Finite Group Databasebeta

← #32982647

Subgroups — #32982647

Order 18688.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#22620881C211yesmaximal, characteristic
2#22618871C211yesmaximal, characteristic
3#22610904C211yesmaximal, characteristic
4#22608615C211yesmaximal, characteristic
5#22608496C211yesmaximal, characteristic
6#22607790C211yesmaximal, characteristic
7C73×#1275C211yesmaximal, characteristic
8#17989144C2211yescharacteristic
9C2×#14556591C2211yescharacteristic
10#17988967C412yescomplemented
11#17988938C412yescomplemented
12#17988798—22no—
13#17988745—22no—
14#17988715—22no—
15#17988715—22no—
16#17988613—21no—
17#17988611C2211yescharacteristic
18#17988542—21no—
19#17988453C2211yescharacteristic
20#17988350—21no—
21#17988350—21no—
22#17988334—22no—
23#17988325—22no—
24#431×C73C2211yescharacteristic
25#431×C73C2211yescharacteristic
26C2×#97×C73C2211yescharacteristic
27C22×Dic146—21no—
28C2×Dic292—24no—
29C2×#11924012—21no—
30#14556591—21no—
31#14556591—21no—
32#14556591—21no—
33#14556591C2×C412yes—
34#14556591C2×C411yescharacteristic
35#14556591C2×C411yescharacteristic
36#14556583—21no—
37#14556570—21no—
38#14556570D412yes—
39C4×Dic146—21no—
40C4×Dic146D412yes—
41#14556550—21no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73—21no—
45C4⋊#10596831—21no—
46C4⋊#10596831D412yes—
47#14556520—24no—
48#14556516—24no—
49#14556501—22no—
50C8×Dic73—22no—
51#14556498—21no—
52#14556498—21no—
53C22×C8×C73D411yescharacteristic
54C22×C8×C73D411yescharacteristic
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—21no—
57C4⋊C8×C73—21no—
58C22⋊C8×C73—21no—
59C22⋊C8×C73—21no—
60#97×C73—21no—
61#97×C73—21no—
62C4×C8×C73—21no—
63C4×C8×C73C2×C412yes—
64C2×Dic146—22no—
65C2×Dic146—21no—
66C2×Dic146—41no—
67C2×Dic146—82no—
68C2×Dic146C22⋊C412yes—
69C22×Dic73—41no—
70C22×Dic73—41no—
71Dic292—84no—
72#11924012—41no—
73#11924012—41no—
74C2×#10596831—22no—
75C4⋊Dic73—21no—
76C4⋊Dic73—21no—
77C4⋊Dic73—21no—
78C4⋊Dic73—22no—
79C4⋊Dic73—22no—
80C4⋊Dic73—41no—
81C4⋊Dic73—41no—
82C4⋊Dic73—41no—
83C4⋊Dic73—41no—
84C4⋊Dic73C22⋊C412yes—
85#11924007—22no—
86#11924007—22no—
87C22⋊Dic73—22no—
88C4×Dic73—22no—
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73#4111yescharacteristic
92C2×C8×C73—21no—
93C2×C8×C73—21no—
94C2×C8×C73—22no—
95C2×C8×C73—21no—
96C2×C8×C73—21no—
97C2×C8×C73—21no—
98C2×C8×C73—41no—
99C2×C8×C73Q1612yescomplemented
100C42×C73C22×C411yescharacteristic
101C42×C73C2×D411yescharacteristic
102C42×C73C2×D411yescharacteristic
103C42×C73#4111yescharacteristic
104C23×C73#12111yescharacteristic
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73C2×C22⋊C411yescharacteristic
108C2×C4×C73C4×D411yescharacteristic
109C2×C4×C73C4×D411yescharacteristic
110C2×C4×C73#12011yescharacteristic
111C2×C4×C73#12111yescharacteristic
112C2×C4×C73#12311yescharacteristic
113C2×C4×C73C2×Q1611yescharacteristic
114C2×C4×C73#13511yescharacteristic
115C2×C4×C73#13711yescharacteristic
116C2×C4×C73#13711yescharacteristic
117C2×Dic73—22no—
118C2×Dic73—22no—
119C2×Dic73—42no—
120C2×Dic73—82no—
121C2×Dic73—81no—
122C2×Dic73—81no—
123C2×Dic73—81no—
124Dic146—42no—
125Dic146—44no—
126Dic146—81no—
127Dic146—82no—
128C8×C73—22no—
129C8×C73—42no—
130C8×C73—41no—
131C8×C73—81no—
132#10596831—42no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73#38611yescharacteristic
136C22×C73#46211yescharacteristic
137C22×C73#46411yescharacteristic
138C4×C73—21no—
139C4×C73—21no—
140C4×C73—21no—
141C4×C73—21no—
142C4×C73C4×Q1611yescharacteristic
143C4×C73#44111yescharacteristic
144C4×C73#45111yescharacteristic
145C4×C73#45211yescharacteristic
146Dic73—42no—
147Dic73—42no—
148Dic73—82no—
149Dic73—161no—
150#9290—731nomaximal, Sylow
151C2×C73—21no—
152C2×C73—21no—
153C2×C73#187811yescharacteristic
