Finite Group Databasebeta

← #33002835

Subgroups — #33002835

Order 18688.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#22618998C211yesmaximal, characteristic
2#22610315C211yesmaximal, characteristic
3#22610153C212yesmaximal, complemented
4#22608061C212yesmaximal, complemented
5C73×#2132C211yesmaximal, characteristic
6#17989354C2211yescharacteristic
7#17988665—22no—
8D4×#10596831—22no—
9#17988640—21no—
10#17988636—22no—
11#17988604—21no—
12#17988561C2211yescharacteristic
13#17988554—21no—
14#17988554C2212yes—
15#17988351—22no—
16#17988347—21no—
17#532×C73C2211yescharacteristic
18#413×C73C2212yes—
19#356×C73C412yescomplemented
20#355×C73C412yescomplemented
21#14556598—42no—
22C2×#11924012—21no—
23C2×#11924012—22no—
24C2×#11924012—21no—
25C2×#11924012D412yes—
26C22×#10596831—21no—
27C22×#10596831D412yes—
28#14556550—21no—
29#14556527—44no—
30C4⋊#10596831—21no—
31C4⋊#10596831—21no—
32#14556518—42no—
33#14556506—22no—
34C22⋊#10596831—22no—
35C22⋊#10596831—22no—
36C22⋊#10596831—22no—
37C4×#10596831—22no—
38C2×#41×C73D411yescharacteristic
39C2×#41×C73C2311yescharacteristic
40C2×M16×C73—22no—
41C4⋊Q8×C73C2×C411yescharacteristic
42#127×C73Q811yescharacteristic
43#124×C73C2×C411yescharacteristic
44#124×C73Q811yescharacteristic
45#121×C73—22no—
46C4×D4×C73—22no—
47C2×C42×C73D411yescharacteristic
48#101×C73C2×C414yes—
49#100×C73—22no—
50#11924012—22no—
51#11924012—24no—
52#11924012—41no—
53#11924012—42no—
54#11924012—81no—
55C2×#10596831—22no—
56C2×#10596831—22no—
57C2×#10596831—42no—
58C2×#10596831—41no—
59C2×#10596831—42no—
60C2×#10596831—81no—
61C4⋊Dic73—41no—
62C22⋊Dic73—41no—
63C4×Dic73—41no—
64#41×C73—41no—
65#41×C73—41no—
66#41×C73—41no—
67#41×C73—41no—
68C2×Q8×C73C4⋊C411yescharacteristic
69C2×Q8×C73C22×C411yescharacteristic
70C2×D4×C73—22no—
71C2×D4×C73—21no—
72C2×D4×C73C4⋊C411yescharacteristic
73C2×D4×C73C2×Q811yescharacteristic
74C2×D4×C73#4112yes—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73C2×D412yes—
78C22×C4×C73C2×D411yescharacteristic
79M16×C73—24no—
80M16×C73—42no—
81C2×C8×C73—42no—
82C4⋊C4×C73—22no—
83C22⋊C4×C73—22no—
84C22⋊C4×C73—22no—
85C42×C73—21no—
86C42×C73C4⋊C411yescharacteristic
87C42×C73C4⋊C411yescharacteristic
88C23×C73—21no—
89C23×C73C22⋊Q812yes—
90C23×C73#12311yescharacteristic
91Q8×C73—41no—
92Q8×C73—41no—
93D4×C73—41no—
94D4×C73—42no—
95D4×C73—41no—
96D4×C73—44no—
97D4×C73—81no—
98C2×C4×C73—21no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73—22no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—42no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73C2×C4⋊C411yescharacteristic
109C2×C4×C73C4×D412yes—
110C2×C4×C73#12011yescharacteristic
111C2×Dic73—81no—
112C8×C73—44no—
113#10596831—42no—
114#10596831—42no—
115#10596831—42no—
116C22×C73—21no—
117C22×C73—22no—
118C22×C73—41no—
119C22×C73—42no—
120C22×C73—42no—
121C22×C73—41no—
122C22×C73#38511yescharacteristic
123C4×C73—22no—
124C4×C73—22no—
125C4×C73—21no—
126C4×C73—41no—
127C4×C73—41no—
128C4×C73B(4,𝔽2)11yescharacteristic
129C4×C73#45811yescharacteristic
130Dic73—161no—
131#12975—731nomaximal, Sylow
132C2×C73—21no—
