Finite Group Databasebeta

← #34013488

Subgroups — #34013488

Order 18688.

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#22626898C211yesmaximal, characteristic, complemented
2#22626173C211yesmaximal, characteristic, complemented
3#22626173C211yesmaximal, characteristic, complemented
4#22626167C211yesmaximal, characteristic, complemented
5#22624979C211yesmaximal, characteristic, complemented
6#22624973C211yesmaximal, characteristic, complemented
7C2×#17988709C211yesmaximal, characteristic
8#17989645—22no—
9#17989645—22no—
10C2×#14556665—22no—
11C2×#14556665C2211yescharacteristic
12C2×#14556665C2211yescharacteristic
13C2×#14556665C2211yescharacteristic
14C2×#14556665C2211yescharacteristic
15#17989485—22no—
16#17989485—22no—
17#17989483—22no—
18C2×#14556598C2211yescharacteristic
19C2×#14556598C2211yescharacteristic
20#17988709—22no—
21#17988709—22no—
22C2×#14556527C411yescharacteristic
23C2×#14556518C411yescharacteristic
24C2×#14556518C411yescharacteristic
25C2×#14556501C2211yescharacteristic
26C2×#135×C73C411yescharacteristic
27#14556665—24no—
28#14556665—21no—
29#14556665—21no—
30#14556665—21no—
31#14556665—21no—
32#14556665—21no—
33#14556665—21no—
34#14556665—21no—
35#14556665D412yes—
36C2×#11924031—24no—
37C2×#11924031—21no—
38C2×#11924031—21no—
39C2×#11924031—21no—
40C2×#11924031D412yes—
41#14556598—21no—
42#14556598—21no—
43#14556598—21no—
44#14556598—21no—
45#14556598—21no—
46#14556598—21no—
47#14556598D412yes—
48#14556598D412yes—
49C2×#11924012—21no—
50C2×#11924012—21no—
51C2×#11924012C2×C411yescharacteristic
52C2×#11924012C2×C411yescharacteristic
53C22×#10596831—21no—
54C22×#10596831—21no—
55C22×#10596831C2311yescharacteristic
56C2×C4×Dic73C2×C411yescharacteristic
57#14556527—22no—
58#14556518—22no—
59#14556518—22no—
60#14556501—21no—
61#14556501—21no—
62C2×#41×C73C2×C411yescharacteristic
63C2×#41×C73C2×C411yescharacteristic
64#135×C73—22no—
65#135×C73—22no—
66C2×Q16×C73C811yescharacteristic
67C2×SD16×C73C811yescharacteristic
68C2×SD16×C73C811yescharacteristic
69C2×D8×C73C811yescharacteristic
70C22×C8×C73C2×C411yescharacteristic
71#11924031—22no—
72#11924031—22no—
73#11924031—41no—
74#11924031—41no—
75C22×Dic73—21no—
76#11924012—21no—
77#11924012—22no—
78#11924012—22no—
79#11924012—21no—
80#11924012—21no—
81#11924012—41no—
82#11924012—41no—
83#11924012C22⋊C412yes—
84C2×#10596831—21no—
85C2×#10596831—22no—
86C2×#10596831—22no—
87C2×#10596831—21no—
88C2×#10596831—41no—
89C2×#10596831—41no—
90C2×#10596831C2×D411yescharacteristic
91C2×#10596831C2×D411yescharacteristic
92C2×#10596831C2×D411yescharacteristic
93C2×#10596831C2×D411yescharacteristic
94C2×#10596831#4111yescharacteristic
95C2×#10596831#4111yescharacteristic
96C4×Dic73—21no—
97C4×Dic73—21no—
98#41×C73—21no—
99#41×C73—21no—
100#41×C73—21no—
101#41×C73—21no—
102#41×C73—21no—
103#41×C73—21no—
104#41×C73C22⋊C412yes—
105#41×C73M1612yes—
106C2×Q8×C73C2×C811yescharacteristic
107C2×Q8×C73C2×C811yescharacteristic
108C2×D4×C73—21no—
109C2×D4×C73—21no—
110C2×D4×C73C2×C811yescharacteristic
111C2×D4×C73C2×C811yescharacteristic
112C22×C4×C73—21no—
113C22×C4×C73—21no—
114C22×C4×C73C22×C411yescharacteristic
115Q16×C73—22no—
116SD16×C73—22no—
117SD16×C73—22no—
118D8×C73—22no—
119C2×C8×C73—21no—
120C2×C8×C73—21no—
121C2×C8×C73C2×C811yescharacteristic
122C2×C8×C73C2×C811yescharacteristic
123C23×C73—21no—
124C23×C73—21no—
125C23×C73#13111yescharacteristic
126Q8×C73—21no—
127Q8×C73—21no—
128Q8×C73—21no—
129Q8×C73C22⋊C812yes—
130D4×C73—22no—
131D4×C73—22no—
132D4×C73—21no—
133D4×C73—21no—
134D4×C73—21no—
135D4×C73—41no—
136D4×C73—41no—
137D4×C73C22⋊C812yes—
138C2×C4×C73—21no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—21no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73C2×C22⋊C411yescharacteristic
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73C22×C811yescharacteristic
148C2×C4×C73C2×M1611yescharacteristic
149C2×C4×C73#13111yescharacteristic
150C2×Dic73—41no—
151C2×Dic73—81no—
152C8×C73—21no—
153C8×C73—21no—
154#10596831—22no—
155#10596831—41no—
156#10596831#12011yescharacteristic
157#10596831#12111yescharacteristic
158#10596831#12111yescharacteristic
159#10596831#12311yescharacteristic
160C22×C73—22no—
161C22×C73—41no—
162C22×C73—41no—
163C22×C73—81no—
164C22×C73—81no—
165C22×C73#40811yescharacteristic
166C22×C73C8×D411yescharacteristic
