Finite Group Databasebeta

← #33240693

Subgroups — #33240693

Order 18688.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620470C211yesmaximal, characteristic, complemented
2#22612095C212yesmaximal, complemented
3#22612088C212yesmaximal, complemented
4#22612083C212yesmaximal, complemented
5#22612068C212yesmaximal, complemented
6#22611717C211yesmaximal, characteristic, complemented
7C2×#17988412C211yesmaximal, characteristic, complemented
8#22611690C211yesmaximal, characteristic, complemented
9#22611643C211yesmaximal, characteristic
10#22611638C211yesmaximal, characteristic
11C73×#2951C211yesmaximal, characteristic
12#17989215—21no—
13C2×#14556531C2211yescharacteristic
14C22×#11924003C2211yescharacteristic
15#17988996—22no—
16#17988992C2212yes—
17#17988989C2212yes—
18#17988629—21no—
19#17988625C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21#17988414C2212yes—
22#17988412C2212yes—
23#17988412C2212yes—
24#17988412C2212yes—
25C4⋊#11924003C2212yes—
26C4⋊#11924003C2212yes—
27#17988391C2211yescharacteristic
28#17988391C2212yes—
29#17988391C2211yescharacteristic
30#17988388C2211yescharacteristic
31#17988388C2212yes—
32#17988388C2211yescharacteristic
33C2×#135×C73C2211yescharacteristic
34#520×C73—21no—
35#440×C73—22no—
36C4×Q16×C73C2212yes—
37C4×D8×C73C2212yes—
38C2×#107×C73C2211yescharacteristic
39#417×C73—21no—
40#405×C73C2211yescharacteristic
41#14556591—22no—
42#14556590—41no—
43#14556583—22no—
44#14556582—41no—
45#14556568—42no—
46#14556550—21no—
47C2×C4⋊Dic73—21no—
48#14556531—22no—
49C2×#11924003—22no—
50C2×#11924003C2311yescharacteristic
51C2×#11924003C2311yescharacteristic
52C2×#11924003C2312yes—
53#14556525—21no—
54#14556525—22no—
55#14556525C2312yes—
56#14556525C2311yescharacteristic
57#14556525C2311yescharacteristic
58#14556523—21no—
59#14556523—22no—
60C2×#41×C73—21no—
61#135×C73—22no—
62#135×C73Q814yescomplemented
63C2×Q16×C73C2311yescharacteristic
64C2×SD16×C73—21no—
65C2×D8×C73C2311yescharacteristic
66C2×M16×C73D411yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C4×Q8×C73—22no—
69C4×D4×C73—22no—
70C4×D4×C73—42no—
71#117×C73—41no—
72#117×C73D411yescharacteristic
73C2×C4⋊C4×C73—21no—
74C2×C4⋊C4×C73—41no—
75#107×C73—21no—
76#107×C73C2312yes—
77#103×C73—22no—
78#102×C73—22no—
79#97×C73D412yes—
80C4×C8×C73C2312yes—
81C2×Dic146—42no—
82C22×Dic73—41no—
83C4⋊Dic73—21no—
84C4⋊Dic73—22no—
85C4⋊Dic73—21no—
86C4⋊Dic73—41no—
87C4⋊Dic73—42no—
88#11924007—42no—
89#11924007—42no—
90C22⋊Dic73—41no—
91C4×Dic73—41no—
92#11924003—21no—
93#11924003—21no—
94#11924003—22no—
95#41×C73—24no—
96#41×C73—42no—
97C2×Q8×C73—21no—
98C2×D4×C73—21no—
99C2×D4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73—42no—
103C22×C4×C73C2×D411yescharacteristic
104Q16×C73—21no—
105Q16×C73C2×Q812yes—
106SD16×C73—24no—
107D8×C73—21no—
108D8×C73C2×Q812yes—
109M16×C73—22no—
110C2×C8×C73C2×D412yes—
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2×Q812yes—
113C2×C8×C73#4112yes—
114C2×C8×C73C2411yescharacteristic
115C4⋊C4×C73—22no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—42no—
118C4⋊C4×C73D811yescharacteristic
119C4⋊C4×C73D811yescharacteristic
120C22⋊C4×C73—42no—
121C22⋊C4×C73D812yes—
122C42×C73—42no—
123C42×C73C2×D412yes—
124C23×C73—41no—
125C23×C73C2×D811yescharacteristic
126Q8×C73—22no—
127Q8×C73—41no—
128D4×C73—22no—
129D4×C73—44no—
130D4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—44no—
133C2×C4×C73—42no—
134C2×C4×C73—162no—
135C2×C4×C73—161no—
136C2×C4×C73C22⋊Q812yes—
137C2×C4×C73C2×D812yes—
138C2×C4×C73C2×D812yes—
139C2×C4×C73C2×D811yescharacteristic
140C2×C4×C73#13712yes—
141C2×C4×C73C22×D411yescharacteristic
142C2×Dic73—41no—
143C2×Dic73—41no—
144C2×Dic73—42no—
145C2×Dic73—161no—
146Dic146—162no—
147C8×C73—22no—
148C8×C73C22⋊Q812yes—
149C8×C73C22×Q811yescharacteristic
150C8×C73C2×#4111yescharacteristic
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—42no—
154C22×C73—81no—
155C22×C73—81no—
156C22×C73#47412yes—
157C22×C73C22×D811yescharacteristic
158C4×C73—22no—
159C4×C73—22no—
160C4×C73—42no—
161C4×C73—82no—
162C4×C73#47412yes—
163C4×C73C2×C22⋊Q811yescharacteristic
164C4×C73C2×#13711yescharacteristic
165Dic73—81no—
166Dic73—82no—
