Finite Group Databasebeta

← #32995226

Subgroups — #32995226

Order 18688.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#22618628C211yesmaximal, characteristic
2#22611540C211yesmaximal, characteristic, complemented
3C22⋊C16×D73C211yesmaximal, characteristic, complemented
4#22607872C211yesmaximal, characteristic, complemented
5#22607872C211yesmaximal, characteristic, complemented
6#22607857C211yesmaximal, characteristic, complemented
7C73×#1315C211yesmaximal, characteristic, complemented
8C2×#14556598C411yescharacteristic
9C22×C8×D73C2211yescharacteristic
10#17989094C411yescharacteristic
11C2×#14556530—21no—
12C2×C16×D73—21no—
13#17988661—21no—
14#17988661—21no—
15#17988650—21no—
16#17988650—21no—
17#17988613C2211yescharacteristic
18#17988550C411yescharacteristic
19#17988545C411yescharacteristic
20C22⋊#11924003C2211yescharacteristic
21C22⋊#11924003C2211yescharacteristic
22#17988366—21no—
23#17988366—21no—
24#408×C73C2211yescharacteristic
25C22⋊C16×C73C2211yescharacteristic
26C22⋊C16×C73C2211yescharacteristic
27C22×C4×D73C2×C411yescharacteristic
28#14556598—42no—
29C2×#11924012—21no—
30C2×#11924012C2×C411yescharacteristic
31C22×#10596831—21no—
32C22×#10596831C2×C411yescharacteristic
33C2×C8×D73—42no—
34C2×C8×D73D411yescharacteristic
35C2×C8×D73D411yescharacteristic
36C2×C8×D73D411yescharacteristic
37C2×C8×D73D411yescharacteristic
38#14556572—21no—
39#14556567C811yescharacteristic
40#14556567C811yescharacteristic
41#14556562C811yescharacteristic
42#14556562C811yescharacteristic
43#14556559—21no—
44C4×C22⋊D73—21no—
45C4×C22⋊D73—21no—
46#14556550C2×C411yescharacteristic
47#14556530—42no—
48C2×#11924003—21no—
49C2×#11924003—21no—
50C16×D73—42no—
51#14556520—21no—
52#14556520—21no—
53#14556503—21no—
54#14556503—21no—
55#14556501—21no—
56C8×Dic73—21no—
57C2×#41×C73C2×C411yescharacteristic
58C2×M16×C73C2×C411yescharacteristic
59C22×C8×C73C2311yescharacteristic
60C2×C16×C73—21no—
61C2×C16×C73—21no—
62C22⋊C8×C73—21no—
63C22⋊C8×C73—21no—
64C23×D73M1611yescharacteristic
65C2×C22⋊D73—21no—
66C2×C22⋊D73—21no—
67C22×Dic73M1611yescharacteristic
68C2×C4×D73—41no—
69C2×C4×D73—41no—
70C2×C4×D73C22⋊C411yescharacteristic
71C2×C4×D73C22⋊C411yescharacteristic
72C2×C4×D73C2×C811yescharacteristic
73C2×C4×D73C2×C811yescharacteristic
74#11924012—42no—
75#11924012—41no—
76C2×#10596831—22no—
77C2×#10596831—21no—
78C2×#10596831—42no—
79C2×#10596831C22⋊C411yescharacteristic
80C2×#10596831C22⋊C411yescharacteristic
81C8×D73—22no—
82C8×D73—22no—
83C8×D73—41no—
84C8×D73—41no—
85C4⋊Dic73C2×C811yescharacteristic
86C4⋊Dic73C2×C811yescharacteristic
87#11924007—21no—
88#11924007—21no—
89C22⋊Dic73—21no—
90C22⋊Dic73—21no—
91C22⋊Dic73—21no—
92#11924005—21no—
93#11924005—21no—
94C4×Dic73—21no—
95#11924003—41no—
96#11924003—41no—
97#41×C73—41no—
98#41×C73—41no—
99C2×Q8×C73C2×C811yescharacteristic
100C2×D4×C73—21no—
101C2×D4×C73C2×C811yescharacteristic
102C22×C4×C73—21no—
103C22×C4×C73C22×C411yescharacteristic
104M16×C73—41no—
105C2×C8×C73—21no—
106C2×C8×C73—41no—
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C16×C73—41no—
110C16×C73—41no—
111C23×C73—21no—
112C23×C73C2×M1611yescharacteristic
113C22×D73—22no—
114C22×D73—41no—
115C22×D73—41no—
116C22×D73C22⋊C811yescharacteristic
117C22×D73C22⋊C811yescharacteristic
118Q8×C73—41no—
119D4×C73—41no—
120D4×C73—42no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73C2×C22⋊C411yescharacteristic
127C2×C4×C73C22×C811yescharacteristic
128C22⋊D73—42no—
129C22⋊D73—42no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—41no—
133C2×Dic73C22⋊C811yescharacteristic
134C2×Dic73C22⋊C811yescharacteristic
135C4×D73—21no—
136C4×D73—21no—
137C4×D73—21no—
138C4×D73—21no—
139C4×D73—41no—
140C4×D73—41no—
141C8×C73—21no—
142C8×C73—21no—
143C8×C73—41no—
144#10596831—21no—
145#10596831—41no—
146#10596831#9912yes—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—21no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73C2×C22⋊C811yescharacteristic
154C2×D73—21no—
155C2×D73—21no—
156C2×D73—21no—
157C2×D73—21no—
158C2×D73—21no—
159C2×D73—44no—
160C2×D73#35012yes—