154C2×C73#190211yescharacteristic
155C2×C73#191011yescharacteristic
156#3209—731no—
157#2961—731no—
158#1993—731no—
159#1549—731no—
160#1518—731no—
161#1286—731no—
162#1275D7311yescharacteristic
163C73#929011yescharacteristic, complemented, Sylow
164#533—731no—
165C2×C4⋊Q8—731no—
166#494—732no—
167#487—732no—
168#462—1462no—
169#451—1462no—
170#446—1462no—
171#446—1462no—
172#431C2×D7311yescharacteristic
173#431C2×D7311yescharacteristic
174#424—1461no—
175#423—731no—
176#405—1461no—
177C2×#97C2×D7311yescharacteristic
178#382—731no—
179#340—1461no—
180#340—1461no—
181#332—1462no—
182#328—1462no—
183C22×Q8—1461no—
184C2×Q16—1464no—
185C2×M16—1461no—
186C22×C8D29211yescharacteristic
187C22×C8D29211yescharacteristic
188C4⋊Q8—731no—
189C4⋊Q8—731no—
190C4⋊Q8—732no—
191C4⋊Q8—1461no—
192C4⋊Q8—1461no—
193C4⋊Q8—1461no—
194#125—1461no—
195C22⋊Q8—732no—
196C22⋊Q8—1461no—
197C4×Q8—732no—
198C4×Q8—1461no—
199#117—1461no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4—1461no—
203C2×C42C22×D7311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—732no—
207C4⋊C8—1464no—
208C4⋊C8—1461no—
209#103—1464no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212#97—21no—
213#97—21no—
214#97—1462no—
215C4×C8—21no—
216C4×C8—1462no—
217C4×C8C4×D7312yes—
218#95—1461no—
219#95—1461no—
220C2×Q8—2921no—
221C2×Q8—5842no—
222C2×Q8—732no—
223C2×Q8—1462no—
224C2×Q8—1461no—
225C22×C4—2921no—
226C22×C4—2921no—
227C22×C4D4×D7311yescharacteristic
228C22×C4C2×D29211yescharacteristic
229C22×C4#1192402911yescharacteristic
230Q16—5844no—
231M16—2921no—
232M16—2921no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—41no—
240C2×C8—1462no—
241C2×C8Dic29212yescomplemented
242C4⋊C4—2921no—
243C4⋊C4—2921no—
244C4⋊C4—2921no—
245C4⋊C4—2921no—
246C4⋊C4—732no—
247C4⋊C4—1462no—
248C4⋊C4—1462no—
249C4⋊C4—1461no—
250C4⋊C4—1461no—
251C4⋊C4—1461no—
252C4⋊C4—1462no—
253C4⋊C4—1462no—
254C22⋊C4—1462no—
255C42—1462no—
256C42C2×C4×D7311yescharacteristic
257C42D4×D7311yescharacteristic
258C42D4×D7311yescharacteristic
259C42#1192402811yescharacteristic
260C23#1455656611yescharacteristic
261Q8—2922no—
262Q8—2924no—
263Q8—5841no—
264Q8—5842no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—2922no—
268C2×C4—5842no—
269C2×C4—5841no—
270C2×C4—5841no—
271C2×C4—5841no—
272C2×C4—1462no—
273C2×C4—1462no—
274C2×C4C22⋊C4×D7311yescharacteristic
275C2×C4#1455655311yescharacteristic
276C2×C4C4×D29211yescharacteristic
277C2×C4#1455656111yescharacteristic
278C2×C4#1455656411yescharacteristic
279C2×C4#1455657611yescharacteristic
280C2×C4C2×Dic29211yescharacteristic
281C2×C4#1455661311yescharacteristic
282C2×C4#1455662411yescharacteristic
283C2×C4#1455662411yescharacteristic
284C8—22no—
285C8—42no—
286C8—41no—
287C8—81no—
288C8—2922no—
289C22—21no—
290C22—21no—
291C22#1798846511yescharacteristic
292C22#1798879811yescharacteristic
293C22#1798881011yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—2922no—
299C4—2922no—
300C4—5842no—
301C4—11681no—
302C4C4×Dic29211yescharacteristic
303C4#1798869511yescharacteristic
304C4#1798874511yescharacteristic
305C4#1798875111yescharacteristic
306C2—21no—
307C2—21no—
308C2#2261027111yescharacteristic
309C2#2261040511yescharacteristic
310C2#2261045711yescharacteristic
subgroupG/H
centerC22#17988465
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1275C2
socleC22×C73#386
prime radicalC23×C73#121
cosocleC2×C42×C73C23
hypercenter#1275D73
hypocenterC73#9290
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
10. #17988967C4semidirect
11. #17988938C4semidirect
99. C2×C8×C73Q16semidirect
163. C73#9290semidirect
241. C2×C8Dic292semidirect

Maximal subgroups

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

subgroupindex
#226208812
#226188712
#226109042
#226086152
#226084962
#226077902
C73×#12752
#929073