Finite Group Databasebeta

← #33002909

Subgroups — #33002909

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623033C211yesmaximal, characteristic, complemented
2#22618634C211yesmaximal, characteristic
3#22611551C211yesmaximal, characteristic, complemented
4C2×#17988369C211yesmaximal, characteristic, complemented
5#22608018C211yesmaximal, characteristic, complemented
6#22608016C211yesmaximal, characteristic, complemented
7C2×#349×C73C211yesmaximal, characteristic, complemented
8C2×#14556597C2211yescharacteristic
9#17989098C411yescharacteristic
10#17989005—22no—
11#17989003—22no—
12C2×#14556531C2211yescharacteristic
13C2×#14556531C2211yescharacteristic
14M32×D73—22no—
15#17988663—22no—
16#17988656—22no—
17#17988613C411yescharacteristic
18#17988553C411yescharacteristic
19#17988551C411yescharacteristic
20#17988542C2211yescharacteristic
21#17988369—22no—
22#17988368—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×#11924011D412yes—
35C2×C8×D73D412yes—
36#14556574C812yes—
37#14556570—21no—
38#14556565—21no—
39#14556560C812yes—
40C4×C22⋊D73—22no—
41#14556550C2×C411yescharacteristic
42#14556531—21no—
43#14556531—21no—
44#14556531—21no—
45#14556531D412yescomplemented
46#14556530—24no—
47C2×#11924003—21no—
48C2×#11924003—21no—
49C16×D73—24no—
50#14556520—22no—
51C22⋊#10596831—22no—
52#14556503—22no—
53#14556501—21no—
54C8×Dic73—21no—
55C23×C4×C73C2×C411yescharacteristic
56C2×M16×C73—21no—
57C2×M16×C73—21no—
58C2×M16×C73—21no—
59C2×M16×C73—21no—
60C2×M16×C73—21no—
61C2×M16×C73—21no—
62C22×C8×C73C2×C411yescharacteristic
63C22×C8×C73C2311yescharacteristic
64M32×C73—21no—
65M32×C73—21no—
66M32×C73—21no—
67M32×C73D412yescomplemented
68C2×C16×C73—21no—
69C2×C16×C73—21no—
70#11924028—21no—
71#11924028—21no—
72#11924028—21no—
73#11924028—21no—
74C2×Dic146M1611yescharacteristic
75C2×D292M1611yescharacteristic
76C2×C22⋊D73—22no—
77C2×C22⋊D73C2×C812yes—
78C2×C4×D73C22⋊C412yes—
79#11924012—21no—
80#11924012—21no—
81#11924012—21no—
82#11924012—21no—
83#11924011—22no—
84#11924011—22no—
85C2×#10596831—21no—
86C2×#10596831C22⋊C412yes—
87C8×D73—22no—
88C8×D73—22no—
89C4⋊Dic73—21no—
90#11924007—22no—
91C22⋊Dic73—22no—
92C22⋊Dic73C2×C812yes—
93#11924005—22no—
94C4×Dic73—21no—
95#11924003—22no—
96#11924003—41no—
97C24×C73C2×C811yescharacteristic
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73C2×C811yescharacteristic
105C22×C4×C73C22×C411yescharacteristic
106M16×C73—41no—
107M16×C73—41no—
108M16×C73—41no—
109M16×C73—41no—
110C2×C8×C73—21no—
111C2×C8×C73—21no—
112C2×C8×C73—21no—
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73C2×D411yescharacteristic
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73#4111yescharacteristic
118C2×C8×C73#4111yescharacteristic
119C16×C73—22no—
120C16×C73—41no—
121C23×C73—21no—
122C23×C73—21no—
123C23×C73—21no—
124C23×C73—81no—
125C23×C73C22×C811yescharacteristic
126C22×D73C22⋊C812yes—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73C2×C22⋊C411yescharacteristic
135C2×C4×C73C4×D411yescharacteristic
136C2×C4×C73C4×D411yescharacteristic
137C2×C4×C73C2×M1611yescharacteristic
138C2×C4×C73#13111yescharacteristic
139C2×C4×C73#13111yescharacteristic
140C22⋊D73—22no—
141C22⋊D73—22no—
142C22⋊D73—82no—
143C2×Dic73—22no—
144C2×Dic73C22⋊C812yes—
145D292—21no—
146D292—21no—
147C4×D73—22no—
148C4×D73—22no—
149Dic146—21no—
150Dic146—21no—
151C8×C73—41no—
152C8×C73#12011yescharacteristic
153C8×C73#12111yescharacteristic
154C8×C73#12111yescharacteristic
155C8×C73#12311yescharacteristic
156#10596831—22no—
157#10596831—41no—
158C22×C73—41no—
159C22×C73—41no—
160C22×C73—41no—
161C22×C73—41no—
162C22×C73—81no—
163C22×C73—81no—
164C22×C73C2×C22⋊C811yescharacteristic
165C22×C73C8×D411yescharacteristic
166C22×C73C8×D411yescharacteristic
167C2×D73—22no—
168C2×D73—42no—
169C4×C73—41no—
170C4×C73#38611yescharacteristic
171C4×C73#40811yescharacteristic
172C4×C73#43511yescharacteristic
173C4×C73#43511yescharacteristic
174Dic73—22no—
175Dic73—42no—
176#12989—731nomaximal, Sylow
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—21no—
180C2×C73—81no—
