Finite Group Databasebeta

← #33005152

Subgroups — #33005152

Order 18688.

446 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988825C211yesmaximal, characteristic
2C2×#17988415C211yesmaximal, characteristic, complemented
3C2×#17988391C211yesmaximal, characteristic, complemented
4C2×#17988366C211yesmaximal, characteristic
5#22609957C211yesmaximal, characteristic
6#22608022C211yesmaximal, characteristic, complemented
7C2×#358×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9C2×#14556567—21no—
10C22×#11924003C2211yescharacteristic
11#17988825—22no—
12#17988825—22no—
13C2×#14556525C2211yescharacteristic
14C2×#14556525C2211yescharacteristic
15C2×#14556515—21no—
16#17988444—21no—
17#17988415—22no—
18#17988391—22no—
19#17988366—22no—
20#17988349—21no—
21C22×Q16×C73C2211yescharacteristic
22C22×C16×C73C2211yescharacteristic
23C2×#107×C73C2211yescharacteristic
24#358×C73—22no—
25C22×C4×D73D411yescharacteristic
26C22×#10596831D411yescharacteristic
27C2×C8×D73—21no—
28C2×C8×D73—21no—
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73—21no—
33#14556567—24no—
34#14556567—42no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37C2×#11924007—41no—
38C2×#11924005—41no—
39C2×#11924005—41no—
40C2×#11924003—21no—
41C2×#11924003—21no—
42C2×#11924003—21no—
43#14556525—21no—
44#14556525—21no—
45#14556525—21no—
46#14556525D412yes—
47#14556515—24no—
48#14556498—41no—
49C22×Q8×C73—21no—
50C2×Q16×C73—21no—
51C2×Q16×C73—21no—
52C2×Q16×C73—21no—
53C2×Q16×C73—21no—
54C2×Q16×C73—21no—
55C2×Q16×C73D412yes—
56C22×C8×C73C2311yescharacteristic
57C2×C4⋊C4×C73—21no—
58C2×C16×C73—21no—
59C2×C16×C73—21no—
60C2×C16×C73D412yes—
61#107×C73—21no—
62#107×C73—21no—
63C23×D73D811yescharacteristic
64C22×Dic73—41no—
65C22×Dic73—41no—
66C22×Dic73D811yescharacteristic
67C2×C4×D73—21no—
68C2×C4×D73—21no—
69C2×C4×D73—21no—
70C2×C4×D73—21no—
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×#10596831—21no—
74C2×#10596831—21no—
75C2×#10596831—21no—
76C8×D73—41no—
77C8×D73—41no—
78C8×D73—41no—
79C8×D73—41no—
80C4⋊Dic73—22no—
81C4⋊Dic73—41no—
82C4⋊Dic73—41no—
83C4⋊Dic73—41no—
84#11924007—44no—
85#11924005—44no—
86#11924005—82no—
87#11924003—41no—
88C2×Q8×C73—22no—
89C2×Q8×C73—41no—
90C2×Q8×C73—41no—
91C2×Q8×C73—41no—
92C2×Q8×C73—41no—
93C2×Q8×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73C2×D411yescharacteristic
97Q16×C73—22no—
98Q16×C73—22no—
99Q16×C73—41no—
100Q16×C73—41no—
101C2×C8×C73C2×D411yescharacteristic
102C2×C8×C73C2×D411yescharacteristic
103C2×C8×C73C2×D411yescharacteristic
104C2×C8×C73#4111yescharacteristic
105C2×C8×C73#4111yescharacteristic
106C2×C8×C73#4111yescharacteristic
107C4⋊C4×C73—41no—
108C4⋊C4×C73—41no—
109C16×C73—22no—
110C23×C73C2×D811yescharacteristic
111C22×D73—21no—
112C22×D73—21no—
113C22×D73—21no—
114C22×D73—21no—
115C22×D73—21no—
116C22×D73—21no—
117C22×D73SD3212yes—
118Q8×C73—42no—
119Q8×C73—42no—
