Finite Group Databasebeta

← #1696×D73 #33068614

Subgroups — #1696×D73 #33068614

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D73C211yesmaximal, characteristic
2#477×D73C212yesmaximal
3#446×D73C211yesmaximal, characteristic
4#22609460C211yesmaximal, characteristic, complemented
5#22609459C211yesmaximal, characteristic, complemented
6#22609458C212yesmaximal, complemented
7#22609457C212yesmaximal, complemented
8#22609456C211yesmaximal, characteristic, complemented
9#334×D73C211yesmaximal, characteristic
10#333×D73C212yesmaximal
11C73×#1696C211yesmaximal, characteristic, complemented
12C4⋊Q8×D73C2212yes—
13C4×Q8×D73C2211yescharacteristic
14#17988992C2211yescharacteristic
15#17988991—21no—
16#17988991—21no—
17#17988990C2212yes—
18#17988883C2212yes—
19#17988882C2212yes—
20#17988881—22no—
21#17988880—22no—
22#17988879C2212yes—
23#17988718C2211yescharacteristic
24#17988717C2211yescharacteristic
25#17988716—21no—
26#17988715—21no—
27#17988714C2211yescharacteristic
28#107×D73—21no—
29#107×D73—21no—
30#107×D73—22no—
31C4⋊C8×D73—21no—
32C4⋊C8×D73C2212yes—
33C4⋊C8×D73C2211yescharacteristic
34#103×D73—22no—
35#97×D73—21no—
36C4×C8×D73C2211yescharacteristic
37#17988338C2211yescharacteristic
38#17988337C2212yes—
39#17988336C2212yes—
40#17988335C2212yes—
41#17988335C2212yes—
42#500×C73C2211yescharacteristic
43#477×C73C2212yes—
44#446×C73C2211yescharacteristic
45#334×C73C2211yescharacteristic
46#333×C73C2212yes—
47C2×Q8×D73—22no—
48C2×Q8×D73D411yescharacteristic
49C2×C8×D73—22no—
50C2×C8×D73—21no—
51C2×C8×D73D411yescharacteristic
52C2×C8×D73D411yescharacteristic
53#14556591C2312yes—
54#14556590—22no—
55#14556590—22no—
56#14556589—22no—
57Q8×Dic73D411yescharacteristic
58C4×Dic146—21no—
59C4×Dic146C2311yescharacteristic
60#14556556—21no—
61#14556556D411yescharacteristic
62C4⋊C4×D73—21no—
63C4⋊C4×D73—22no—
64C4⋊C4×D73—22no—
65C4⋊C4×D73D411yescharacteristic
66C42×D73—21no—
67C42×D73C2311yescharacteristic
68#14556525—22no—
69#14556525—21no—
70#14556525—21no—
71#14556524—22no—
72#14556524—22no—
73#14556524—22no—
74#14556524—22no—
75C4⋊#10596831—21no—
76C4⋊#10596831C2312yes—
77C4⋊#10596831C2311yescharacteristic
78#14556520—21no—
79#14556520—22no—
80#14556520—21no—
81#14556520—21no—
82#14556516—22no—
83#14556515—22no—
84#14556514—22no—
85#14556502—21no—
86#14556501—21no—
87#14556501—21no—
88C4×#10596831C2311yescharacteristic
89C8×Dic73D411yescharacteristic
90C8×Dic73D411yescharacteristic
91C4⋊Q8×C73C2312yes—
92C4×Q8×C73C2311yescharacteristic
93#107×C73—21no—
94#107×C73—21no—
95#107×C73—22no—
96C4⋊C8×C73—21no—
97C4⋊C8×C73C2312yes—
98C4⋊C8×C73C2311yescharacteristic
99#103×C73—22no—
100#97×C73—21no—
101C4×C8×C73C2311yescharacteristic
102C2×Dic146—21no—
103C2×Dic146—22no—
104C2×Dic146—42no—
105C2×Dic146C2×D411yescharacteristic
106Q8×D73—24no—
107Q8×D73—84no—
108C2×C4×D73—21no—
109C2×C4×D73—21no—
110C2×C4×D73—42no—
111C2×C4×D73C2×D411yescharacteristic
112C2×C4×D73C2×D411yescharacteristic
113C2×C4×D73#4111yescharacteristic
114C2×#10596831—21no—
115C2×#10596831—22no—
116C2×#10596831C2×D411yescharacteristic
117C2×#10596831C2×D411yescharacteristic
118C8×D73—22no—
119C8×D73—42no—
120C8×D73—44no—
121C8×D73Q1614yes—
122C4⋊Dic73—21no—
