Finite Group Databasebeta

← #34008202

Subgroups — #34008202

Order 18688.

450 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625864C211yesmaximal, characteristic
2C4×#14556658C211yesmaximal, characteristic
3#22624809C212yesmaximal
4#22624809C212yesmaximal
5#22618290C211yesmaximal, characteristic
6C2×#14556658—21no—
7C2×#14556658C2212yes—
8C2×#14556658C2212yes—
9C2×C4⋊#10596833—21no—
10C2×C4×#10596833—21no—
11#17989453—21no—
12C2×#14556581—21no—
13#17988486C411yescharacteristic
14#17988485C411yescharacteristic
15#17988484—21no—
16#17988483C411yescharacteristic
17#17988482—21no—
18#17988456C2211yescharacteristic
19#17988456C2211yescharacteristic
20#17988451—21no—
21C42×Dic73C2211yescharacteristic
22#389×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25#14556658—22no—
26#14556658—24no—
27#14556658—42no—
28#14556658Q814yes—
29C4⋊#10596833—42no—
30C4×#10596833—42no—
31#14556581—42no—
32#14556581—42no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×#11924007—21no—
37C2×#11924007—21no—
38C2×#11924007—21no—
39C2×#11924007—21no—
40C2×#11924007—21no—
41C2×#11924007—21no—
42C2×C4×Dic73C2×C411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73D411yescharacteristic
46C2×C4×Dic73D411yescharacteristic
47C2×C4×Dic73C2311yescharacteristic
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C4⋊C4×C73C2×C411yescharacteristic
54C2×C42×C73C2×C411yescharacteristic
55#95×C73—21no—
56C2×#10596833—24no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73C22⋊C411yescharacteristic
66C22×Dic73C2×D411yescharacteristic
67C22×Dic73C2×D411yescharacteristic
68C4⋊Dic73—42no—
69C4⋊Dic73—41no—
70C4⋊Dic73—41no—
71#11924007—42no—
72#11924007—42no—
73#11924007—41no—
74#11924007—41no—
75#11924007—41no—
76#11924007—41no—
77#11924007—42no—
78#11924007—41no—
79#11924007—41no—
80C4×Dic73—22no—
81C4×Dic73—21no—
82C4×Dic73—21no—
83C4×Dic73—41no—
84C4×Dic73D812yes—
85C4×Dic73SD1612yes—
86C4×Dic73SD1612yes—
87C4×Dic73Q1612yes—
88C4×Dic73C2×Q812yes—
89C4×Dic73#4112yes—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22×C411yescharacteristic
95C4⋊C4×C73—41no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—42no—
98C42×C73—41no—
99C23×C73C2×C22⋊C411yescharacteristic
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73#10212yes—
113C2×C4×C73#10312yes—
114C2×C4×C73C4×Q812yes—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—41no—
120C2×Dic73—41no—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73#10212yes—
126C2×Dic73#10312yes—
127C2×Dic73C22⋊Q812yes—
128C2×Dic73#12411yescharacteristic
129C2×Dic73C4⋊Q811yescharacteristic
130C2×Dic73C2×D811yescharacteristic
131C2×Dic73C2×SD1611yescharacteristic
132C2×Dic73C2×SD1611yescharacteristic
133C2×Dic73C2×Q1611yescharacteristic
134C2×Dic73#13512yes—
135C2×Dic73AGL(1,ℤ/8)11yescharacteristic
136C2×Dic73#13711yescharacteristic
137#10596833—44no—
138#10596833—82no—
139C22×C73#39111yescharacteristic
140C22×C73#41311yescharacteristic
141C22×C73#41311yescharacteristic
142C22×C73C2×#10211yescharacteristic
143C22×C73C2×#10311yescharacteristic
144C22×C73#41611yescharacteristic
145C22×C73#41711yescharacteristic
146C4×C73—22no—
147C4×C73—22no—
148C4×C73—22no—
149C4×C73—81no—
150C4×C73—81no—
151Dic73—22no—
152Dic73—22no—
153Dic73—81no—
154Dic73—81no—
155Dic73#47812yes—
156Dic73#47912yes—
157Dic73#48611yescharacteristic
158Dic73#48711yescharacteristic
159Dic73#49811yescharacteristic
160Dic73#50011yescharacteristic
161#9900—731nomaximal, Sylow
162C2×C73#154011yescharacteristic
163C2×C73#154211yescharacteristic
164C2×C73#154711yescharacteristic
