Finite Group Databasebeta

← #32977757

Subgroups — #32977757

Order 18688.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609923C211yesmaximal, characteristic, complemented
2#22608616C212yesmaximal, complemented
3#22608356C212yesmaximal, complemented
4#22608252C211yesmaximal, characteristic, complemented
5C2×#555×C73C211yesmaximal, characteristic
6#17988640C2212yes—
7#17988611C2212yes—
8C2×#14556508—21no—
9#17988356—21no—
10#17988333—24no—
11D4⋊#10596831—24no—
12#17988311—22no—
13#555×C73—24no—
14#555×C73—24no—
15C2×C4⋊Q8×C73C2211yescharacteristic
16C2×#124×C73—22no—
17C2×#121×C73—21no—
18C2×#121×C73C412yes—
19C2×C4×D4×C73C2211yescharacteristic
20C2×C4×D4×C73C2211yescharacteristic
21C2×#117×C73—21no—
22C2×#117×C73C412yes—
23C2×#11924012—22no—
24C4⋊#10596831—22no—
25C4⋊#10596831—22no—
26C4⋊#10596831—22no—
27#14556508—42no—
28C22⋊#10596831—22no—
29C4×#10596831—22no—
30C22×Q8×C73—21no—
31C22×D4×C73—22no—
32C22×D4×C73C2×C411yescharacteristic
33C22×D4×C73D411yescharacteristic
34C23×C4×C73—21no—
35C23×C4×C73C2×C412yes—
36C4⋊Q8×C73—22no—
37C4⋊Q8×C73D414yes—
38#124×C73—216no—
39#121×C73—28no—
40#121×C73—24no—
41#121×C73—24no—
42C4×D4×C73—22no—
43C4×D4×C73—22no—
44C4×D4×C73—22no—
45C4×D4×C73D414yes—
46#117×C73—28no—
47#117×C73—24no—
48#117×C73—24no—
49C2×C4⋊C4×C73—21no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C412yes—
52C2×C4⋊C4×C73D411yescharacteristic
53C2×C22⋊C4×C73—22no—
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73D412yes—
57C2×C42×C73—22no—
58C2×C42×C73C2311yescharacteristic
59#11924012—42no—
60#11924012—42no—
61C2×#10596831—42no—
62C2×#10596831—42no—
63C24×C73—21no—
64C24×C73C22⋊C412yes—
65C2×Q8×C73—24no—
66C2×Q8×C73—82no—
67C2×D4×C73—28no—
68C2×D4×C73—21no—
69C2×D4×C73—22no—
70C2×D4×C73—22no—
71C2×D4×C73—21no—
72C2×D4×C73—21no—
73C2×D4×C73—42no—
74C2×D4×C73—84no—
75C2×D4×C73C22⋊C412yes—
76C22×C4×C73—21no—
77C22×C4×C73—22no—
78C22×C4×C73—22no—
79C22×C4×C73—22no—
80C22×C4×C73—24no—
81C22×C4×C73—44no—
82C22×C4×C73—42no—
83C22×C4×C73—41no—
84C22×C4×C73—41no—
85C22×C4×C73C22⋊C412yes—
86C22×C4×C73C22×C411yescharacteristic
87C22×C4×C73C2×D411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C4⋊C4×C73—21no—
90C4⋊C4×C73—22no—
91C4⋊C4×C73—24no—
92C4⋊C4×C73—22no—
93C4⋊C4×C73C22⋊C412yes—
94C4⋊C4×C73C22⋊C414yes—
95C22⋊C4×C73—28no—
96C22⋊C4×C73—28no—
97C22⋊C4×C73—24no—
98C22⋊C4×C73—24no—
99C42×C73—28no—
100C42×C73C2×D412yes—
101C42×C73C2×D412yes—
102C23×C73—22no—
103C23×C73—22no—
104C23×C73—22no—
105C23×C73—22no—
106C23×C73—42no—
107C23×C73—42no—
108C23×C73—42no—
109C23×C73—42no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73C2×C22⋊C411yescharacteristic
113Q8×C73—84no—
114D4×C73—24no—
115D4×C73—42no—
116D4×C73—42no—
117D4×C73—42no—
118D4×C73—88no—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—22no—
122C2×C4×C73—24no—
123C2×C4×C73—41no—
124C2×C4×C73—44no—
125C2×C4×C73—44no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—44no—
129C2×C4×C73—44no—
130C2×C4×C73—44no—
131C2×C4×C73—42no—
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73C2×C22⋊C411yescharacteristic
134C2×C4×C73#12012yes—
135C2×C4×C73#12012yes—
136C2×C4×C73#13512yes—
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C73#13711yescharacteristic
139#10596831—82no—
140#10596831—82no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—42no—
144C22×C73—42no—
145C22×C73—42no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—82no—
150C22×C73—82no—
151C22×C73—82no—
152C22×C73—82no—
153C22×C73#37911yescharacteristic
154C22×C73#41611yescharacteristic
155C22×C73#41711yescharacteristic