120Q8×C73—81no—
121Q8×C73—81no—
122C2×C4×C73—42no—
123C2×C4×C73—81no—
124C2×C4×C73—81no—
125C2×C4×C73—81no—
126C2×C4×C73—81no—
127C2×C4×C73—81no—
128C2×C4×C73#12011yescharacteristic
129C2×C4×C73#12111yescharacteristic
130C2×C4×C73#12311yescharacteristic
131C2×C4×C73#13511yescharacteristic
132C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C73#13711yescharacteristic
134C2×Dic73—21no—
135C2×Dic73—21no—
136C2×Dic73—21no—
137C2×Dic73—42no—
138C2×Dic73—81no—
139C2×Dic73—81no—
140C2×Dic73—81no—
141C2×Dic73—81no—
142C2×Dic73—81no—
143C4×D73—41no—
144C4×D73—41no—
145C4×D73—41no—
146C4×D73—41no—
147C8×C73#12111yescharacteristic
148C8×C73#12111yescharacteristic
149C8×C73#12311yescharacteristic
150C8×C73#12411yescharacteristic
151#10596831—41no—
152C22×C73#44711yescharacteristic
153C22×C73#46611yescharacteristic
154C22×C73#48011yescharacteristic
155C22×C73C2×SD3211yescharacteristic
156C22×C73#50811yescharacteristic
157C22×C73#51011yescharacteristic
158C22×C73#51011yescharacteristic
159C2×D73—22no—
160C2×D73—22no—
161C2×D73—22no—
162C2×D73—41no—
163C2×D73—41no—
164C2×D73—41no—
165C2×D73—41no—
166C4×C73—82no—
167C4×C73—161no—
168C4×C73#39411yescharacteristic
169C4×C73#45211yescharacteristic
170C4×C73#46811yescharacteristic
171C4×C73#48211yescharacteristic
172Dic73—41no—
173Dic73—82no—
174Dic73—161no—
175#13382—731nomaximal, Sylow
176C2×C73#203311yescharacteristic
177C2×C73#218511yescharacteristic
178C2×C73#218911yescharacteristic
179C2×C73#221711yescharacteristic
180C2×C73#221911yescharacteristic
181C2×C73#223511yescharacteristic
182C2×C73#223711yescharacteristic
183D73—42no—
184C2×#467—731no—
185C2×#367—731no—
186C2×#358—731no—
187C2×#358D7311yescharacteristic, complemented
188C2×C22⋊C16—731no—
189#1808—731no—
190#1380—731no—
191C73#1338211yescharacteristic, complemented, Sylow
192C22×Q16C2×D7311yescharacteristic
193C23×C8—731no—
194C2×C22⋊Q8—1461no—
195C22×C16—731no—
196C22×C16C2×D7311yescharacteristic
197#467—1462no—
198#467—1462no—
199C2×#107—731no—
200C2×#107—731no—
201C2×#107C2×D7311yescharacteristic
202C2×#103—1461no—
203#380—1461no—
204#367—1462no—
205#358—22no—
206#358—1462no—
207C22⋊C16—1462no—
208#340—1461no—
209C22×Q8—21no—
210C23×C4—731no—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16D29212yes—
217C22×C8—731no—
218C22×C8—1461no—
219C22×C8—1461no—
220C22×C8—1461no—
221C22×C8—1461no—
222C22×C8—1461no—
223C22×C8—1461no—
224C22×C8C22×D7311yescharacteristic
225C22⋊Q8—2922no—
226C22⋊Q8—1464no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—2921no—
229C2×C4⋊C4—1461no—
230C2×C4⋊C4—1461no—
231C2×C22⋊C4—2921no—
232C2×C22⋊C4—2921no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—1461no—
236C2×C16—1461no—
237C2×C16—1461no—
238C2×C16C22⋊D7312yes—
239#107—21no—
240#107—21no—
241#107—732no—
242#107—1461no—
243#107—1461no—
244#107—1461no—
245#103—1464no—
246#95—2921no—
247C24—731no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—2921no—
