Finite Group Databasebeta

← #32989115

Subgroups — #32989115

Order 18688.

416 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 95 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#22613225C211yesmaximal, characteristic
2#22610913C211yesmaximal, characteristic
3#22608430C211yesmaximal, characteristic
4#22607834C211yesmaximal, characteristic
5#22607833C211yesmaximal, characteristic
6#22607759C211yesmaximal, characteristic
7C73×#1503C211yesmaximal, characteristic
8C2×C4×Dic146—21no—
9#17988640C2211yescharacteristic
10#17988611C2211yescharacteristic
11#17988521—21no—
12#17988520—21no—
13#17988508—21no—
14#17988507—21no—
15C4×#11924007—21no—
16C4×C4⋊Dic73C2211yescharacteristic
17#17988433C2211yescharacteristic
18#17988346—21no—
19#17988346—21no—
20#17988324—21no—
21#17988324—21no—
22#533×C73C2211yescharacteristic
23#431×C73C2211yescharacteristic
24C4×C4⋊C4×C73C2211yescharacteristic
25#328×C73—21no—
26#328×C73—21no—
27C22×Dic146—21no—
28C2×#11924012—21no—
29C4×Dic146—42no—
30C4×Dic146—42no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73—21no—
33C2×#11924007—21no—
34C2×#11924007—21no—
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×C4×Dic73—21no—
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C22⋊#10596831—21no—
44C4×#10596831—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49#14556498D411yescharacteristic
50#14556498D411yescharacteristic
51C4⋊Q8×C73Q811yescharacteristic, complemented
52C4⋊Q8×C73Q811yescharacteristic, complemented
53#125×C73—21no—
54C22⋊Q8×C73—21no—
55C22⋊Q8×C73—21no—
56C4×Q8×C73—21no—
57C4×Q8×C73—21no—
58#117×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C42×C73—21no—
61C2×C42×C73C2311yescharacteristic
62C4⋊C8×C73—21no—
63C22⋊C8×C73—21no—
64C4×C8×C73—21no—
65#95×C73D411yescharacteristic
66#95×C73D411yescharacteristic
67C2×Dic146—42no—
68C2×Dic146—42no—
69C2×Dic146—42no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C22×Dic73—21no—
73C22×Dic73—21no—
74#11924012—41no—
75#11924012—41no—
76C2×#10596831—41no—
77C2×#10596831—41no—
78C4⋊Dic73—42no—
79C4⋊Dic73—42no—
80#11924007—42no—
81#11924007—42no—
82#11924007—42no—
83#11924007—42no—
84C4×Dic73—21no—
85C4×Dic73—21no—
86C4×Dic73—41no—
87C4×Dic73—41no—
88C4×Dic73SD1612yes—
89C4×Dic73Q1612yes—
90C2×Q8×C73—21no—
91C2×Q8×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73C2×D411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C2×C8×C73—41no—
98C4⋊C4×C73—21no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73—21no—
101C4⋊C4×C73—21no—
102C4⋊C4×C73—42no—
103C4⋊C4×C73—41no—
104C4⋊C4×C73—41no—
105C22⋊C4×C73—41no—
106C42×C73—41no—
107C42×C73—41no—
108C42×C73—41no—
109C42×C73C2×Q811yescharacteristic
110C42×C73#4111yescharacteristic
111C42×C73#4111yescharacteristic
112C42×C73#4111yescharacteristic
113C23×C73#12011yescharacteristic
114Q8×C73—81no—
115Q8×C73—81no—
116C2×C4×C73—22no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73C22⋊Q811yescharacteristic
127C2×C4×C73C22⋊Q811yescharacteristic
128C2×C4×C73C22⋊Q811yescharacteristic
129C2×C4×C73#12411yescharacteristic
130C2×C4×C73#12411yescharacteristic
131C2×C4×C73#12411yescharacteristic
132C2×C4×C73C2×SD1611yescharacteristic
133C2×C4×C73C2×Q1611yescharacteristic
134C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C73#13711yescharacteristic
136C2×Dic73—22no—
137C2×Dic73—22no—
138C2×Dic73—22no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—41no—
144C2×Dic73—41no—
145C2×Dic73—41no—
146C2×Dic73—41no—
147C2×Dic73—41no—
148Dic146—48no—
149C8×C73—81no—
150#10596831—81no—
151#10596831—81no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73#39311yescharacteristic
155C22×C73#45011yescharacteristic
156C22×C73#45111yescharacteristic
157C4×C73—21no—
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—21no—
161C4×C73—42no—
162C4×C73—81no—
163C4×C73—81no—
164C4×C73—81no—
165C4×C73#47511yescharacteristic
166C4×C73#47711yescharacteristic
167C4×C73#48711yescharacteristic
168C4×C73#48911yescharacteristic
169Dic73—42no—
170Dic73—42no—
171Dic73—42no—
172Dic73—81no—
173#10371—731nomaximal, Sylow