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125C4⋊Dic73C2×D411yescharacteristic
126#11924007—21no—
127#11924007—21no—
128#11924007—21no—
129#11924007—42no—
130#11924007—42no—
131C4×Dic73—21no—
132C4×Dic73—21no—
133C4×Dic73—21no—
134C4×Dic73C2×D411yescharacteristic
135C4×Dic73C2×D411yescharacteristic
136C4×Dic73#4111yescharacteristic
137C2×Q8×C73—22no—
138C2×Q8×C73C2×D411yescharacteristic
139C2×C8×C73—21no—
140C2×C8×C73—22no—
141C2×C8×C73C2×D411yescharacteristic
142C2×C8×C73C2×D411yescharacteristic
143C4⋊C4×C73—21no—
144C4⋊C4×C73—22no—
145C4⋊C4×C73—22no—
146C4⋊C4×C73C2×D411yescharacteristic
147C42×C73—21no—
148C42×C73C2411yescharacteristic
149C22×D73#12111yescharacteristic
150Q8×C73—22no—
151Q8×C73—82no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—42no—
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C22×D411yescharacteristic
157C2×C4×C73C2×#4111yescharacteristic
158C2×Dic73—21no—
159C2×Dic73—21no—
160C2×Dic73—42no—
161C2×Dic73#12111yescharacteristic
162C2×Dic73#12111yescharacteristic
163C2×Dic73#12111yescharacteristic
164C4×D73—22no—
165C4×D73—42no—
166C4×D73—42no—
167C4×D73—84no—
168C4×D73C2×Q1612yes—
169C4×D73#13712yes—
170C4×D73#13714yes—
171Dic146—22no—
172Dic146—42no—
173Dic146—84no—
174Dic146—82no—
175C8×C73—21no—
176C8×C73—41no—
177C8×C73—42no—
178C8×C73C2×Q1612yes—
179#10596831—21no—
180#10596831—41no—
181#10596831—42no—
182#10596831C2×Q1612yes—
183C22×C73C2×#12111yescharacteristic
184C2×D73#46212yes—
185C2×D73#47012yes—
186C2×D73#47212yes—
187C4×C73—21no—
188C4×C73—41no—
189C4×C73—41no—
190C4×C73—82no—
191C4×C73C22×Q1611yescharacteristic
192C4×C73C2×#13711yescharacteristic
193C4×C73C2×#13712yes—
194Dic73—21no—
195Dic73—41no—
196Dic73—41no—
197Dic73—82no—
198Dic73#46212yes—
199Dic73#47012yes—
200C2×#1696—731nomaximal, Sylow
201C2×C73C2×#46211yescharacteristic
202C2×C73C2×#47011yescharacteristic
203C2×C73C2×#47211yescharacteristic
204D73#169614yesdirect product factor
205C2×#500—731no—
206C2×#477—732no—
207C2×#446—731no—
208#1696—731no—
209#1696—732no—
210#1696—731no—
211#1696—731no—
212#1696—732no—
213#1696D7311yescharacteristic, direct product factor
214C2×#334—731no—
215C2×#333—732no—
216C73C2×#169611yescharacteristic, complemented, Sylow
217C2×C4⋊Q8—732no—
218C2×C4×Q8—731no—
219#500—732no—
220#500—731no—
221#500—1461no—
222#500—1461no—
223#500C2×D7311yescharacteristic
224#477—732no—
225#477—732no—
226#477—732no—
227#477—1462no—
228#477—1462no—
229#477C2×D7312yes—
230#446—731no—
231#446—731no—
232#446—731no—
233#446—1461no—
234#446—1461no—
235#446C2×D7311yescharacteristic
236C2×#107—1462no—
237C2×#107—1461no—
238C2×#107—1461no—
239C2×C4⋊C8—732no—
240C2×C4⋊C8—731no—
241C2×C4⋊C8—1461no—
242C2×#103—1462no—
243C2×#97—1461no—
244C2×C4×C8—731no—
245#334—732no—
246#334—731no—
247#334C2×D7311yescharacteristic
248#333—732no—
249#333—732no—
250#333—732no—
251#333C2×D7312yes—
252C22×Q8—731no—
253C22×Q8—1462no—
254C22×C8—731no—
255C22×C8—731no—
256C22×C8—1461no—
257C22×C8—1462no—
258C4⋊Q8—732no—
259C4⋊Q8—1462no—
260C4⋊Q8—1462no—
261C4⋊Q8—1462no—
262C4⋊Q8C22×D7312yes—
263C4×Q8—731no—
264C4×Q8—731no—
265C4×Q8—731no—
266C4×Q8—1461no—
267C4×Q8—1461no—
268C4×Q8C22×D7311yescharacteristic
269C2×C4⋊C4—731no—
270C2×C4⋊C4—1462no—
271C2×C4⋊C4—1462no—