161C4×C73—21no—
162C4×C73—41no—
163C4×C73C2×#9911yescharacteristic
164C4×C73#41311yescharacteristic
165Dic73—21no—
166Dic73—21no—
167Dic73—41no—
168Dic73—41no—
169#11470—731nomaximal, Sylow
170C2×C73—41no—
171C2×C73—81no—
172C2×C73#145911yescharacteristic
173C2×C73C2×#35011yescharacteristic
174C2×C73#211711yescharacteristic
175D73—22no—
176D73—41no—
177#2929—731no—
178#2112—731no—
179C2×C22⋊C16—731no—
180#1323—731no—
181#1323—731no—
182#1315—731no—
183#1315D7311yescharacteristic, complemented
184C73#1147011yescharacteristic, complemented, Sylow
185C2×#131—731no—
186C23×C8—731no—
187#525—731no—
188C2×M32—1461no—
189C22×C16—1461no—
190#435—1461no—
191#435—1461no—
192C8×D4—1461no—
193C8×D4—1461no—
194#424—731no—
195#408C2×D7311yescharacteristic
196#407—731no—
197#405—731no—
198C22⋊C16—731no—
199C22⋊C16—731no—
200C22⋊C16—1461no—
201C22⋊C16—1461no—
202C22⋊C16C2×D7311yescharacteristic
203C22⋊C16C2×D7311yescharacteristic
204C2×#41C4×D7311yescharacteristic
205C23×C4—731no—
206#131—2922no—
207C2×M16—731no—
208C2×M16—1461no—
209C2×M16C4×D7311yescharacteristic
210C22×C8—2922no—
211C22×C8—731no—
212C22×C8—731no—
213C22×C8—731no—
214C22×C8—731no—
215C22×C8—731no—
216C22×C8—1461no—
217C22×C8C22×D7311yescharacteristic
218#123—1461no—
219C22⋊Q8—731no—
220C22⋊Q8—731no—
221#121—731no—
222#121—731no—
223#120—1461no—
224C4×D4—1461no—
225C4×D4—1461no—
226#117—731no—
227M32—2922no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—2922no—
231C2×C16—1461no—
232C2×C16—1461no—
233C4⋊C8—1461no—
234C4⋊C8—1461no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—1461no—
238C22⋊C8—1461no—
239#97—1461no—
240C4×C8—1461no—
241C24—731no—
242#41—41no—
243#41—41no—
244C2×Q8C8×D7311yescharacteristic
245C2×D4—21no—
246C2×D4—1461no—
247C2×D4—1461no—
248C2×D4C8×D7311yescharacteristic
249C22×C4—21no—
250C22×C4—2921no—
251C22×C4—2921no—
252C22×C4—731no—
253C22×C4—731no—
254C22×C4—731no—
255C22×C4—731no—
256C22×C4—731no—
257C22×C4C2×C4×D7311yescharacteristic
258M16—41no—
259M16—2922no—
260M16—2921no—
261C2×C8—21no—
262C2×C8—41no—
263C2×C8—2921no—
264C2×C8—2921no—
265C2×C8—2922no—
266C2×C8—731no—
267C2×C8—731no—
268C2×C8—1462no—
269C2×C8—1462no—
270C2×C8—1461no—
271C2×C8—1462no—
272C2×C8D4×D7311yescharacteristic
273C2×C8D4×D7311yescharacteristic
274C4⋊C4—731no—
275C4⋊C4—731no—
276C4⋊C4—1461no—
277C4⋊C4—1461no—
278C22⋊C4—1461no—
279C22⋊C4—1461no—
280C22⋊C4—1461no—
281C22⋊C4—1461no—
282C22⋊C4—1461no—
283C42—1461no—
284C16—41no—
285C16—41no—
286C16—2921no—
287C16—2921no—
288C23—21no—
289C23—2921no—
290C23—2921no—
291C23—731no—
292C23—731no—
293C23—1462no—
294C23M16×D7311yescharacteristic
295Q8—41no—
296D4—41no—
297D4—42no—
298D4—2922no—
299D4—2922no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—731no—
309C2×C4—731no—
310C2×C4—1461no—
311C2×C4—1461no—
312C2×C4—1461no—
313C2×C4—1461no—
314C2×C4—1461no—
315C2×C4—1461no—
316C2×C4C22⋊C4×D7311yescharacteristic
317C2×C4C2×C8×D7311yescharacteristic
318C8—21no—
319C8—21no—
320C8—41no—
321C8—2921no—
322C8—732no—
323C8—1461no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—2924no—
331C22—732no—
332C22—1461no—
333C22—1461no—
334C22—1461no—
335C22—1461no—
336C22—1461no—
337C22C22⋊C8×D7311yescharacteristic
338C4—21no—
339C4—41no—
340C4—2921no—
341C4—2921no—
342C4—1461no—
343C4—1461no—
344C4#99×D7311yescharacteristic
345C4#1798856811yescharacteristic
346C2—41no—
347C2—81no—
348C2—2921no—
349C2—1462no—
350C2#2260824611yescharacteristic
351C2#350×D7311yescharacteristic
352C2#2261155711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1315C2
socleC22×C73C2×C22⋊C8
prime radical#17989094C4
cosocleC22×C8×C73C23
hypercenter#1315D73
hypocenterC73#11470
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #22611540C2semidirect
3. C22⋊C16×D73C2semidirect
4. #22607872C2semidirect
5. #22607872C2semidirect
6. #22607857C2semidirect
7. C73×#1315C2semidirect
183. #1315D73semidirect
184. C73#11470semidirect

Maximal subgroups

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

subgroupindex
#226186282
#226115402
C22⋊C16×D732
#226078722
#226078572
C73×#13152
#1147073