167Dic73—81no—
168#33633—731nomaximal, Sylow
169C2×C73—22no—
170C2×C73—82no—
171C2×C73C2×#47411yescharacteristic
172C2×C73#341511yescharacteristic
173C2×C73#341711yescharacteristic
174#3164—731no—
175#2951D7311yescharacteristic
176#2229—732no—
177#2228—732no—
178#2227—732no—
179#2224—732no—
180#2159—731no—
181C2×#366—731no—
182#2151—731no—
183#2139—731no—
184#2138—731no—
185C73#3363311yescharacteristic, complemented, Sylow
186C2×#135C2×D7311yescharacteristic
187#544—1461no—
188#520—21no—
189C2×M32—731no—
190C22×C16—731no—
191#501—1462no—
192#500—732no—
193#499—732no—
194#440—22no—
195C4×Q16C2×D7312yes—
196C4×D8C2×D7312yes—
197#428—1461no—
198#427—731no—
199C2×#107—731no—
200C2×#107C2×D7311yescharacteristic
201#417—21no—
202#405C2×D7311yescharacteristic
203#367—732no—
204#366—732no—
205#366—732no—
206#366—732no—
207C4⋊C16—732no—
208C4⋊C16—732no—
209#358—731no—
210#358—731no—
211#358—732no—
212#357—731no—
213#357—731no—
214#357—732no—
215C2×#41—21no—
216#135—22no—
217#135Dic14614yescomplemented
218C2×Q16C22×D7311yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D7311yescharacteristic
221C2×M16C22⋊D7311yescharacteristic
222C22×C8C22×D7311yescharacteristic
223C4⋊Q8—2921no—
224C4⋊Q8—1462no—
225#125—2921no—
226#125—1462no—
227C22⋊Q8—2922no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—42no—
231#117—41no—
232#117—1461no—
233#117C22⋊D7311yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—1461no—
237M32—1462no—
238C2×C16—731no—
239C2×C16—731no—
240C2×C16—732no—
241C2×C16—1462no—
242#107—21no—
243#107—732no—
244#107—731no—
245#107—731no—
246#107—1461no—
247#107—1462no—
248#107C22×D7312yes—
249#106—1461no—
250#106—1462no—
251#103—22no—
252#102—22no—
253#97C22⋊D7312yes—
254C4×C8C22×D7312yes—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—2922no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—2921no—
265C22×C4C2×C22⋊D7311yescharacteristic
266Q16—21no—
267Q16C2×Dic14612yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic14612yes—
271M16—22no—
272C2×C8C2×C22⋊D7312yes—
273C2×C8C2×C22⋊D7311yescharacteristic
274C2×C8C2×Dic14612yes—
275C2×C8#1192402812yes—
276C2×C8C23×D7311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—2922no—
281C4⋊C4—2922no—
282C4⋊C4—2921no—
283C4⋊C4—2922no—
284C4⋊C4—1461no—
285C4⋊C4—1462no—
286C4⋊C4—1461no—
287C4⋊C4#1192401311yescharacteristic
288C4⋊C4#1192401311yescharacteristic
289C22⋊C4—42no—
290C22⋊C4—2921no—
291C22⋊C4#1192401312yes—
292C42—42no—
293C42—2921no—
294C42C2×C22⋊D7312yes—
295C16—1461no—
296C16—1461no—
297C16—1462no—
298C23—41no—
299C23C2×#1192401311yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—11682no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—162no—
310C2×C4—161no—
311C2×C4—2921no—
312C2×C4—2921no—
313C2×C4—2922no—
314C2×C4—11681no—
315C2×C4#1455657012yes—
316C2×C4C2×#1192401312yes—
317C2×C4C2×#1192401312yes—
318C2×C4C2×#1192401311yescharacteristic
319C2×C4#1455662112yes—
320C2×C4C22×C22⋊D7311yescharacteristic
321C8—22no—
322C8#1455657012yes—
323C8C22×Dic14611yescharacteristic
324C8C2×#1192402811yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1798886712yes—
331C22C22×#1192401311yescharacteristic
332C4—22no—
333C4—22no—
334C4—42no—
335C4—82no—
336C4—5842no—
337C4—5841no—
338C4—5841no—
339C4#1798886712yes—
340C4C2×#1455657011yescharacteristic
341C4C2×#1455662111yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1798886711yescharacteristic
345C2#2262307011yescharacteristic
346C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2951C2
socleC22×C73C22×D8
prime radicalC2×#135×C73C22
hypercenter#2951D73
hypocenterC73#33633
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #22620470C2semidirect
2. #22612095C2semidirect
3. #22612088C2semidirect
4. #22612083C2semidirect
5. #22612068C2semidirect
6. #22611717C2semidirect
7. C2×#17988412C2semidirect
8. #22611690C2semidirect
62. #135×C73Q8semidirect
185. C73#33633semidirect
217. #135Dic146semidirect

Maximal subgroups

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

subgroupindex
#226204702
#226120952
#226120882
#226120832
#226120682
#226117172
C2×#179884122
#226116902
#226116432
#226116382
C73×#29512
#3363373