Finite Group Databasebeta

← #32974992

Subgroups — #32974992

Order 18688.

436 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 85 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#22617893C211yesmaximal, characteristic
2#22611069C211yesmaximal, characteristic
3#22611031C211yesmaximal, characteristic
4#22607756C211yesmaximal, characteristic
5#22607756C211yesmaximal, characteristic
6#22607754C211yesmaximal, characteristic
7C73×#2057C211yesmaximal, characteristic
8C22×#11924018—21no—
9C2×#14556515—21no—
10C2×#14556514—21no—
11#17988520—21no—
12#17988519—21no—
13#17988507—21no—
14#17988493C2211yescharacteristic
15#17988493C2211yescharacteristic
16#17988491—21no—
17#17988486—21no—
18#17988473—21no—
19#17988349—21no—
20#17988342C2211yescharacteristic
21#17988342C2211yescharacteristic
22C2×C4⋊Q8×C73C2211yescharacteristic
23C2×#107×C73—21no—
24#389×C73C2211yescharacteristic
25#340×C73—21no—
26#336×C73C2211yescharacteristic
27C22×Dic146D411yescharacteristic
28C22×Dic146D411yescharacteristic
29C2×#11924018—42no—
30C2×#11924018—42no—
31C2×#11924018—42no—
32C22×#10596831—21no—
33C22×#10596831—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73D411yescharacteristic
37C2×C4⋊Dic73D411yescharacteristic
38C2×#11924007—21no—
39C2×#11924007—21no—
40C2×C4×Dic73—21no—
41#14556515—42no—
42#14556514—42no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50C22×Q8×C73—21no—
51C22×C8×C73—21no—
52C4⋊Q8×C73—42no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73D411yescharacteristic
57C2×C4⋊C4×C73D411yescharacteristic
58C2×C42×C73C2311yescharacteristic
59#107×C73—42no—
60#95×C73—21no—
61C2×Dic146—21no—
62C2×Dic146—21no—
63C2×Dic146—21no—
64C2×Dic146—41no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C2×Dic146—41no—
68C2×Dic146Q1612yescomplemented
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C22×Dic73—21no—
73#11924018—48no—
74C2×#10596831—22no—
75C2×#10596831—41no—
76C2×#10596831—41no—
77C2×#10596831—41no—
78C2×#10596831—41no—
79C2×#10596831—41no—
80C4⋊Dic73—41no—
81C4⋊Dic73—41no—
82C4⋊Dic73—81no—
83C4⋊Dic73—81no—
84#11924007—81no—
85#11924007—81no—
86C4×Dic73—81no—
87C2×Q8×C73—22no—
88C2×Q8×C73—41no—
89C2×Q8×C73—41no—
90C2×Q8×C73—41no—
91C2×Q8×C73—41no—
92C2×Q8×C73—41no—
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73—41no—
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C2×C8×C73—22no—
100C2×C8×C73—41no—
101C2×C8×C73—41no—
102C4⋊C4×C73—21no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—22no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73Q1612yescomplemented
111C42×C73—41no—
112C23×C73#12011yescharacteristic
113Q8×C73—42no—
114Q8×C73—42no—
115Q8×C73—81no—
116Q8×C73—81no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—22no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—42no—
129C2×C4×C73—41no—
130C2×C4×C73—81no—
131C2×C4×C73—81no—
132C2×C4×C73C2×Q1611yescharacteristic
133C2×C4×C73C2×Q1611yescharacteristic
134C2×C4×C73#13511yescharacteristic
135C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C73#13711yescharacteristic
137C2×C4×C73#13711yescharacteristic
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—
148C2×Dic73—41no—
149C2×Dic73—41no—
150Dic146—42no—
151Dic146—42no—
152Dic146—81no—
153Dic146—81no—
154C8×C73—42no—
155#10596831—42no—
156#10596831—81no—
157C22×C73#44911yescharacteristic
158C22×C73#45111yescharacteristic
159C22×C73#45111yescharacteristic
160C22×C73#45311yescharacteristic
161C22×C73#45511yescharacteristic
162C22×C73#45611yescharacteristic
163C22×C73B(4,𝔽2)11yescharacteristic
164C4×C73—21no—
165C4×C73—21no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—42no—
171C4×C73—81no—
172C4×C73—82no—
173Dic73—42no—
174Dic73—81no—
175Dic73—81no—
176Dic73—81no—
177#7784—731nomaximal, Sylow
178C2×C73#160011yescharacteristic
179C2×C73#160211yescharacteristic
180C2×C73#160511yescharacteristic