156C4×C73—22no—
157C4×C73—21no—
158C4×C73—44no—
159C4×C73—42no—
160C4×C73—42no—
161C4×C73#41312yes—
162C4×C73#44912yes—
163C4×C73#45212yes—
164#8397—731nomaximal, Sylow
165C2×C73—21no—
166C2×C73—21no—
167C2×C73—82no—
168C2×C73—82no—
169C2×C73#178511yescharacteristic
170C2×C73#178811yescharacteristic
171C2×C73#179511yescharacteristic
172C2×#555D7311yescharacteristic
173#1800—731no—
174#1549—732no—
175#1486—732no—
176#1460—731no—
177C73#839711yescharacteristic, complemented, Sylow
178#555—24no—
179#555—24no—
180C2×C4⋊Q8C2×D7311yescharacteristic
181C2×#124—22no—
182C2×#121—21no—
183C2×#121Dic7312yes—
184C2×C4×D4C2×D7311yescharacteristic
185C2×C4×D4C2×D7311yescharacteristic
186C2×#117—21no—
187C2×#117Dic7312yes—
188#431—732no—
189#423—732no—
190C2×#100—1461no—
191#343—1461no—
192#332—1464no—
193D4⋊C8—1464no—
194#323—1462no—
195C22×Q8—21no—
196C22×D4—22no—
197C22×D4C2×Dic7311yescharacteristic
198C22×D4C22⋊D7311yescharacteristic
199C23×C4—21no—
200C23×C4C2×Dic7312yes—
201C2×M16—1462no—
202C4⋊Q8—22no—
203C4⋊Q8C22⋊D7314yes—
204#124—216no—
205#121—28no—
206#121—24no—
207#121—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4C22⋊D7314yes—
212#117—28no—
213#117—24no—
214#117—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×Dic7311yescharacteristic
217C2×C4⋊C4C2×Dic7312yes—
218C2×C4⋊C4C22⋊D7311yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4C22⋊D7312yes—
223C2×C42—22no—
224C2×C42C22×D7311yescharacteristic
225C4⋊C8—1462no—
226C4⋊C8—1462no—
227C4⋊C8—1462no—
228#100—2922no—
229C22⋊C8—1462no—
230C4×C8—1462no—
231C24—21no—
232C24C22⋊Dic7312yes—
233C2×Q8—24no—
234C2×Q8—82no—
235C2×D4—28no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—84no—
243C2×D4C22⋊Dic7312yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C22⋊Dic7312yes—
254C22×C4C22×Dic7311yescharacteristic
255C22×C4C2×C22⋊D7311yescharacteristic
256C22×C4C2×C22⋊D7311yescharacteristic
257M16—2922no—
258M16—2922no—
259C2×C8—2922no—
260C2×C8—2922no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4C22⋊Dic7312yes—
266C4⋊C4C22⋊Dic7314yes—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42—28no—
272C42C2×C22⋊D7312yes—
273C42C2×C22⋊D7312yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic7311yescharacteristic
285Q8—84no—
286D4—24no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—88no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4C2×C22⋊Dic7311yescharacteristic
305C2×C4C2×C22⋊Dic7311yescharacteristic
306C2×C4#1455656012yes—
307C2×C4#1455656012yes—
308C2×C4#1455661112yes—
309C2×C4#1455661611yescharacteristic
310C2×C4#1455662111yescharacteristic
311C8—5842no—
312C8—5842no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#1798844311yescharacteristic
326C22#1798858311yescharacteristic
327C22#1798858811yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4#1798857112yes—
334C4#1798873212yes—
335C4#1798874912yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—82no—
340C2#2260985911yescharacteristic
341C2#2260987111yescharacteristic
342C2#2260990111yescharacteristic
subgroupG/H
centerC22#17988443
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
Fitting, prime radicalC2×#555×C73C2
socleC22×C73#379
cosocleC2×C42×C73C23
hypercenterC2×#555D73
hypocenterC73#8397
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #22609923C2semidirect
2. #22608616C2semidirect
3. #22608356C2semidirect
4. #22608252C2semidirect
177. C73#8397semidirect

Maximal subgroups

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

subgroupindex
#226099232
#226086162
#226083562
#226082522
C2×#555×C732
#839773