharacteristic
46C2×SD16×C71C811yescharacteristic
47C2×SD16×C71C811yescharacteristic
48C2×D8×C71C811yescharacteristic
49#131×C71—21no—
50#131×C71—21no—
51#131×C71—21no—
52#131×C71—21no—
53#131×C71—21no—
54#131×C71—21no—
55#131×C71D412yes—
56#131×C71D412yes—
57C2×M16×C71—21no—
58C2×M16×C71—21no—
59C2×M16×C71C2×C411yescharacteristic
60C2×M16×C71C2×C411yescharacteristic
61C22×C8×C71—21no—
62C22×C8×C71—21no—
63C22×C8×C71C2×C411yescharacteristic
64C22×C8×C71C2311yescharacteristic
65C2×C42×C71C2×C411yescharacteristic
66#108×C71—22no—
67#104×C71—22no—
68#104×C71—22no—
69#97×C71—21no—
70#97×C71—21no—
71C71⋊C16—22no—
72C71⋊C16—22no—
73C71⋊C16—41no—
74C71⋊C16—41no—
75#41×C71—21no—
76#41×C71—21no—
77#41×C71—21no—
78#41×C71—21no—
79#41×C71—21no—
80#41×C71—21no—
81#41×C71C22⋊C412yes—
82#41×C71M1612yes—
83C2×Q8×C71C2×C811yescharacteristic
84C2×Q8×C71C2×C811yescharacteristic
85C2×D4×C71—21no—
86C2×D4×C71—21no—
87C2×D4×C71C2×C811yescharacteristic
88C2×D4×C71C2×C811yescharacteristic
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71C22×C411yescharacteristic
93Q16×C71—22no—
94SD16×C71—22no—
95SD16×C71—22no—
96D8×C71—22no—
97M16×C71—21no—
98M16×C71—22no—
99M16×C71—22no—
100M16×C71—21no—
101M16×C71—21no—
102M16×C71—41no—
103M16×C71—41no—
104M16×C71C22⋊C412yes—
105C2×C8×C71—21no—
106C2×C8×C71—21no—
107C2×C8×C71—21no—
108C2×C8×C71—22no—
109C2×C8×C71—22no—
110C2×C8×C71—21no—
111C2×C8×C71—41no—
112C2×C8×C71—41no—
113C2×C8×C71C2×C811yescharacteristic
114C2×C8×C71C2×C811yescharacteristic
115C2×C8×C71C2×D411yescharacteristic
116C2×C8×C71C2×D411yescharacteristic
117C2×C8×C71C2×D411yescharacteristic
118C2×C8×C71C2×D411yescharacteristic
119C2×C8×C71#4111yescharacteristic
120C2×C8×C71#4111yescharacteristic
121C42×C71—21no—
122C42×C71—21no—
123C23×C71—21no—
124C23×C71—21no—
125C23×C71#13111yescharacteristic
126Q8×C71—21no—
127Q8×C71—21no—
128Q8×C71—21no—
129Q8×C71C22⋊C812yes—
130D4×C71—22no—
131D4×C71—22no—
132D4×C71—21no—
133D4×C71—21no—
134D4×C71—21no—
135D4×C71—41no—
136D4×C71—41no—
137D4×C71C22⋊C812yes—
138C2×C4×C71—21no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—21no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—81no—
146C2×C4×C71C2×C22⋊C411yescharacteristic
147C2×C4×C71C4×D411yescharacteristic
148C2×C4×C71C4×D411yescharacteristic
149C2×C4×C71C22×C811yescharacteristic
150C2×C4×C71C2×M1611yescharacteristic
151C2×C4×C71#13111yescharacteristic
152C8×C71—22no—
153C8×C71—21no—
154C8×C71—21no—
155C8×C71—41no—
156C8×C71#12011yescharacteristic
157C8×C71#12111yescharacteristic
158C8×C71#12111yescharacteristic
159C8×C71#12311yescharacteristic
160C22×C71—22no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71#40811yescharacteristic
166C22×C71C8×D411yescharacteristic
167C22×C71#43511yescharacteristic
168C4×C71—22no—
169C4×C71—41no—
170C4×C71—81no—
171C4×C71#38611yescharacteristic
172C4×C71C2×C22⋊C811yescharacteristic
173C4×C71C8×D411yescharacteristic
174C4×C71#43511yescharacteristic
175#12986—711nomaximal, Sylow
176C2×C71—21no—
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—42no—
180C2×C71—81no—
181C2×C71#186911yescharacteristic
182#3407—711no—
183C2×#444D7111yescharacteristic
184#2117—711no—
185#2117—711no—
186#2115—711no—
187#1377—711no—
188#1375—711no—
189C71#1298611yescharacteristic, complemented, Sylow
190C2×#135Dic7111yescharacteristic
191C2×#131C2×D7111yescharacteristic
192C2×#131C2×D7111yescharacteristic
193#504—1422no—
194#504—1422no—
195C2×M32—711no—
196C2×M32—711no—
197C2×M32—711no—
198C2×M32—711no—
199C2×M32—1422no—
200#444—22no—
201#444—22no—
202C2×#108Dic7111yescharacteristic
203C2×#104Dic7111yescharacteristic
