Finite Group Databasebeta

← #33002251

Subgroups — #33002251

Order 18688.

642 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 99 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#22612422C211yesmaximal, characteristic
2#22609859C211yesmaximal, characteristic
3#22608057C214yesmaximal, complemented
4C73×#1785C211yesmaximal, characteristic
5C2×#14556550C411yescharacteristic
6C22×C4×Dic73C411yescharacteristic
7C2×#14556510C2212yes—
8C2×#14556508—21no—
9#17988551—21no—
10#17988494C2212yes—
11#17988490C412yes—
12#17988467—21no—
13#17988467—21no—
14C4×C22⋊Dic73—21no—
15C4×C22⋊Dic73—21no—
16C22×#41×C73C2211yescharacteristic
17C2×#101×C73C2212yes—
18C2×#100×C73—21no—
19#407×C73—21no—
20C23×Dic73—21no—
21C2×#11924012—22no—
22#14556550—42no—
23C2×C4⋊Dic73C2×C412yes—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73—21no—
27C2×C4×Dic73—21no—
28C2×C4×Dic73—21no—
29C2×C4×Dic73—42no—
30C2×C4×Dic73C2×C412yes—
31#14556510—24no—
32#14556508—42no—
33C22⋊#10596831—42no—
34#14556498—22no—
35#14556498—22no—
36C2×#41×C73—24no—
37C2×#41×C73—42no—
38C2×#41×C73—42no—
39C22×Q8×C73C2311yescharacteristic
40C22×D4×C73C2×C411yescharacteristic
41C22×D4×C73D411yescharacteristic
42C22×D4×C73D411yescharacteristic
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73D411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C2×M16×C73—22no—
47#101×C73—24no—
48#100×C73—42no—
49C22⋊C8×C73—42no—
50C22×Dic73—22no—
51C22×Dic73—22no—
52C22×Dic73—44no—
53C22×Dic73—42no—
54#11924012—42no—
55#11924012—42no—
56C2×#10596831—42no—
57C4⋊Dic73—42no—
58C22⋊Dic73—44no—
59C22⋊Dic73—44no—
60C22⋊Dic73—42no—
61C4×Dic73—42no—
62C4×Dic73—42no—
63C4×Dic73—42no—
64C24×C73C22⋊C411yescharacteristic
65C24×C73C22⋊C411yescharacteristic
66C24×C73C22⋊C411yescharacteristic
67#41×C73—416no—
68C2×Q8×C73—41no—
69C2×Q8×C73—41no—
70C2×Q8×C73#4114yes—
71C2×D4×C73—22no—
72C2×D4×C73—22no—
73C2×D4×C73—22no—
74C2×D4×C73—22no—
75C2×D4×C73—22no—
76C2×D4×C73—41no—
77C2×D4×C73—41no—
78C2×D4×C73—41no—
79C2×D4×C73—41no—
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—41no—
88C22×C4×C73—41no—
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93M16×C73—42no—
94M16×C73—42no—
95C2×C8×C73—42no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—42no—
105C23×C73—42no—
106C23×C73—42no—
107C23×C73#9912yes—
108C23×C73C2×C22⋊C411yescharacteristic
109Q8×C73—44no—
110D4×C73—44no—
111D4×C73—44no—
112D4×C73—44no—
113C2×C4×C73—28no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—44no—
123C2×C4×C73—44no—
124C2×C4×C73#9912yes—
125C2×C4×C73#11712yes—
126C2×C4×C73#12312yes—
127C2×C4×C73#12312yes—
128C2×Dic73—24no—
129C2×Dic73—42no—
130C2×Dic73—48no—
131C2×Dic73—42no—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C8×C73—82no—
136#10596831—82no—
137C22×C73—22no—
138C22×C73—22no—
139C22×C73—42no—
140C22×C73—42no—
141C22×C73—42no—
142C22×C73—42no—
143C22×C73—42no—
144C22×C73—41no—
145C22×C73—41no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73—81no—
149C22×C73—81no—
150C22×C73#38111yescharacteristic
151C22×C73C2×#9911yescharacteristic
152C22×C73C2×#9911yescharacteristic
153C4×C73—22no—
154C4×C73—22no—
155C4×C73—22no—
156C4×C73—22no—
157Dic73—44no—
158Dic73—82no—
