Finite Group Databasebeta

← #33004193

Subgroups — #33004193

Order 18688.

420 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618634C211yesmaximal, characteristic
2#22611696C211yesmaximal, characteristic, complemented
3#22611560C211yesmaximal, characteristic, complemented
4#22611558C211yesmaximal, characteristic, complemented
5C2×#17988369C211yesmaximal, characteristic, complemented
6#22611535C211yesmaximal, characteristic, complemented
7C2×#349×C73C211yesmaximal, characteristic, complemented
8C2×#14556597C2211yescharacteristic
9#17989098C411yescharacteristic
10C2×#14556531C2211yescharacteristic
11C2×#14556531C2211yescharacteristic
12#17988663—22no—
13#17988656—22no—
14#17988613C411yescharacteristic
15#17988553C411yescharacteristic
16#17988551C411yescharacteristic
17#17988542C2211yescharacteristic
18#17988407—22no—
19#17988372—22no—
20#17988370—22no—
21#17988369—22no—
22#17988364—22no—
23C22×M16×C73C2211yescharacteristic
24C2×M32×C73C2211yescharacteristic
25C2×M32×C73C2211yescharacteristic
26#349×C73—22no—
27C2×#11924028C2×C411yescharacteristic
28#14556597—21no—
29#14556597—21no—
30#14556597—21no—
31#14556597—21no—
32C2×#11924012C2×C411yescharacteristic
33C2×#11924012C2×C411yescharacteristic
34C2×#11924011—21no—
35C2×C8×D73—21no—
36#14556574—21no—
37#14556570—21no—
38#14556565—21no—
39#14556560—21no—
40C4×C22⋊D73—22no—
41#14556550C2×C411yescharacteristic
42#14556531—21no—
43#14556531—21no—
44#14556531—21no—
45#14556531D412yescomplemented
46C2×#11924003—21no—
47C2×#11924003—21no—
48#14556520—22no—
49C22⋊#10596831—22no—
50#14556503—22no—
51#14556501D412yes—
52C8×Dic73D412yes—
53C23×C4×C73C2×C411yescharacteristic
54C2×M16×C73—21no—
55C2×M16×C73—21no—
56C2×M16×C73—21no—
57C2×M16×C73—21no—
58C2×M16×C73—21no—
59C2×M16×C73—21no—
60C22×C8×C73C2×C411yescharacteristic
61C22×C8×C73C2311yescharacteristic
62M32×C73—21no—
63M32×C73—21no—
64M32×C73—21no—
65M32×C73D412yescomplemented
66C2×C16×C73—21no—
67C2×C16×C73—21no—
68#11924028—21no—
69#11924028—21no—
70#11924028—21no—
71#11924028—21no—
72C2×Dic146M1611yescharacteristic
73C2×D292M1611yescharacteristic
74C2×C22⋊D73—21no—
75C2×C22⋊D73—22no—
76C2×C4×D73—21no—
77#11924012—21no—
78#11924012—21no—
79#11924012—21no—
80#11924012—21no—
81#11924011—41no—
82#11924011—41no—
83C2×#10596831—21no—
84C2×#10596831C22⋊C412yes—
85C8×D73—41no—
86C8×D73—41no—
87C4⋊Dic73—21no—
88#11924007—22no—
89C22⋊Dic73—22no—
90C22⋊Dic73—21no—
91#11924005—22no—
92C4×Dic73C22⋊C412yes—
93#11924003—22no—
94#11924003—41no—
95C24×C73M1611yescharacteristic
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73M1611yescharacteristic
103C22×C4×C73C22×C411yescharacteristic
104M16×C73—41no—
105M16×C73—41no—
106M16×C73—41no—
107M16×C73—41no—
108C2×C8×C73—21no—
109C2×C8×C73—21no—
110C2×C8×C73—21no—
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73#4111yescharacteristic
116C2×C8×C73#4111yescharacteristic
117C16×C73—22no—
118C16×C73—41no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73—81no—
123C23×C73C2×M1611yescharacteristic
124C22×D73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73C4×D411yescharacteristic
134C2×C4×C73C4×D411yescharacteristic
135C2×C4×C73C2×M1611yescharacteristic
136C2×C4×C73#13111yescharacteristic
137C2×C4×C73#13111yescharacteristic
138C22⋊D73—41no—
139C22⋊D73—41no—
140C22⋊D73—82no—
141C2×Dic73—22no—
142C2×Dic73—21no—
143D292—21no—
144D292—21no—
145C4×D73—41no—
146C4×D73—41no—
147Dic146—21no—
148Dic146—21no—
149C8×C73—41no—
150C8×C73#12111yescharacteristic
151C8×C73#12111yescharacteristic
152C8×C73#12411yescharacteristic
153C8×C73#12711yescharacteristic
154#10596831—22no—
155#10596831—41no—
156C22×C73—41no—
157C22×C73—41no—
158C22×C73—41no—
159C22×C73—41no—
160C22×C73—81no—
161C22×C73—81no—
162C22×C73#40711yescharacteristic
163C22×C73#43511yescharacteristic
164C22×C73#43511yescharacteristic
165C2×D73—42no—
166C2×D73—41no—
167C4×C73—41no—
168C4×C73#39011yescharacteristic
169C4×C73#40811yescharacteristic
170C4×C73#43511yescharacteristic
171C4×C73#43511yescharacteristic
172Dic73—42no—
173Dic73—41no—
174#13201—731nomaximal, Sylow
175C2×C73—21no—
