Finite Group Databasebeta

← #32996491

Subgroups — #32996491

Order 18688.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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#22615508C211yesmaximal, characteristic
2#22609670C211yesmaximal, characteristic
3#22607962C212yesmaximal, complemented
4#22607938C212yesmaximal, complemented
5C73×#1464C211yesmaximal, characteristic
6C22×#11924012—21no—
7C2×C4×Dic146C411yescharacteristic
8#17988517C2211yescharacteristic
9#17988512C411yescharacteristic
10#17988507C2211yescharacteristic
11#17988499C411yescharacteristic
12#17988475—21no—
13#17988471—21no—
14#17988468—21no—
15#17988460—21no—
16#17988455C411yescharacteristic
17#17988356—21no—
18#17988356C2212yes—
19C2×#117×C73C2211yescharacteristic
20#432×C73—21no—
21#324×C73C2212yes—
22#323×C73—21no—
23C22×Dic146C2×C411yescharacteristic
24C2×#11924012—24no—
25C2×#11924012—22no—
26C2×#11924012—41no—
27C2×#11924012—42no—
28C22×#10596831—41no—
29C4×Dic146—41no—
30C4×Dic146—41no—
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×#11924007—21no—
33C2×#11924007—21no—
34C2×#11924007—21no—
35C2×C22⋊Dic73—21no—
36C2×C22⋊Dic73—21no—
37C2×C4×Dic73—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498C2×C411yescharacteristic
45#14556498C2×C411yescharacteristic
46C23×C4×C73D411yescharacteristic
47#117×C73—41no—
48#117×C73—41no—
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C4⋊C4×C73C2311yescharacteristic
51C2×C22⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73D411yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54C2×C42×C73D411yescharacteristic
55C4⋊C8×C73—41no—
56C22⋊C8×C73—22no—
57#97×C73—41no—
58C2×Dic146—41no—
59C2×Dic146—42no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64#11924012—24no—
65#11924012—44no—
66#11924012—42no—
67C2×#10596831—44no—
68C2×#10596831—42no—
69C4⋊Dic73—41no—
70#11924007—42no—
71#11924007—41no—
72#11924007—41no—
73#11924007—41no—
74#11924007—41no—
75C22⋊Dic73—42no—
76C22⋊Dic73—42no—
77C4×Dic73—42no—
78C24×C73C22⋊C411yescharacteristic
79C22×C4×C73—22no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C2×C8×C73—42no—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—41no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C42×C73—21no—
96C42×C73—21no—
97C42×C73—21no—
98C42×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73#10412yes—
104C23×C73C2×C22⋊C411yescharacteristic
105C2×C4×C73—22no—
106C2×C4×C73—22no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73#9911yescharacteristic
122C2×C4×C73#9911yescharacteristic
123C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C73#13711yescharacteristic
125C2×Dic73—22no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137Dic146—42no—
138Dic146—42no—
139C8×C73—82no—
140#10596831—44no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—22no—
144C22×C73—42no—
145C22×C73—41no—
146C22×C73—41no—
147C22×C73—81no—
148C22×C73C2×#9911yescharacteristic
149C22×C73#41711yescharacteristic
150C22×C73C2×#10411yescharacteristic
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—41no—
159Dic73—42no—
160Dic73—81no—
161Dic73—81no—
162Dic73—81no—
163#11763—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—42no—
167C2×C73#150411yescharacteristic
168C2×C73#212011yescharacteristic
169C2×C73#213511yescharacteristic
170#2599—731no—
171#1742—731no—
172#1464D7311yescharacteristic
173#1354—732no—
174#1346—732no—
175C73#1176311yescharacteristic, complemented, Sylow
176C22×M16—1461no—
177C2×C4×Q8—731no—
178C2×#117C2×D7311yescharacteristic
179#432—21no—
180#398—731no—
181#397—731no—
182#395—731no—
183#393—731no—
184#388—1461no—
185#387—1461no—
186#387—1461no—
187#385—1461no—
188#383—731no—
189#343—732no—
190#343—1461no—
191#324C2×D7312yes—
192#323—21no—
193C22×Q8—731no—
194C23×C4D29211yescharacteristic
195C2×M16—2921no—
196C2×M16—2922no—
197C2×M16—1464no—
198C2×M16—1462no—
199C22×C8—2921no—
200C4×Q8—2921no—
201C4×Q8—2921no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4—1461no—
208C2×C4⋊C4D29211yescharacteristic
209C2×C4⋊C4C22×D7311yescharacteristic
210C2×C22⋊C4—1461no—
211C2×C22⋊C4—1461no—
212C2×C22⋊C4C4×D7311yescharacteristic
213C2×C22⋊C4C22⋊D7311yescharacteristic
214C2×C42—1461no—
215C2×C42C4×D7311yescharacteristic
216C2×C42C22⋊D7311yescharacteristic
217C4⋊C8—41no—
218C22⋊C8—22no—
219#97—41no—
220#95—731no—
221#95—731no—
222#95—1461no—
223#95—1461no—
224#95—1461no—
225#95—1461no—
226#95—1461no—
227#95—1461no—
228C24#1192400511yescharacteristic
229C2×Q8—2921no—
230C2×Q8—2922no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1461no—
237C22×C4—1461no—
238C22×C4—1461no—
239C22×C4—1461no—
240C22×C4#1192400511yescharacteristic
241C22×C4#1192400511yescharacteristic
242C22×C4#1192400511yescharacteristic
243C22×C4C2×C4×D7311yescharacteristic
244C22×C4C2×C22⋊D7311yescharacteristic
245C22×C4C2×D29211yescharacteristic
246M16—2924no—
247M16—2922no—
248M16—1464no—
249C2×C8—42no—
250C2×C8—2924no—
251C2×C8—2922no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2921no—
255C4⋊C4—2922no—
256C4⋊C4—2921no—
257C4⋊C4—2921no—
258C4⋊C4—2921no—
259C4⋊C4—2921no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2922no—
263C22⋊C4—2922no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—2922no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1455651712yes—
274C23C2×#1192400511yescharacteristic
275Q8—2922no—
276Q8—2922no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—2921no—
294C2×C4—2921no—
295C2×C4—2921no—
296C2×C4—2921no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—1462no—
305C2×C4#1455650511yescharacteristic
306C2×C4#1455650511yescharacteristic
307C2×C4#1455661911yescharacteristic
308C2×C4#1455662411yescharacteristic
309C8—82no—
310C8—2924no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1455650511yescharacteristic
319C22#1798858911yescharacteristic
320C22C2×#1455651711yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—2922no—
330C4—5841no—
331C4—5841no—
332C4—5841no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2260843611yescharacteristic
337C2#2261157011yescharacteristic
338C2#2261162511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1464C2
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1464D73
hypocenterC73#11763
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #22607962C2semidirect
4. #22607938C2semidirect
175. C73#11763semidirect

Maximal subgroups

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

subgroupindex
#226155082
#226096702
#226079622
#226079382
C73×#14642
#1176373