181C2×C73#163311yescharacteristic
182C2×C73#164411yescharacteristic
183C2×C73#164811yescharacteristic
184C2×C73#166111yescharacteristic
185#2806—731no—
186#2057D7311yescharacteristic
187#2024—731no—
188#2016—731no—
189#1272—731no—
190#1272—731no—
191#1271—731no—
192C73#778411yescharacteristic, complemented, Sylow
193C22×Q16—1461no—
194C2×C4⋊Q8C2×D7311yescharacteristic
195C2×#107—21no—
196C2×#103—1461no—
197C2×#103—1461no—
198#398—1461no—
199#398—1461no—
200#395—1461no—
201#391—731no—
202#391—731no—
203#391—1461no—
204#389—1461no—
205#389C2×D7311yescharacteristic
206#387—1461no—
207#340—21no—
208#340—1461no—
209#336—731no—
210#336—731no—
211#336C2×D7311yescharacteristic
212C22×Q8—21no—
213C22×Q8—731no—
214C22×Q8—731no—
215C2×Q16—2922no—
216C2×Q16—2922no—
217C2×Q16—2922no—
218C22×C8—21no—
219C22×C8—1461no—
220C22×C8—1461no—
221C4⋊Q8—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—731no—
226C2×C4⋊C4—731no—
227C2×C4⋊C4—1461no—
228C2×C4⋊C4—1461no—
229C2×C4⋊C4—1461no—
230C2×C4⋊C4—1461no—
231C2×C4⋊C4C22⋊D7311yescharacteristic
232C2×C4⋊C4C22⋊D7311yescharacteristic
233C2×C42—1461no—
234C2×C42C22×D7311yescharacteristic
235#107—42no—
236#103—2922no—
237#103—2922no—
238#95—21no—
239#95—1461no—
240#95—1461no—
241#95—1461no—
242#95—1461no—
243#95—1461no—
244#95—1461no—
245#95—1461no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—2921no—
253C2×Q8—2921no—
254C2×Q8—2921no—
255C2×Q8—2921no—
256C2×Q8—732no—
257C2×Q8—1461no—
258C2×Q8—1461no—
259C2×Q8—1461no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—1461no—
264C22×C4—1461no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4D4×D7311yescharacteristic
268C22×C4D4×D7311yescharacteristic
269C22×C4C2×C22⋊D7311yescharacteristic
270Q16—2928no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—2921no—
275C2×C8—2921no—
276C2×C8—2921no—
277C2×C8—2921no—
278C2×C8—2921no—
279C2×C8—1462no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—2921no—
289C4⋊C4—2921no—
290C4⋊C4—5841no—
291C4⋊C4—5841no—
292C4⋊C4—5841no—
293C4⋊C4—5841no—
294C4⋊C4#1192401812yescomplemented
295C42—41no—
296C42—5841no—
297C23#1455655911yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—81no—
301Q8—81no—
302Q8—2922no—
303Q8—2922no—
304Q8—5841no—
305Q8—5841no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—2921no—
322C2×C4—2921no—
323C2×C4—2921no—
324C2×C4—2921no—
325C2×C4—2921no—
326C2×C4—2921no—
327C2×C4—2921no—
328C2×C4—2921no—
329C2×C4—2921no—
330C2×C4—2921no—
331C2×C4—2921no—
332C2×C4—1462no—
333C2×C4Q16×D7311yescharacteristic
334C2×C4C2×#1192401811yescharacteristic
335C2×C4#1455661011yescharacteristic
336C2×C4#1455661511yescharacteristic
337C2×C4#1455662111yescharacteristic
338C2×C4#1455662311yescharacteristic
339C8—42no—
340C8—2922no—
341C8—5841no—
342C22#1798872911yescharacteristic
343C22#1798874111yescharacteristic
344C22#1798874211yescharacteristic
345C22#1798875311yescharacteristic
346C22#1798876311yescharacteristic
347C22#1798876911yescharacteristic
348C22#1798877311yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—81no—
357C4—82no—
358C4—2922no—
359C4—5841no—
360C4—5841no—
361C4—5841no—
362C2#2260887411yescharacteristic
363C2#2260888611yescharacteristic
364C2#2260890411yescharacteristic
365C2#2260907011yescharacteristic
366C2#2260914411yescharacteristic
367C2#2260917011yescharacteristic
368C2#2260924611yescharacteristic
subgroupG/H
centerC23#14556559
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2057C2
socle, prime radicalC23×C73#120
hypercenter#2057D73
hypocenterC73#7784
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
68. C2×Dic146Q16semidirect
110. C4⋊C4×C73Q16semidirect
192. C73#7784semidirect
294. C4⋊C4#11924018semidirect

Maximal subgroups

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

subgroupindex
#226178932
#226110692
#226110312
#226077562
#226077542
C73×#20572
#778473