Finite Group Databasebeta

← #32981011

Subgroups — #32981011

Order 18688.

812 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#17988523C211yesmaximal, characteristic
2#22612545C211yesmaximal, characteristic
3#22610022C212yesmaximal, complemented
4#22607811C212yesmaximal, complemented
5C2×#323×C73C211yesmaximal, characteristic
6C22×C4⋊Dic73C2211yescharacteristic
7C22×#11924007—21no—
8C22×C22⋊Dic73—21no—
9C22×C22⋊Dic73—21no—
10#17988523—22no—
11#17988523C414yescomplemented
12#17988462—24no—
13#17988447—21no—
14#17988447C2212yes—
15C2×#14556498C2211yescharacteristic
16#17988350—24no—
17C22×C22⋊C4×C73C2211yescharacteristic
18C2×C22⋊C8×C73C2212yes—
19#323×C73—22no—
20C23×Dic73—22no—
21C23×Dic73—21no—
22C23×Dic73—21no—
23C2×C4⋊Dic73—22no—
24C2×C4⋊Dic73—22no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73C2×C412yescomplemented
28C2×C4⋊Dic73D414yes—
29C2×#11924007—24no—
30C2×#11924007—44no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—24no—
33C2×C22⋊Dic73—22no—
34C2×C22⋊Dic73—42no—
35C2×C22⋊Dic73—44no—
36C2×C22⋊Dic73—42no—
37#14556498—22no—
38#14556498—21no—
39#14556498—22no—
40#14556498C2×C412yescomplemented
41C25×C73D411yescharacteristic
42C23×C4×C73D411yescharacteristic
43C23×C4×C73C2311yescharacteristic
44C22×C8×C73—22no—
45C2×C22⋊C4×C73—22no—
46C2×C22⋊C4×C73—22no—
47C2×C22⋊C4×C73C2×C412yes—
48C2×C22⋊C4×C73D412yes—
49C22⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51C22×Dic73—24no—
52C22×Dic73—22no—
53C22×Dic73—22no—
54C22×Dic73—24no—
55C22×Dic73—42no—
56C22×Dic73—41no—
57C22×Dic73—41no—
58C22×Dic73—42no—
59C22×Dic73—44no—
60C22×Dic73—42no—
61C22×Dic73—41no—
62C22×Dic73—41no—
63C22×Dic73—82no—
64C22×Dic73—82no—
65C4⋊Dic73—24no—
66C4⋊Dic73—42no—
67C4⋊Dic73—44no—
68C4⋊Dic73—82no—
69#11924007—48no—
70C22⋊Dic73—48no—
71C22⋊Dic73—44no—
72C22⋊Dic73—42no—
73C22⋊Dic73—42no—
74C24×C73—21no—
75C24×C73—21no—
76C24×C73—21no—
77C24×C73—21no—
78C24×C73—42no—
79C24×C73—42no—
80C24×C73C22⋊C412yes—
81C24×C73SD1612yes—
82C24×C73Q1612yes—
83C24×C73C2×D411yescharacteristic
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—42no—
87C22×C4×C73—42no—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D412yes—
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73#4112yes—
93C2×C8×C73—44no—
94C2×C8×C73—42no—
95C22⋊C4×C73—24no—
96C22⋊C4×C73—24no—
97C23×C73—22no—
98C23×C73—22no—
99C23×C73—22no—
100C23×C73—22no—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73—22no—
104C23×C73—22no—
105C23×C73—42no—
106C23×C73—42no—
107C23×C73—44no—
108C23×C73—44no—
109C23×C73—42no—
110C23×C73—42no—
111C23×C73—82no—
112C23×C73—82no—
113C23×C73—81no—
114C23×C73—81no—
115C23×C73#10314yes—
116C23×C73C2×C22⋊C411yescharacteristic
117C23×C73#12011yescharacteristic
118C23×C73#12311yescharacteristic
119C23×C73C2×SD1611yescharacteristic
120C23×C73C2×Q1611yescharacteristic
121C23×C73AGL(1,ℤ/8)11yescharacteristic
122C23×C73#13711yescharacteristic
123C2×C4×C73—24no—
124C2×C4×C73—22no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—42no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—81no—
131C2×C4×C73C4×D412yes—
132C2×C4×C73#12112yes—
133C2×Dic73—24no—
134C2×Dic73—44no—
135C2×Dic73—44no—
136C2×Dic73—42no—
137C2×Dic73—42no—
138C2×Dic73—42no—
139C2×Dic73—42no—
140C2×Dic73—42no—
141C2×Dic73—44no—
142C2×Dic73—44no—
143C2×Dic73—44no—
144C2×Dic73—82no—
145C2×Dic73—88no—
146C2×Dic73—82no—
147C8×C73—82no—
148C22×C73—22no—
149C22×C73—22no—
150C22×C73—22no—
151C22×C73—22no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—24no—
157C22×C73—44no—
158C22×C73—44no—
159C22×C73—44no—
160C22×C73—42no—
161C22×C73—42no—
162C22×C73—82no—
163C22×C73—82no—
164C22×C73—82no—
165C22×C73#38611yescharacteristic
166C22×C73C2×#10311yescharacteristic
167C22×C73#41711yescharacteristic
168C22×C73#45011yescharacteristic
169C22×C73#45111yescharacteristic
170C22×C73#45312yes—
171C22×C73#45812yes—
172C22×C73#48311yescharacteristic
173C22×C73#48411yescharacteristic
174C4×C73—22no—
175C4×C73—22no—
176C4×C73—42no—
177Dic73—44no—
178Dic73—84no—
179Dic73—82no—
180#9021—731nomaximal, Sylow