133C2×C73—41no—
134C2×C73—42no—
135C2×C73—81no—
136C2×C73#186711yescharacteristic
137#2979—731no—
138#2132D7311yescharacteristic
139#1884—731no—
140#1858—732no—
141#1395—732no—
142C73#1297511yescharacteristic, complemented, Sylow
143#568—731no—
144#532C2×D7311yescharacteristic
145#436—1462no—
146C8×D4—1462no—
147#431—1461no—
148#430—1462no—
149#421—1461no—
150#413C2×D7312yes—
151#410—731no—
152#408—732no—
153#408—1461no—
154#356Dic7312yescomplemented
155#355Dic7312yescomplemented
156#341—1462no—
157#339—1461no—
158C2×#41D29211yescharacteristic
159C2×#41C22×D7311yescharacteristic
160#131—2922no—
161C2×M16—22no—
162C2×M16—732no—
163C2×M16—1462no—
164C2×M16—1461no—
165C2×M16—1461no—
166C22×C8—732no—
167C22×C8—1461no—
168C4⋊Q8C2×Dic7311yescharacteristic
169#127Dic14611yescharacteristic
170#124Dic14611yescharacteristic
171#124C2×Dic7311yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—1461no—
175C2×C42D29211yescharacteristic
176#108—2924no—
177C4⋊C8—1461no—
178C4⋊C8—1461no—
179#104—2922no—
180#101C2×Dic7314yes—
181#100—22no—
182#99—1462no—
183C22⋊C8—1462no—
184C22⋊C8—1462no—
185C22⋊C8—1462no—
186C4×C8—1462no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊Dic7311yescharacteristic
192C2×Q8C22×Dic7311yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊Dic7311yescharacteristic
196C2×D4C2×Dic14611yescharacteristic
197C2×D4#1192402712yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4D4×D7312yes—
201C22×C4C2×D29211yescharacteristic
202M16—24no—
203M16—42no—
204M16—2921no—
205M16—2922no—
206M16—5841no—
207M16—1462no—
208M16—1464no—
209C2×C8—42no—
210C2×C8—2921no—
211C2×C8—2922no—
212C2×C8—2922no—
213C2×C8—5841no—
214C2×C8—1462no—
215C2×C8—1462no—
216C4⋊C4—22no—
217C4⋊C4—2921no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—2921no—
221C42—21no—
222C42—2921no—
223C42C4⋊Dic7311yescharacteristic
224C42C4⋊Dic7311yescharacteristic
225C23—21no—
226C23#1455656812yes—
227C23#1455657511yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—44no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—5841no—
246C2×C4C2×C4⋊Dic7311yescharacteristic
247C2×C4D4×Dic7312yes—
248C2×C4#1455656111yescharacteristic
249C8—44no—
250C8—2922no—
251C8—2922no—
252C8—2922no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#1798846211yescharacteristic
260C4—22no—
261C4—22no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—11681no—
266C4#1798877511yescharacteristic
267C4#1798877811yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2#2261021411yescharacteristic
subgroupG/H
centerC4#17988775
commutatorC2×Q8×C73C22×C4
FrattiniC2×#41C22×D73
FittingC73×#2132C2
socleC2×C73#1867
prime radical#532×C73C22
cosocleC2×#41×C73C23
hypercenter#2132D73
hypocenterC73#12975
LCSG ▷ C2×Q8×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #22610153C2semidirect
4. #22608061C2semidirect
19. #356×C73C4semidirect
20. #355×C73C4semidirect
142. C73#12975semidirect
154. #356Dic73semidirect
155. #355Dic73semidirect

Maximal subgroups

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

subgroupindex
#226189982
#226103152
#226101532
#226080612
C73×#21322
#1297573