174C2×C73—21no—
175C2×C73—21no—
176C2×C73#202411yescharacteristic
177C2×C73#203111yescharacteristic
178C2×C73#203111yescharacteristic
179#2399—731no—
180#1995—731no—
181#1503—731no—
182#1503D7311yescharacteristic
183#1303—731no—
184#1303—731no—
185#1274—731no—
186C73#1037111yescharacteristic, complemented, Sylow
187#533C2×D7311yescharacteristic
188C2×C4×Q8—1461no—
189#431—731no—
190#431C2×D7311yescharacteristic
191#423—731no—
192#398—1461no—
193#398—1461no—
194#395—1461no—
195#395—1461no—
196C4×C4⋊C4—731no—
197C4×C4⋊C4—1461no—
198C4×C4⋊C4C2×D7311yescharacteristic
199#376—731no—
200#339—1461no—
201#339—1461no—
202#328—21no—
203#328—21no—
204#328—1461no—
205#328—1461no—
206C22×Q8—1461no—
207C2×M16—1461no—
208C4⋊Q8Dic14611yescharacteristic, complemented
209C4⋊Q8Dic14611yescharacteristic, complemented
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—2922no—
216C4×Q8—2922no—
217#117—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—1461no—
220C2×C4⋊C4—1461no—
221C2×C4⋊C4—1461no—
222C2×C4⋊C4—1461no—
223C2×C4⋊C4—1461no—
224C2×C4⋊C4—1461no—
225C2×C42—21no—
226C2×C42—731no—
227C2×C42—731no—
228C2×C42—1461no—
229C2×C42C22×D7311yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—1461no—
232C4⋊C8—1461no—
233C4⋊C8—1461no—
234C22⋊C8—21no—
235C22⋊C8—1461no—
236C4×C8—21no—
237C4×C8—1461no—
238#95—731no—
239#95—731no—
240#95—1461no—
241#95—1461no—
242#95—1461no—
243#95—1461no—
244#95C22⋊D7311yescharacteristic
245#95C22⋊D7311yescharacteristic
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—2922no—
249C2×Q8—2922no—
250C2×Q8—2922no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—1461no—
254C22×C4—1461no—
255C22×C4—1461no—
256C22×C4—1461no—
257C22×C4D4×D7311yescharacteristic
258C22×C4D4×D7311yescharacteristic
259C22×C4C2×C22⋊D7311yescharacteristic
260M16—2921no—
261M16—2921no—
262C2×C8—41no—
263C2×C8—2921no—
264C2×C8—2921no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—2922no—
273C4⋊C4—2922no—
274C4⋊C4—2922no—
275C4⋊C4—2922no—
276C4⋊C4—2922no—
277C4⋊C4—2922no—
278C22⋊C4—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—2921no—
283C42—2921no—
284C42—732no—
285C42—732no—
286C42—1461no—
287C42—1461no—
288C42C2×Dic14611yescharacteristic
289C42#1192402711yescharacteristic
290C42#1192402711yescharacteristic
291C42#1192402811yescharacteristic
292C23#1455655911yescharacteristic
293Q8—81no—
294Q8—81no—
295Q8—2928no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—2921no—
309C2×C4—2921no—
310C2×C4—2921no—
311C2×C4—2921no—
312C2×C4—2921no—
313C2×C4—2921no—
314C2×C4—2921no—
315C2×C4—1462no—
316C2×C4—1462no—
317C2×C4—1462no—
318C2×C4#1455656811yescharacteristic
319C2×C4#1455656811yescharacteristic
320C2×C4#1455657011yescharacteristic
321C2×C4#1455657711yescharacteristic
322C2×C4#1455657711yescharacteristic
323C2×C4#1455657811yescharacteristic
324C2×C4SD16×D7311yescharacteristic
325C2×C4Q16×D7311yescharacteristic
326C2×C4#1455661811yescharacteristic
327C2×C4#1455662311yescharacteristic
328C8—81no—
329C8—5841no—
330C8—5841no—
331C22—21no—
332C22—21no—
333C22#1798849911yescharacteristic
334C22#1798873611yescharacteristic
335C22#1798874111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—2922no—
345C4—2922no—
346C4—2922no—
347C4—5841no—
348C4#1798887011yescharacteristic
349C4#1798888011yescharacteristic
350C4#1798893611yescharacteristic
351C4#1798894811yescharacteristic
352C2—21no—
353C2—21no—
354C2#2261107011yescharacteristic
355C2#2261110311yescharacteristic
356C2#2261110511yescharacteristic
subgroupG/H
centerC2×C4#14556577
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1503C2
socleC22×C73#393
prime radicalC23×C73#120
hypercenter#1503D73
hypocenterC73#10371
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
51. C4⋊Q8×C73Q8semidirect
52. C4⋊Q8×C73Q8semidirect
186. C73#10371semidirect
208. C4⋊Q8Dic146semidirect
209. C4⋊Q8Dic146semidirect

Maximal subgroups

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

subgroupindex
#226132252
#226109132
#226084302
#226078342
#226078332
#226077592
C73×#15032
#1037173