Finite Group Databasebeta

← #32983152

Subgroups — #32983152

Order 18688.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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#22619879C211yesmaximal, characteristic
2#22610915C211yesmaximal, characteristic
3#22610030C211yesmaximal, characteristic
4#22608450C211yesmaximal, characteristic
5#22607837C211yesmaximal, characteristic
6#22607817C211yesmaximal, characteristic
7C73×#2347C211yesmaximal, characteristic
8C2×#14556570C2211yescharacteristic
9#17988928—22no—
10#17988922—21no—
11#17988917—21no—
12#17988630C412yescomplemented
13#17988595C412yescomplemented
14#17988551C2211yescharacteristic
15#17988551C2211yescharacteristic
16C4×C22⋊Dic73C2211yescharacteristic
17#17988353—22no—
18#17988346—22no—
19#17988323—21no—
20#17988323—21no—
21C2×#117×C73C2211yescharacteristic
22#389×C73—21no—
23#388×C73—22no—
24#383×C73—21no—
25#380×C73C2211yescharacteristic
26C4×C4⋊C4×C73—21no—
27C4×C4⋊C4×C73—21no—
28C4×C22⋊C4×C73C2211yescharacteristic
29C22×Dic146C2×C411yescharacteristic
30C2×#11924012—21no—
31C2×#11924012C2×C412yes—
32#14556570—21no—
33#14556570—21no—
34#14556570—21no—
35#14556570—21no—
36C2×C4⋊Dic73C2×C411yescharacteristic
37C2×#11924007—21no—
38C2×C22⋊Dic73—21no—
39C2×C22⋊Dic73—21no—
40C2×C4×Dic73—21no—
41#14556524—22no—
42#14556522—22no—
43#14556514—22no—
44#14556514—22no—
45C22⋊#10596831—21no—
46C22⋊#10596831—21no—
47C22⋊#10596831—21no—
48C22⋊#10596831—21no—
49#14556498—21no—
50C23×C4×C73C2311yescharacteristic
51#117×C73—21no—
52#117×C73—21no—
53#117×C73C2×C412yes—
54#117×C73D412yes—
55C2×C4⋊C4×C73—22no—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73D411yescharacteristic
58C2×C4⋊C4×C73D411yescharacteristic
59C2×C22⋊C4×C73—21no—
60C2×C22⋊C4×C73—21no—
61C2×C22⋊C4×C73D412yes—
62C2×C42×C73—21no—
63C2×C42×C73—21no—
64C2×C42×C73—22no—
65#95×C73—22no—
66#95×C73—22no—
67#95×C73—21no—
68#95×C73D411yescharacteristic
69#95×C73D411yescharacteristic
70C2×Dic146—21no—
71C2×Dic146—21no—
72C2×Dic146—42no—
73C22×Dic73—21no—
74C22×Dic73—21no—
75C22×Dic73—41no—
76#11924012—42no—
77#11924012—42no—
78C2×#10596831—22no—
79C2×#10596831—41no—
80C4⋊Dic73—21no—
81C4⋊Dic73—21no—
82#11924007—41no—
83#11924007—41no—
84C22⋊Dic73—22no—
85C22⋊Dic73—41no—
86C22⋊Dic73—41no—
87C22⋊Dic73—41no—
88C4×Dic73—42no—
89C24×C73C2×D411yescharacteristic
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—22no—
95C22×C4×C73—42no—
96C22×C4×C73C22×C411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2×D411yescharacteristic
100C22×C4×C73#4111yescharacteristic
101C22×C4×C73#4111yescharacteristic
102C4⋊C4×C73—21no—
103C4⋊C4×C73—21no—
104C4⋊C4×C73—42no—
105C4⋊C4×C73—44no—
106C4⋊C4×C73C22⋊C412yes—
107C4⋊C4×C73C22⋊C412yes—
108C22⋊C4×C73—22no—
109C22⋊C4×C73—22no—
110C22⋊C4×C73—42no—
111C22⋊C4×C73—42no—
112C42×C73—22no—
113C42×C73—22no—
114C42×C73—42no—
115C42×C73—82no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73C4×D411yescharacteristic
121C23×C73#12011yescharacteristic
122C23×C73#12111yescharacteristic
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—22no—
128C2×C4×C73—22no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—42no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—42no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73C2×C22⋊C411yescharacteristic
144C2×C4×C73C4×D411yescharacteristic
145C2×C4×C73#12111yescharacteristic
146C2×C4×C73#12311yescharacteristic
147C2×Dic73—42no—
148C2×Dic73—41no—
149C2×Dic73—41no—
150C2×Dic73—81no—
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73—81no—
154Dic146—42no—
155Dic146—81no—
156#10596831—42no—
157#10596831—81no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—81no—
167C22×C73—81no—
168C22×C73#38611yescharacteristic
169C22×C73#45511yescharacteristic
170C22×C73#45611yescharacteristic
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—42no—
175C4×C73—42no—