159#12861—731nomaximal, Sylow
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—42no—
163C2×C73—81no—
164C2×C73—81no—
165C2×C73#181911yescharacteristic
166C2×C73#213212yes—
167#2297—731no—
168#1785—731no—
169#1785D7311yescharacteristic
170#1394—734no—
171C73#1286111yescharacteristic, complemented, Sylow
172C22×#41C2×D7311yescharacteristic
173C2×#117—731no—
174C22×C42—731no—
175C2×#101—732no—
176C2×#101C2×D7312yes—
177C2×#100—21no—
178C2×#100—1461no—
179#407—21no—
180#407—1461no—
181#391—732no—
182#390—732no—
183#386—1461no—
184#386—1461no—
185C4×C22⋊C4—1461no—
186C4×C22⋊C4—1461no—
187C2×#41—24no—
188C2×#41—42no—
189C2×#41—42no—
190C22×Q8C22×D7311yescharacteristic
191C22×D4C4×D7311yescharacteristic
192C22×D4D29211yescharacteristic
193C22×D4C22⋊D7311yescharacteristic
194C23×C4—1461no—
195C23×C4C4×D7311yescharacteristic
196C23×C4D29211yescharacteristic
197C23×C4C22⋊D7311yescharacteristic
198C2×M16—22no—
199C2×M16—1462no—
200#117—2922no—
201C2×C4⋊C4—732no—
202C2×C22⋊C4—1461no—
203C2×C22⋊C4—1462no—
204C2×C22⋊C4—1462no—
205C2×C42—2922no—
206C2×C42—732no—
207C2×C42—1461no—
208C2×C42—1461no—
209#101—24no—
210#101—1464no—
211#100—42no—
212#100—2922no—
213C22⋊C8—42no—
214C22⋊C8—2922no—
215#95—1462no—
216#95—1462no—
217C24#1192400511yescharacteristic
218C24#1192400511yescharacteristic
219C24#1192400511yescharacteristic
220#41—416no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8#1192402714yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—2922no—
243C22×C4—2924no—
244C22×C4—1462no—
245C22×C4—1462no—
246C22×C4#1192400511yescharacteristic
247C22×C4C2×C4×D7311yescharacteristic
248C22×C4C2×C22⋊D7311yescharacteristic
249C22×C4C2×D29211yescharacteristic
250M16—42no—
251M16—42no—
252M16—2922no—
253M16—2922no—
254C2×C8—42no—
255C2×C8—2922no—
256C4⋊C4—2922no—
257C22⋊C4—2922no—
258C22⋊C4—2924no—
259C22⋊C4—2924no—
260C42—2922no—
261C42—2922no—
262C42—2922no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23#1455650512yes—
275C23C2×#1192400511yescharacteristic
276Q8—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280C2×C4—28no—
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—44no—
290C2×C4—44no—
291C2×C4—2922no—
292C2×C4—2928no—
293C2×C4—2922no—
294C2×C4—2922no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—1464no—
298C2×C4#1455650512yes—
299C2×C4#1455654712yes—
300C2×C4#1455657212yes—
301C2×C4#1455657512yes—
302C8—82no—
303C8—5842no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—81no—
316C22—81no—
317C22#1798844811yescharacteristic
318C22C2×#1455650511yescharacteristic
319C22C2×#1455650511yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—2924no—
325C4—5842no—
326C2—21no—
327C2—21no—
328C2—42no—
329C2—81no—
330C2—81no—
331C2#2261000311yescharacteristic
332C2#2261161412yes—
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC73×#1785C2
socleC22×C73C2×#99
prime radicalC22×#41×C73C22
cosocleC22×Q8×C73C23
hypercenter#1785D73
hypocenterC73#12861
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #22608057C2semidirect
171. C73#12861semidirect

Maximal subgroups

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

subgroupindex
#226124222
#226098592
#226080572
C73×#17852
#1286173