165C2×C73#155011yescharacteristic
166C2×C73#198911yescharacteristic
167C2×C73#199011yescharacteristic
168C2×C73#199811yescharacteristic
169#2859—731no—
170#1842—731no—
171C4×C4⋊C8—731no—
172#1296—732no—
173#1296—732no—
174C73#990011yescharacteristic, complemented, Sylow
175C2×#125—1461no—
176C2×C4⋊C8—732no—
177C2×C4⋊C8—732no—
178C2×C4⋊C8—1461no—
179C2×C4⋊C8—1461no—
180C2×C4×C8—1461no—
181#389—731no—
182#389—731no—
183#389—731no—
184#389—1461no—
185#389—1461no—
186#389#6440811yescharacteristic
187#384—731no—
188#384—731no—
189#382—1461no—
190C43—731no—
191#336—1461no—
192C22×C8—1462no—
193C22×C8—1462no—
194#125—2922no—
195#125—2922no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—731no—
198C2×C4⋊C4—731no—
199C2×C4⋊C4—1461no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4—1461no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4—1461no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4C2×#6440811yescharacteristic
207C2×C4⋊C4C2×#6440811yescharacteristic
208C2×C42—731no—
209C2×C42—731no—
210C2×C42—731no—
211C2×C42—731no—
212C2×C42—731no—
213C2×C42—731no—
214C2×C42C2×#6440811yescharacteristic
215C4⋊C8—2922no—
216C4⋊C8—2922no—
217C4⋊C8—734no—
218C4⋊C8—1464no—
219C4⋊C8—1462no—
220C4×C8—2922no—
221#95—21no—
222#95—1461no—
223#95—1461no—
224#95—1461no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—731no—
231C22×C4—1461no—
232C22×C4—1461no—
233C22×C4C22⋊#6440811yescharacteristic
234C22×C4C22⋊#6440811yescharacteristic
235C22×C4C22×#6440811yescharacteristic
236C2×C8—2922no—
237C2×C8—2922no—
238C2×C8—2922no—
239C2×C8—2922no—
240C2×C8—2922no—
241C2×C8—1464no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—2921no—
246C4⋊C4—2921no—
247C4⋊C4—2922no—
248C4⋊C4—2921no—
249C4⋊C4—2921no—
250C4⋊C4—2921no—
251C4⋊C4—2921no—
252C4⋊C4—2921no—
253C4⋊C4—2921no—
254C4⋊C4—2922no—
255C4⋊C4—2922no—
256C4⋊C4—2922no—
257C42—41no—
258C42—2921no—
259C42—732no—
260C42—732no—
261C42—732no—
262C42—732no—
263C42—732no—
264C42—732no—
265C42—1461no—
266C42—1462no—
267C42—1461no—
268C23C2×C22⋊#6440811yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—2921no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—731no—
288C2×C4—732no—
289C2×C4—731no—
290C2×C4—731no—
291C2×C4—731no—
292C2×C4—731no—
293C2×C4—731no—
294C2×C4—732no—
295C2×C4—731no—
296C2×C4—731no—
297C2×C4—732no—
298C2×C4—732no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1461no—
303C2×C4#1455665312yes—
304C2×C4#1455665412yes—
305C2×C4Q8×#6440812yes—
306C8—2924no—
307C8—5842no—
308C22(C2×#64408)⋊Q811yescharacteristic
309C22#1798957311yescharacteristic
310C22#1798957411yescharacteristic
311C22C2×#1455665311yescharacteristic
312C22C2×#1455665411yescharacteristic
313C22#1798957711yescharacteristic
314C22#1798957811yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—81no—
319C4—81no—
320C4—5841no—
321C4—5841no—
322C4—732no—
323C4—732no—
324C4—731no—
325C4—731no—
326C4—731no—
327C4—731no—
328C4—1462no—
329C4—1462no—
330C2#2262542911yescharacteristic
331C2#2262543211yescharacteristic
332C2#2262544211yescharacteristic
333C2#2262544911yescharacteristic
334C2#2262611111yescharacteristic
335C2#2262611211yescharacteristic
336C2#2262612511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#389×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#389#64408
hypocenterC73#9900
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
174. C73#9900semidirect

Maximal subgroups

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

subgroupindex
#226258642
C4×#145566582
#226248092
#226182902
#990073