176C4×C73—42no—
177C4×C73—82no—
178C4×C73—81no—
179C4×C73#41012yes—
180Dic73—82no—
181Dic73—81no—
182Dic73—81no—
183#9362—731nomaximal, Sylow
184C2×C73—21no—
185C2×C73—21no—
186C2×C73—41no—
187C2×C73—41no—
188C2×C73#188211yescharacteristic
189C2×C73#189911yescharacteristic
190C2×C73#190611yescharacteristic
191#3095—731no—
192#2347D7311yescharacteristic
193#1996—731no—
194#1826—731no—
195#1508—731no—
196#1305—731no—
197#1297—731no—
198C73#936211yescharacteristic, complemented, Sylow
199C2×C22⋊Q8—731no—
200C2×#117C2×D7311yescharacteristic
201#485—1462no—
202#484—1461no—
203#483—1461no—
204#428—732no—
205#419—732no—
206#407—731no—
207#407—731no—
208#389—21no—
209#388—22no—
210#383—21no—
211#380C2×D7311yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C22⋊C4—731no—
215C4×C22⋊C4C2×D7311yescharacteristic
216#342—1462no—
217#339—1462no—
218#328—1461no—
219#328—1461no—
220C22×Q8—731no—
221C23×C4C22×D7311yescharacteristic
222C2×M16—732no—
223C2×M16—1461no—
224C22⋊Q8—1461no—
225C22⋊Q8—1461no—
226C22⋊Q8—1461no—
227C22⋊Q8—1461no—
228#117—21no—
229#117—21no—
230#117C4×D7312yes—
231#117D29212yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—731no—
235C2×C4⋊C4—1461no—
236C2×C4⋊C4C22⋊D7311yescharacteristic
237C2×C4⋊C4C22⋊D7311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1461no—
241C2×C22⋊C4—1461no—
242C2×C22⋊C4C22⋊D7312yes—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—1461no—
247#107—1462no—
248#106—1462no—
249#103—1462no—
250#103—1462no—
251C22⋊C8—1461no—
252C22⋊C8—1461no—
253C22⋊C8—1461no—
254C22⋊C8—1461no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—1461no—
259#95C22⋊D7311yescharacteristic
260#95C22⋊D7311yescharacteristic
261C24C2×C22⋊D7311yescharacteristic
262C2×Q8—2922no—
263C2×Q8—1461no—
264C2×Q8—1461no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—2921no—
272C22×C4—1461no—
273C22×C4—1461no—
274C22×C4C2×C4×D7311yescharacteristic
275C22×C4C2×C22⋊D7311yescharacteristic
276C22×C4C2×C22⋊D7311yescharacteristic
277C22×C4C2×D29211yescharacteristic
278C22×C4#1192402811yescharacteristic
279C22×C4#1192402811yescharacteristic
280M16—2922no—
281M16—2922no—
282C2×C8—2921no—
283C2×C8—1462no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—2921no—
289C4⋊C4—2921no—
290C4⋊C4—1461no—
291C4⋊C4—1461no—
292C4⋊C4#1192400512yes—
293C4⋊C4#1192400512yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—2921no—
299C22⋊C4—2921no—
300C22⋊C4—2921no—
301C22⋊C4—1462no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—82no—
306C42—2922no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23C4×C22⋊D7311yescharacteristic
312C23#1455656011yescharacteristic
313C23#1455656511yescharacteristic
314Q8—2922no—
315Q8—5841no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—2922no—
337C2×C4—2921no—
338C2×C4—2921no—
339C2×C4—5841no—
340C2×C4—5841no—
341C2×C4—5841no—
342C2×C4—5841no—
343C2×C4C2×#1192400511yescharacteristic
344C2×C4C4×C22⋊D7311yescharacteristic
345C2×C4#1455656511yescharacteristic
346C2×C4#1455657411yescharacteristic
347C8—2922no—
348C8—5841no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1798846611yescharacteristic
360C22#1798876511yescharacteristic
361C22#1798877011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—5842no—
371C4—5841no—
372C4—5841no—
373C4#1798856012yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2261030011yescharacteristic
379C2#2261038911yescharacteristic
380C2#2261043211yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2347C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2347D73
hypocenterC73#9362
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
12. #17988630C4semidirect
13. #17988595C4semidirect
198. C73#9362semidirect

Maximal subgroups

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

subgroupindex
#226198792
#226109152
#226100302
#226084502
#226078372
#226078172
C73×#23472
#936273