204C2×#104Dic7111yescharacteristic
205C2×#97C2×D7111yescharacteristic
206#350—1422no—
207#350—1422no—
208#349—1422no—
209C2×#41C2×Dic7111yescharacteristic
210C2×#41C2×Dic7111yescharacteristic
211#135—22no—
212#135—22no—
213C2×Q16C71⋊C811yescharacteristic
214C2×SD16C71⋊C811yescharacteristic
215C2×SD16C71⋊C811yescharacteristic
216C2×D8C71⋊C811yescharacteristic
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131C22⋊D7112yes—
224#131C22⋊D7112yes—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16C2×Dic7111yescharacteristic
228C2×M16C2×Dic7111yescharacteristic
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C2×Dic7111yescharacteristic
232C22×C8C22×D7111yescharacteristic
233C2×C42C2×Dic7111yescharacteristic
234M32—712no—
235M32—1421no—
236M32—1421no—
237M32—1421no—
238M32—1421no—
239M32—1424no—
240M32—1421no—
241M32—1421no—
242M32—1421no—
243C2×C16—712no—
244C2×C16—1421no—
245C2×C16—1421no—
246C2×C16—1424no—
247C2×C16—1421no—
248#108—22no—
249#104—22no—
250#104—22no—
251#97—21no—
252#97—21no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41C22⋊Dic7112yes—
260#41#1176584912yes—
261C2×Q8C2×C71⋊C811yescharacteristic
262C2×Q8C2×C71⋊C811yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×C71⋊C811yescharacteristic
266C2×D4C2×C71⋊C811yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C22×Dic7111yescharacteristic
271Q16—22no—
272SD16—22no—
273SD16—22no—
274D8—22no—
275M16—21no—
276M16—22no—
277M16—22no—
278M16—21no—
279M16—21no—
280M16—41no—
281M16—41no—
282M16C22⋊Dic7112yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8C2×C71⋊C811yescharacteristic
292C2×C8C2×C71⋊C811yescharacteristic
293C2×C8D4×D7111yescharacteristic
294C2×C8D4×D7111yescharacteristic
295C2×C8C2×C22⋊D7111yescharacteristic
296C2×C8C2×C22⋊D7111yescharacteristic
297C2×C8#1176586411yescharacteristic
298C2×C8#1176586411yescharacteristic
299C42—21no—
300C42—21no—
301C16—2841no—
302C16—2841no—
303C16—1422no—
304C16—1422no—
305C23—21no—
306C23—21no—
307C23#1455613511yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8C22⋊(C71⋊C8)12yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319D4C22⋊(C71⋊C8)12yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4C2×C22⋊Dic7111yescharacteristic
329C2×C4D4×Dic7111yescharacteristic
330C2×C4D4×Dic7111yescharacteristic
331C2×C4C22×C71⋊C811yescharacteristic
332C2×C4C2×#1176584911yescharacteristic
333C2×C4#1455613511yescharacteristic
334C8—22no—
335C8—21no—
336C8—21no—
337C8—41no—
338C8#1455609611yescharacteristic
339C8#1455610011yescharacteristic
340C8#1455610011yescharacteristic
341C8#1455610911yescharacteristic
342C22—22no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22#1798596611yescharacteristic
348C22D4×C71⋊C811yescharacteristic
349C22#1798607111yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4#1798587911yescharacteristic
354C4C2×C22⋊(C71⋊C8)11yescharacteristic
355C4D4×C71⋊C811yescharacteristic
356C4#1798607111yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2258863311yescharacteristic
subgroupG/H
centerC4#17985879
commutatorC2×C4×C71C22×C8
FrattiniC22×C8C22×D71
FittingC2×#444×C71C2
socleC2×C71#1869
prime radicalC2×#135×C71C4
cosocleC22×C8×C71C23
hypercenterC2×#444D71
hypocenterC71#12986
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
1. #22601443C2semidirect
2. #22589968C2semidirect
3. #22589968C2semidirect
4. #22589961C2semidirect
5. #22586424C2semidirect
6. #22586420C2semidirect
189. C71#12986semidirect

Maximal subgroups

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

subgroupindex
#226014432
#225899682
#225899612
#225864242
#225864202
C2×#444×C712
#1298671