272C2×C4⋊C4—1461no—
273C2×C42—731no—
274C2×C42—1461no—
275#107—21no—
276#107—21no—
277#107—22no—
278#107—1462no—
279#107—1462no—
280#107—1462no—
281#107—1462no—
282#107—1461no—
283#107—1461no—
284#107—1462no—
285C4⋊C8—21no—
286C4⋊C8—731no—
287C4⋊C8—732no—
288C4⋊C8—1461no—
289C4⋊C8—1461no—
290C4⋊C8—1462no—
291C4⋊C8—1461no—
292C4⋊C8—1461no—
293C4⋊C8C22×D7311yescharacteristic
294C4⋊C8C22×D7312yes—
295#103—22no—
296#103—1462no—
297#103—1462no—
298#103—1462no—
299#97—21no—
300#97—1461no—
301#97—1461no—
302#97—1461no—
303C4×C8—731no—
304C4×C8—731no—
305C4×C8—731no—
306C4×C8C22×D7311yescharacteristic
307C2×Q8—22no—
308C2×Q8—2922no—
309C2×Q8—5844no—
310C2×Q8—731no—
311C2×Q8—1461no—
312C2×Q8—1464no—
313C2×Q8—1462no—
314C2×Q8D4×D7311yescharacteristic
315C22×C4—2922no—
316C22×C4—731no—
317C22×C4—731no—
318C22×C4—731no—
319C22×C4—1461no—
320C22×C4—1461no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—2922no—
324C2×C8—2924no—
325C2×C8—731no—
326C2×C8—734no—
327C2×C8—731no—
328C2×C8—1462no—
329C2×C8—1461no—
330C2×C8—1462no—
331C2×C8D4×D7311yescharacteristic
332C2×C8D4×D7311yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—2922no—
337C4⋊C4—2922no—
338C4⋊C4—731no—
339C4⋊C4—1461no—
340C4⋊C4—1461no—
341C4⋊C4—1461no—
342C4⋊C4—1462no—
343C4⋊C4—1462no—
344C4⋊C4—1461no—
345C4⋊C4D4×D7311yescharacteristic
346C42—21no—
347C42—731no—
348C42—731no—
349C42—731no—
350C42—1461no—
351C42—1461no—
352C42—1461no—
353C42C23×D7311yescharacteristic
354C23—731no—
355Q8—22no—
356Q8—82no—
357Q8—2922no—
358Q8—5842no—
359Q8—5844no—
360Q8—1462no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—2922no—
365C2×C4—2922no—
366C2×C4—2922no—
367C2×C4—5844no—
368C2×C4—731no—
369C2×C4—731no—
370C2×C4—732no—
371C2×C4—734no—
372C2×C4—732no—
373C2×C4—731no—
374C2×C4—1462no—
375C2×C4—1461no—
376C2×C4—1461no—
377C2×C4C2×D4×D7311yescharacteristic
378C2×C4C2×D4×D7311yescharacteristic
379C2×C4#41×D7311yescharacteristic
380C8—21no—
381C8—42no—
382C8—41no—
383C8—2922no—
384C8—2921no—
385C8—732no—
386C8—1461no—
387C8Q16×D7312yes—
388C22—732no—
389C22—732no—
390C22—732no—
391C22#121×D7311yescharacteristic
392C4—21no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—2921no—
397C4—2921no—
398C4—5842no—
399C4—732no—
400C4—732no—
401C4—1461no—
402C4C2×Q16×D7311yescharacteristic
403C4#137×D7312yes—
404C4#137×D7311yescharacteristic
405C2—734no—
406C2#462×D7311yescharacteristic
407C2#470×D7311yescharacteristic
408C2#472×D7311yescharacteristic
subgroupG/H
centerC22#121×D73
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#1696C2
socleC22×C73C2×#121
prime radicalC22×D73#121
hypercenter#1696D73
hypocenterC73C2×#1696
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #1696

Complements

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

NH ≅ G/Ntype
4. #22609460C2semidirect
5. #22609459C2semidirect
6. #22609458C2semidirect
7. #22609457C2semidirect
8. #22609456C2semidirect
11. C73×#1696C2semidirect
204. D73#1696direct (N × H)
213. #1696D73direct (N × H)
216. C73C2×#1696semidirect

Maximal subgroups

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

subgroupindex
#500×D732
#477×D732
#446×D732
#226094602
#226094592
#226094582
#226094572
#226094562
#334×D732
#333×D732
C73×#16962
C2×#169673