167C22×C73#43511yescharacteristic
168C4×C73—22no—
169C4×C73—41no—
170C4×C73#38611yescharacteristic
171C4×C73C2×C22⋊C811yescharacteristic
172C4×C73C8×D411yescharacteristic
173C4×C73#43511yescharacteristic
174Dic73—81no—
175#12986—731nomaximal, Sylow
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—42no—
180C2×C73—81no—
181C2×C73#186911yescharacteristic
182#3407—731no—
183C2×#444—731no—
184#2117—731no—
185#2117—731no—
186#2115—731no—
187#1377—731no—
188#1375—731no—
189C73#1298611yescharacteristic, complemented, Sylow
190C2×#135#6440811yescharacteristic
191C2×#131—731no—
192C2×#131—731no—
193#504—1462no—
194#504—1462no—
195C2×M32—731no—
196C2×M32—731no—
197C2×M32—731no—
198C2×M32—731no—
199C2×M32—1462no—
200#444—1462no—
201#444—1462no—
202C2×#108—731no—
203C2×#104—731no—
204C2×#104—731no—
205C2×#97—731no—
206#350—1462no—
207#350—1462no—
208#349—1462no—
209C2×#41C2×#6440811yescharacteristic
210C2×#41C2×#6440811yescharacteristic
211#135—22no—
212#135—22no—
213C2×Q16#1059683311yescharacteristic
214C2×SD16#1059683311yescharacteristic
215C2×SD16#1059683311yescharacteristic
216C2×D8#1059683311yescharacteristic
217#131—732no—
218#131—732no—
219#131—1461no—
220#131—1461no—
221#131—1461no—
222#131—1461no—
223#131—1461no—
224#131—1461no—
225C2×M16—731no—
226C2×M16—731no—
227C2×M16—1461no—
228C2×M16—1461no—
229C22×C8—731no—
230C22×C8—1461no—
231C22×C8—1461no—
232C22×C8C2×#6440811yescharacteristic
233C2×C42—731no—
234M32—732no—
235M32—1461no—
236M32—1461no—
237M32—1461no—
238M32—1461no—
239M32—1461no—
240M32—1461no—
241M32—1461no—
242M32—1464no—
243C2×C16—732no—
244C2×C16—1461no—
245C2×C16—1461no—
246C2×C16—1461no—
247C2×C16—1464no—
248#108—1462no—
249#104—1462no—
250#104—1462no—
251#97—1461no—
252#97—1461no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41C22⋊#6440812yes—
260#41#1192403712yes—
261C2×Q8C2×#1059683311yescharacteristic
262C2×Q8C2×#1059683311yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×#1059683311yescharacteristic
266C2×D4C2×#1059683311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—1461no—
270C22×C4C22×#6440811yescharacteristic
271Q16—22no—
272SD16—22no—
273SD16—22no—
274D8—22no—
275M16—2921no—
276M16—2921no—
277M16—732no—
278M16—1461no—
279M16—1461no—
280M16—1462no—
281M16—1462no—
282M16—1461no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—2921no—
286C2×C8—2921no—
287C2×C8—731no—
288C2×C8—731no—
289C2×C8—731no—
290C2×C8—731no—
291C2×C8—731no—
292C2×C8—731no—
293C2×C8—1461no—
294C2×C8—1462no—
295C2×C8—1462no—
296C2×C8—1461no—
297C2×C8C2×#1059683311yescharacteristic
298C2×C8C2×#1059683311yescharacteristic
299C42—1461no—
300C42—1461no—
301C16—2921no—
302C16—2921no—
303C16—1462no—
304C16—1462no—
305C23—21no—
306C23—21no—
307C23#1455667811yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8C22⋊#1059683312yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319D4C22⋊#1059683312yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—2921no—
327C2×C4—5841no—
328C2×C4C2×C22⋊#6440811yescharacteristic
329C2×C4D4×#6440811yescharacteristic
330C2×C4D4×#6440811yescharacteristic
331C2×C4C22×#1059683311yescharacteristic
332C2×C4C2×#1192403711yescharacteristic
333C2×C4#1455667811yescharacteristic
334C8—21no—
335C8—21no—
336C8—2921no—
337C8—731no—
338C8—731no—
339C8—731no—
340C8—731no—
341C8—1462no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22#1798956011yescharacteristic
348C22D4×#1059683311yescharacteristic
349C22#1798961811yescharacteristic
350C4—22no—
351C4—41no—
352C4—5841no—
353C4#1798953511yescharacteristic
354C4C2×C22⋊#1059683311yescharacteristic
355C4D4×#1059683311yescharacteristic
356C4#1798961811yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2#2262590411yescharacteristic
subgroupG/H
centerC4#17989535
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting, prime radicalC2×#135×C73C4
socleC2×C73#1869
cosocleC22×#10596831C23
hypercenterC2×#135#64408
hypocenterC73#12986
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #22626898C2semidirect
2. #22626173C2semidirect
3. #22626173C2semidirect
4. #22626167C2semidirect
5. #22624979C2semidirect
6. #22624973C2semidirect
189. C73#12986semidirect

Maximal subgroups

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

subgroupindex
#226268982
#226261732
#226261672
#226249792
#226249732
C2×#179887092
#1298673