181C2×C73#186911yescharacteristic
182D73—42no—
183#3407—731no—
184#2930—731no—
185#2115—731no—
186C2×#349—731no—
187C2×#349D7311yescharacteristic, complemented
188#1378—731no—
189#1378—731no—
190C73#1298911yescharacteristic, complemented, Sylow
191C2×#131—731no—
192C22×M16C2×D7311yescharacteristic
193#525—731no—
194#504—1462no—
195#504—1462no—
196C2×M32—731no—
197C2×M32—731no—
198C2×M32—1462no—
199C2×M32C2×D7311yescharacteristic
200C2×M32C2×D7311yescharacteristic
201#436—1462no—
202#435—1462no—
203#424—731no—
204#408—731no—
205#407—731no—
206#405—731no—
207#349—22no—
208#349—1462no—
209#349—1462no—
210C2×#41—731no—
211C23×C4C4×D7311yescharacteristic
212#131—1461no—
213#131—1461no—
214#131—1461no—
215#131—1461no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—731no—
223C2×M16—732no—
224C2×M16—731no—
225C22×C8—732no—
226C22×C8C4×D7311yescharacteristic
227C22×C8C22×D7311yescharacteristic
228#123—732no—
229C22⋊Q8—1461no—
230#121—1461no—
231#120—732no—
232C4×D4—1462no—
233#117—731no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—732no—
238M32—1464no—
239M32—1461no—
240M32—1461no—
241M32—1461no—
242M32C22⋊D7312yescomplemented
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1464no—
246C2×C16—1461no—
247C2×C16—1461no—
248C4⋊C8—1462no—
249C22⋊C8—1462no—
250C22⋊C8—1462no—
251#97—1461no—
252C4×C8—1461no—
253C24C8×D7311yescharacteristic
254#41—1461no—
255#41—1461no—
256#41—1461no—
257#41—1461no—
258C2×Q8—731no—
259C2×D4—732no—
260C2×D4—731no—
261C2×D4—1462no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—732no—
269C22×C4C8×D7311yescharacteristic
270C22×C4C2×C4×D7311yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—1461no—
276M16—1461no—
277M16—1461no—
278M16—1461no—
279M16—1462no—
280M16—1462no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—732no—
285C2×C8—1461no—
286C2×C8—1462no—
287C2×C8—1462no—
288C2×C8D4×D7311yescharacteristic
289C2×C8D4×D7311yescharacteristic
290C2×C8D4×D7311yescharacteristic
291C2×C8C2×C22⋊D7311yescharacteristic
292C2×C8#1192402711yescharacteristic
293C2×C8#1192402811yescharacteristic
294C4⋊C4—1462no—
295C4⋊C4—1461no—
296C22⋊C4—732no—
297C22⋊C4—1462no—
298C22⋊C4—1462no—
299C42—1461no—
300C16—22no—
301C16—41no—
302C16—2921no—
303C16—1462no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—81no—
308C23—732no—
309C23C2×C8×D7311yescharacteristic
310Q8—1461no—
311Q8—1461no—
312D4—5842no—
313D4—1462no—
314D4—1462no—
315D4—1461no—
316D4—1461no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—732no—
325C2×C4—1462no—
326C2×C4—1462no—
327C2×C4—1462no—
328C2×C4C22⋊C4×D7311yescharacteristic
329C2×C4#1455655111yescharacteristic
330C2×C4C4×C22⋊D7311yescharacteristic
331C2×C4M16×D7311yescharacteristic
332C2×C4#1455659711yescharacteristic
333C2×C4#1455659911yescharacteristic
334C8—41no—
335C8—2921no—
336C8—1462no—
337C8#1455655911yescharacteristic
338C8#1455656211yescharacteristic
339C8#1455656411yescharacteristic
340C8#1455657311yescharacteristic
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—2922no—
348C22—1462no—
349C22C22⋊C8×D7311yescharacteristic
350C22#1798865011yescharacteristic
351C22C8×C22⋊D7311yescharacteristic
352C4—41no—
353C4—2922no—
354C4—1462no—
355C4#1798846311yescharacteristic
356C4#1798855211yescharacteristic
357C4#1798865611yescharacteristic
358C4#1798866111yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—2922no—
364C2#2261022011yescharacteristic
subgroupG/H
centerC4#17988463
commutatorC23×C73C22×C8
FrattiniC22×C8C22×D73
FittingC2×#349×C73C2
socleC2×C73#1869
prime radical#17989098C4
cosocleC22×C8×C73C23
hypercenterC2×#349D73
hypocenterC73#12989
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
1. #22623033C2semidirect
3. #22611551C2semidirect
4. C2×#17988369C2semidirect
5. #22608018C2semidirect
6. #22608016C2semidirect
7. C2×#349×C73C2semidirect
45. #14556531D4semidirect
67. M32×C73D4semidirect
187. C2×#349D73semidirect
190. C73#12989semidirect
242. M32C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
#226230332
#226186342
#226115512
C2×#179883692
#226080182
#226080162
C2×#349×C732
#1298973