Finite Group Databasebeta

← #33004311

Subgroups — #33004311

Order 18688.

472 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 111 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×#17988386C214yesmaximal
2#22609927C211yesmaximal, characteristic
3C4×#14556510C211yesmaximal, characteristic
4C73×#2546C211yesmaximal, characteristic
5#17988611—21no—
6C2×#14556510C2212yes—
7C2×#14556510C2212yes—
8C4×#11924012—21no—
9#17988386—28no—
10#17988356—21no—
11#17988356—21no—
12C2×C4⋊Q8×C73C411yescharacteristic
13C2×C4⋊Q8×C73C2212yes—
14C2×C4×Q8×C73C2211yescharacteristic
15#398×C73—21no—
16#398×C73—21no—
17#391×C73C412yes—
18#389×C73—21no—
19#389×C73—21no—
20C4×C4⋊C4×C73C411yescharacteristic
21C2×#11924012—22no—
22C2×#11924012—22no—
23C4⋊#10596831—41no—
24C4⋊#10596831—41no—
25#14556510—24no—
26#14556510—22no—
27#14556510D414yes—
28#14556502—41no—
29C4×#10596831—41no—
30C22×Q8×C73C2×C412yes—
31C22×Q8×C73C2311yescharacteristic
32C4⋊Q8×C73—24no—
33C4⋊Q8×C73—24no—
34C4⋊Q8×C73—41no—
35C4⋊Q8×C73—41no—
36C4×Q8×C73—41no—
37C4×Q8×C73—41no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73D411yescharacteristic
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C42×C73C2×C412yes—
47C2×C42×C73C2×C411yescharacteristic
48C2×C42×C73D411yescharacteristic
49C2×C42×C73D411yescharacteristic
50#95×C73—21no—
51#95×C73—22no—
52#95×C73—22no—
53#11924012—24no—
54#11924012—42no—
55#11924012—42no—
56#11924012—42no—
57C2×#10596831—42no—
58C2×#10596831—42no—
59C2×Q8×C73—24no—
60C2×Q8×C73—41no—
61C2×Q8×C73—41no—
62C2×Q8×C73—42no—
63C2×Q8×C73—42no—
64C2×Q8×C73C2×D412yes—
65C2×Q8×C73#4112yes—
66C22×C4×C73—22no—
67C22×C4×C73—22no—
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22×C411yescharacteristic
73C22×C4×C73C2×D411yescharacteristic
74C22×C4×C73C2×D411yescharacteristic
75C4⋊C4×C73—24no—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—42no—
80C4⋊C4×C73—41no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—44no—
83C4⋊C4×C73—42no—
84C4⋊C4×C73—42no—
85C42×C73—21no—
86C42×C73—21no—
87C42×C73—21no—
88C42×C73—21no—
89C42×C73—21no—
90C42×C73—21no—
91C42×C73—24no—
92C23×C73C2×C22⋊C411yescharacteristic
93Q8×C73—42no—
94Q8×C73—42no—
95Q8×C73—84no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—22no—
103C2×C4×C73—24no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73#9911yescharacteristic
116C2×C4×C73#9911yescharacteristic
117C2×C4×C73C4×D412yes—
118C2×C4×C73#12112yes—
119C2×C4×C73#12411yescharacteristic
120C2×C4×C73#12711yescharacteristic
121#10596831—44no—
122#10596831—82no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73#39011yescharacteristic
126C22×C73C2×#9911yescharacteristic
127C22×C73#41011yescharacteristic
128C4×C73—21no—
129C4×C73—22no—
130C4×C73—21no—
131C4×C73—22no—
132C4×C73—21no—
133C4×C73—44no—
134C4×C73—41no—
135C4×C73—41no—
136C4×C73—41no—
137C4×C73—82no—
138C4×C73#35612yescomplemented
139#13227—731nomaximal, Sylow
140C2×C73—21no—
141C2×C73—21no—
142C2×C73#196411yescharacteristic
143C2×C73C2×#35611yescharacteristic
144C2×C73#213511yescharacteristic
145#2546D7311yescharacteristic
146C2×#356—734no—
147#1801—731no—
148C4×#101—731no—
149C73#1322711yescharacteristic, complemented, Sylow
150C2×C4⋊Q8Dic7311yescharacteristic
151C2×C4⋊Q8C2×D7312yes—
152C2×C4×Q8C2×D7311yescharacteristic
153#423—1461no—
154C2×#101—732no—
155C2×#101—732no—
156C4×M16—1461no—
157#398—21no—
158#398—21no—
159#391Dic7312yes—
160#389—21no—
161#389—21no—
162C4×C4⋊C4Dic7311yescharacteristic
163#356—1468no—
164#343—1461no—
165#343—1461no—
166C22×Q8C2×Dic7312yes—
167C22×Q8C22×D7311yescharacteristic
168C2×M16—1462no—
169C2×M16—1462no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×Dic7311yescharacteristic
182C2×C4⋊C4C22⋊D7311yescharacteristic
183C2×C4⋊C4C22⋊D7311yescharacteristic
184C2×C42C2×Dic7312yes—
185C2×C42C2×Dic7311yescharacteristic
186C2×C42C22⋊D7311yescharacteristic
187C2×C42C22⋊D7311yescharacteristic
188C4⋊C8—2921no—
189C4⋊C8—2921no—
190#101—734no—
191#101—1464no—
192#101—1462no—
193#97—2921no—
194C4×C8—2921no—
195#95—21no—
196#95—22no—
197#95—22no—
198C2×Q8—24no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8D4×D7312yes—
204C2×Q8#1192402712yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4C22⋊Dic7311yescharacteristic
208C22×C4C22⋊Dic7311yescharacteristic
209C22×C4C22⋊Dic7311yescharacteristic
210C22×C4C22⋊Dic7311yescharacteristic
211C22×C4C22×Dic7311yescharacteristic
212C22×C4C2×C22⋊D7311yescharacteristic
213C22×C4C2×C22⋊D7311yescharacteristic
214M16—2922no—
215M16—2922no—
216M16—2922no—
217M16—1464no—
218C2×C8—2922no—
219C2×C8—2922no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—24no—
237C23C2×C22⋊Dic7311yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4#1455650611yescharacteristic
261C2×C4#1455650611yescharacteristic
262C2×C4D4×Dic7312yes—
263C2×C4#1455656212yes—
264C2×C4#1455657811yescharacteristic
265C2×C4#1455658711yescharacteristic
266C8—2924no—
267C8—5842no—
268C22—21no—
269C22—21no—
270C22#1798849011yescharacteristic
271C22C2×#1455650611yescharacteristic
272C22#1798856111yescharacteristic
273C4—22no—
274C4—21no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4#1798838612yescomplemented
284C2—21no—
285C2—21no—
286C2#2261076811yescharacteristic
287C2C2×#1798838611yescharacteristic
288C2#2261162611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC73×#2546C2
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Q8×C73C23
hypercenter#2546D73
hypocenterC73#13227
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
138. C4×C73#356semidirect
149. C73#13227semidirect
283. C4#17988386semidirect

Maximal subgroups

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

subgroupindex
C2×#179883862
#226099272
C4×#145565102
C73×#25462
#1322773