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—81no—
179C2×C73#195511yescharacteristic
180D73—81no—
181#2930—731no—
182#2153—731no—
183#2117—731no—
184#2117—731no—
185C2×#349—731no—
186C2×#349D7311yescharacteristic, complemented
187#2110—731no—
188C73#1320111yescharacteristic, complemented, Sylow
189C2×#131—731no—
190C22×M16C2×D7311yescharacteristic
191#525—731no—
192C2×M32—731no—
193C2×M32—731no—
194C2×M32C2×D7311yescharacteristic
195C2×M32C2×D7311yescharacteristic
196#436—1462no—
197#435—1462no—
198#424—731no—
199#408—731no—
200#407—731no—
201#405—731no—
202#364—1462no—
203#350—1462no—
204#350—1462no—
205#349—22no—
206#349—1462no—
207#347—1462no—
208C2×#41—731no—
209C23×C4C4×D7311yescharacteristic
210#131—1461no—
211#131—1461no—
212#131—1461no—
213#131—1461no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—731no—
221C2×M16—731no—
222C2×M16—1461no—
223C22×C8—1461no—
224C22×C8C4×D7311yescharacteristic
225C22×C8C22×D7311yescharacteristic
226#123—1461no—
227C22⋊Q8—1461no—
228#121—1461no—
229#120—1461no—
230C4×D4—1462no—
231#117—731no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—732no—
236M32—1461no—
237M32—1461no—
238M32—1461no—
239M32C22⋊D7312yescomplemented
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1461no—
243C2×C16—1461no—
244C4⋊C8—1462no—
245C22⋊C8—1462no—
246C22⋊C8—1462no—
247#97—732no—
248C4×C8—732no—
249C24#1192401111yescharacteristic
250#41—1461no—
251#41—1461no—
252#41—1461no—
253#41—1461no—
254C2×Q8—731no—
255C2×D4—731no—
256C2×D4—1461no—
257C2×D4—1462no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—1461no—
265C22×C4#1192401111yescharacteristic
266C22×C4C2×C4×D7311yescharacteristic
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—2921no—
272M16—2921no—
273M16—1461no—
274M16—1461no—
275M16—1461no—
276M16—1461no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—2921no—
281C2×C8—2921no—
282C2×C8—732no—
283C2×C8—1461no—
284C2×C8D4×D7311yescharacteristic
285C2×C8D4×D7311yescharacteristic
286C2×C8D4×D7311yescharacteristic
287C2×C8C2×C22⋊D7311yescharacteristic
288C2×C8#1192402711yescharacteristic
289C2×C8#1192402811yescharacteristic
290C4⋊C4—1462no—
291C4⋊C4—1461no—
292C22⋊C4—1462no—
293C22⋊C4—1462no—
294C22⋊C4—1461no—
295C42—732no—
296C16—22no—
297C16—41no—
298C16—2921no—
299C16—1462no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—81no—
304C23—1461no—
305C23C2×#1192401111yescharacteristic
306Q8—1461no—
307Q8—1461no—
308D4—2921no—
309D4—2921no—
310D4—5842no—
311D4—1461no—
312D4—1461no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—1462no—
323C2×C4—1461no—
324C2×C4C22⋊C4×D7311yescharacteristic
325C2×C4#1455655111yescharacteristic
326C2×C4C4×C22⋊D7311yescharacteristic
327C2×C4M16×D7311yescharacteristic
328C2×C4#1455659711yescharacteristic
329C2×C4#1455659911yescharacteristic
330C8—41no—
331C8—2921no—
332C8—1462no—
333C8#1455656311yescharacteristic
334C8#1455656411yescharacteristic
335C8#1455657711yescharacteristic
336C8#1455658711yescharacteristic
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—81no—
343C22—2922no—
344C22—2921no—
345C22#1798854911yescharacteristic
346C22#1798865511yescharacteristic
347C22#1798865811yescharacteristic
348C4—41no—
349C4—2922no—
350C4—2921no—
351C4#1798848711yescharacteristic
352C4#1798855211yescharacteristic
353C4#1798865611yescharacteristic
354C4#1798866111yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—5841no—
360C2#2261072711yescharacteristic
subgroupG/H
centerC4#17988487
commutatorC22×C4×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#349×C73C2
socleC2×C73#1955
prime radical#17989098C4
cosocleC22×C8×C73C23
hypercenterC2×#349D73
hypocenterC73#13201
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
2. #22611696C2semidirect
3. #22611560C2semidirect
4. #22611558C2semidirect
5. C2×#17988369C2semidirect
6. #22611535C2semidirect
7. C2×#349×C73C2semidirect
45. #14556531D4semidirect
65. M32×C73D4semidirect
186. C2×#349D73semidirect
188. C73#13201semidirect
239. M32C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
#226186342
#226116962
#226115602
#226115582
C2×#179883692
#226115352
C2×#349×C732
#1320173