181C2×C73—21no—
182C2×C73—21no—
183C2×C73—21no—
184C2×C73—21no—
185C2×C73—44no—
186C2×C73#161312yes—
187C2×C73#161412yes—
188C2×C73#187211yescharacteristic
189C2×C73#188711yescharacteristic
190C2×C73#189011yescharacteristic
191C2×#399—731no—
192#2315—731no—
193#1824—732no—
194C2×#323D7311yescharacteristic
195#1294—732no—
196C73#902111yescharacteristic, complemented, Sylow
197C22×C4⋊C4—731no—
198C22×C4⋊C4—1461no—
199C22×C22⋊C4—1461no—
200C22×C22⋊C4—1461no—
201C22×C22⋊C4C2×D7311yescharacteristic
202C2×C22⋊C8C2×D7312yes—
203#399—734no—
204#399—1462no—
205#385—1464no—
206#380—732no—
207#380—1461no—
208C2×#95—731no—
209#340—1464no—
210#323—22no—
211C25D29211yescharacteristic
212C23×C4—1461no—
213C23×C4—1461no—
214C23×C4—1462no—
215C23×C4D29211yescharacteristic
216C23×C4C22×D7311yescharacteristic
217C22×C8—22no—
218C2×C4⋊C4—2924no—
219C2×C4⋊C4—734no—
220C2×C4⋊C4—732no—
221C2×C4⋊C4—1464no—
222C2×C4⋊C4—1461no—
223C2×C4⋊C4—1462no—
224C2×C4⋊C4—1462no—
225C2×C4⋊C4—1462no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—2922no—
229C2×C22⋊C4—2922no—
230C2×C22⋊C4—2924no—
231C2×C22⋊C4—1462no—
232C2×C22⋊C4—1464no—
233C2×C22⋊C4—1462no—
234C2×C22⋊C4C4×D7312yes—
235C2×C22⋊C4C22⋊D7312yes—
236C22⋊C8—22no—
237C22⋊C8—22no—
238#95—732no—
239#95—1462no—
240#95—1462no—
241#95—1461no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24#1192400512yes—
249C24#1192401712yes—
250C24Dic29212yes—
251C24C2×D29211yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—2922no—
257C22×C4—2921no—
258C22×C4—2921no—
259C22×C4—2922no—
260C22×C4—2924no—
261C22×C4—2922no—
262C22×C4—2921no—
263C22×C4—2921no—
264C22×C4—5842no—
265C22×C4—5842no—
266C22×C4—1462no—
267C22×C4—1464no—
268C22×C4—1464no—
269C22×C4—1462no—
270C22×C4#1192400512yes—
271C22×C4C2×C4×D7311yescharacteristic
272C22×C4D4×D7312yes—
273C22×C4C2×C22⋊D7311yescharacteristic
274C22×C4#1192402712yes—
275C2×C8—44no—
276C2×C8—42no—
277C4⋊C4—2924no—
278C4⋊C4—2928no—
279C4⋊C4—2922no—
280C4⋊C4—5842no—
281C4⋊C4—1464no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—2924no—
285C22⋊C4—2922no—
286C22⋊C4—2922no—
287C22⋊C4—2928no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—82no—
303C23—82no—
304C23—81no—
305C23—81no—
306C23#1455651614yes—
307C23C2×#1192400511yescharacteristic
308C23#1455656111yescharacteristic
309C23#1455657511yescharacteristic
310C23C2×#1192401711yescharacteristic
311C23C2×Dic29211yescharacteristic
312C23#1455661911yescharacteristic
313C23#1455662411yescharacteristic
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—2924no—
323C2×C4—2924no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—2922no—
328C2×C4—2922no—
329C2×C4—2924no—
330C2×C4—2924no—
331C2×C4—2924no—
332C2×C4—5842no—
333C2×C4—5848no—
334C2×C4—5842no—
335C2×C4—1464no—
336C2×C4#1455655112yes—
337C2×C4#1455656312yes—
338C8—82no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—24no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22#1798846411yescharacteristic
357C22C2×#1455651611yescharacteristic
358C22#1798858911yescharacteristic
359C22#1798874011yescharacteristic
360C22#1798874511yescharacteristic
361C22#1798875712yes—
362C22#1798877812yes—
363C22#1798891811yescharacteristic
364C22#1798892311yescharacteristic
365C4—22no—
366C4—22no—
367C4—42no—
368C4—2924no—
369C4—5844no—
370C4—5842no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—44no—
376C2#2260895312yes—
377C2#2260895812yes—
378C2#2261023611yescharacteristic
379C2#2261032711yescharacteristic
380C2#2261034211yescharacteristic
subgroupG/H
centerC23#14556561
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#323×C73C2
socleC23×C73#120
prime radicalC25×C73D4
cosocleC23×C4×C73C23
hypercenterC2×#323D73
hypocenterC73#9021
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #22610022C2semidirect
4. #22607811C2semidirect
11. #17988523C4semidirect
27. C2×C4⋊Dic73C2×C4semidirect
40. #14556498C2×C4semidirect
196. C73#9021semidirect

Maximal subgroups

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

subgroupindex
C2×#179885232
#226125452
#226100222
#226078112
C2×#323×C732
#902173