257C22×C4—2921no—
258C22×C4—731no—
259C22×C4—1461no—
260C22×C4—1461no—
261C22×C4—1461no—
262C22×C4—1461no—
263C22×C4—1461no—
264C22×C4—1461no—
265C22×C4D4×D7311yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—41no—
269Q16—41no—
270C2×C8—2921no—
271C2×C8—2921no—
272C2×C8—2921no—
273C2×C8—2921no—
274C2×C8—1461no—
275C2×C8—1461no—
276C2×C8—1461no—
277C2×C8D4×D7311yescharacteristic
278C2×C8C2×C22⋊D7311yescharacteristic
279C2×C8C2×D29211yescharacteristic
280C2×C8#1192402711yescharacteristic
281C2×C8#1192402711yescharacteristic
282C2×C8#1192402811yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—2924no—
286C4⋊C4—2921no—
287C4⋊C4—2921no—
288C4⋊C4—2921no—
289C4⋊C4—1462no—
290C22⋊C4—2924no—
291C22⋊C4—5842no—
292C16—22no—
293C16—2921no—
294C23—732no—
295C23—1461no—
296C23—1461no—
297C23—1461no—
298C23—1461no—
299C23—1461no—
300C23—1461no—
301C23D8×D7311yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—2921no—
313C2×C4—2921no—
314C2×C4—2921no—
315C2×C4—2921no—
316C2×C4—2922no—
317C2×C4—5841no—
318C2×C4—5841no—
319C2×C4—5841no—
320C2×C4—5841no—
321C2×C4—5841no—
322C2×C4—1461no—
323C2×C4—1461no—
324C2×C4—1461no—
325C2×C4#1455656111yescharacteristic
326C2×C4#1455656211yescharacteristic
327C2×C4#1455657311yescharacteristic
328C2×C4#1455661011yescharacteristic
329C2×C4#1455661811yescharacteristic
330C2×C4#1455662011yescharacteristic
331C8—2921no—
332C8#1455656311yescharacteristic
333C8#1455656511yescharacteristic
334C8#1455657511yescharacteristic
335C8#1455657711yescharacteristic
336C22—2921no—
337C22—2921no—
338C22—2921no—
339C22—2921no—
340C22—1462no—
341C22—1462no—
342C22—1462no—
343C22#1798872211yescharacteristic
344C22#1798882011yescharacteristic
345C22#1798889911yescharacteristic
346C22SD32×D7311yescharacteristic
347C22#1798901811yescharacteristic
348C22#1798902911yescharacteristic
349C22#1798902911yescharacteristic
350C4—82no—
351C4—161no—
352C4—2921no—
353C4—5842no—
354C4—11681no—
355C4#1798850311yescharacteristic
356C4#1798875011yescharacteristic
357C4#1798883011yescharacteristic
358C4#1798890911yescharacteristic
359C2—2922no—
360C2#2261111211yescharacteristic
361C2#2261184911yescharacteristic
362C2#2261187411yescharacteristic
363C2#2261202411yescharacteristic
364C2#2261203611yescharacteristic
365C2#2261212711yescharacteristic
366C2#2261213711yescharacteristic
subgroupG/H
centerC23D8×D73
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC2×#358×C73C2
socleC23×C73C2×D8
prime radicalC23×D73D8
hypercenterC2×#358D73
hypocenterC73#13382
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×#17988415C2semidirect
3. C2×#17988391C2semidirect
6. #22608022C2semidirect
7. C2×#358×C73C2semidirect
187. C2×#358D73semidirect
191. C73#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#179888252
C2×#179884152
C2×#179883912
C2×#179883662
#226099572
#226080222
C2×#358×C732
#1338273