Finite Group Databasebeta

← #33005454

Subgroups — #33005454

Order 18688.

444 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 77 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×#17988386C212yesmaximal
2#22610325C212yesmaximal
3#22610145C211yesmaximal, characteristic
4#22608058C211yesmaximal, characteristic
5C73×#2734C211yesmaximal, characteristic
6#17989065C2211yescharacteristic
7#17988627—21no—
8#17988596—22no—
9C2×#14556510C2212yes—
10C2×#14556510C2211yescharacteristic
11#17988386—24no—
12#17988347—22no—
13#17988347—21no—
14C2×C4⋊Q8×C73C2212yes—
15#398×C73—21no—
16#395×C73—22no—
17#391×C73C2211yescharacteristic
18#389×C73—21no—
19#387×C73—22no—
20C2×#11924012—21no—
21C2×#11924012—22no—
22Q8×Dic73—41no—
23Q8×Dic73—41no—
24#14556550—21no—
25#14556550—22no—
26#14556524—41no—
27#14556522—41no—
28#14556515—42no—
29#14556515—42no—
30#14556510—22no—
31#14556510—21no—
32#14556510—22no—
33#14556510D412yes—
34C22×Q8×C73D412yes—
35C22×Q8×C73C2311yescharacteristic
36C4⋊Q8×C73—22no—
37C4⋊Q8×C73—22no—
38C4⋊Q8×C73—22no—
39C4⋊Q8×C73—22no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C42×C73D412yes—
47C2×C42×C73D411yescharacteristic
48#95×C73—21no—
49#95×C73—22no—
50#95×C73—22no—
51#95×C73—22no—
52#11924012—22no—
53#11924012—41no—
54#11924012—42no—
55#11924012—42no—
56C2×#10596831—41no—
57C2×#10596831—42no—
58C4⋊Dic73—22no—
59C4⋊Dic73—22no—
60C4⋊Dic73—21no—
61C4⋊Dic73—21no—
62C22⋊Dic73—41no—
63C4×Dic73—41no—
64C4×Dic73—42no—
65C2×Q8×C73—22no—
66C2×Q8×C73—22no—
67C2×Q8×C73—41no—
68C2×Q8×C73—41no—
69C2×Q8×C73—42no—
70C2×Q8×C73—42no—
71C2×Q8×C73C2×D411yescharacteristic
72C2×Q8×C73#4111yescharacteristic
73C2×Q8×C73#4112yes—
74C22×C4×C73—21no—
75C22×C4×C73—22no—
76C22×C4×C73—21no—
77C22×C4×C73—22no—
78C22×C4×C73C2×D412yes—
79C22×C4×C73C2×D411yescharacteristic
80C4⋊C4×C73—24no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—42no—
83C4⋊C4×C73—42no—
84C4⋊C4×C73—42no—
85C4⋊C4×C73—42no—
86C4⋊C4×C73—81no—
87C4⋊C4×C73—81no—
88C42×C73—22no—
89C42×C73—22no—
90C42×C73—41no—
91C23×C73#12011yescharacteristic
92Q8×C73—41no—
93Q8×C73—41no—
94Q8×C73—42no—
95Q8×C73—44no—
96Q8×C73—82no—
97C2×C4×C73—21no—
98C2×C4×C73—22no—
99C2×C4×C73—21no—
100C2×C4×C73—22no—
101C2×C4×C73—22no—
102C2×C4×C73—24no—
103C2×C4×C73—41no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—41no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73#12111yescharacteristic
113C2×C4×C73#12112yes—
114C2×C4×C73#12312yes—
115C2×C4×C73#12411yescharacteristic
116C2×Dic73—41no—
117C2×Dic73—41no—
118#10596831—42no—
119#10596831—82no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73#39411yescharacteristic
123C22×C73B(4,𝔽2)11yescharacteristic
124C22×C73#45811yescharacteristic
125C4×C73—21no—
126C4×C73—22no—
127C4×C73—22no—
128C4×C73—21no—
129C4×C73—44no—
130C4×C73—42no—
131C4×C73—81no—
132C4×C73—82no—
133Dic73—81no—
134Dic73—81no—
135#13428—731nomaximal, Sylow
136C2×C73—21no—
137C2×C73—21no—
138C2×C73#204411yescharacteristic
139C2×C73#220311yescharacteristic
140C2×C73#220511yescharacteristic
141#2734D7311yescharacteristic
142C2×#356—732no—
143#1886—732no—
144#1857—731no—
145#1394—731no—
146C73#1342811yescharacteristic, complemented, Sylow
147C2×C4⋊Q8C2×D7312yes—
148#519—731no—
149#428—1461no—
150#419—1462no—
151C2×#101—731no—
152C2×#101—732no—
153#398—21no—
154#395—22no—
155#391C2×D7311yescharacteristic
156#389—21no—
157#387—22no—
158#356—1464no—
159#339—1461no—
160#339—1462no—
161C22×Q8C22⋊D7312yes—
162C22×Q8C22×D7311yescharacteristic
163C2×M16—1462no—
164C2×M16—1461no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4×Q8—2921no—
170C4×Q8—2921no—
171#117—1462no—
172#117—1461no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C22⋊D7311yescharacteristic
179C2×C42C22⋊D7312yes—
180C2×C42C22⋊D7311yescharacteristic
181#107—2921no—
182#106—2921no—
183#103—2922no—
184#103—2922no—
185#101—732no—
186#101—1462no—
187#101—1462no—
188#101—1461no—
189#95—21no—
190#95—22no—
191#95—22no—
192#95—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8D4×D7311yescharacteristic
200C2×Q8#1192402711yescharacteristic
201C2×Q8#1192402712yes—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4C2×C22⋊D7311yescharacteristic
207C22×C4C2×C22⋊D7312yes—
208M16—2921no—
209M16—2922no—
210M16—2922no—
211M16—1462no—
212C2×C8—2921no—
213C2×C8—2922no—
214C4⋊C4—24no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—1462no—
223C4⋊C4—1462no—
224C4⋊C4—1461no—
225C4⋊C4—1461no—
226C22⋊C4—2921no—
227C42—22no—
228C42—22no—
229C42—41no—
230C42—2921no—
231C42—2922no—
232C23#1455656011yescharacteristic
233Q8—41no—
234Q8—41no—
235Q8—42no—
236Q8—44no—
237Q8—82no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—2921no—
254C2×C4—2921no—
255C2×C4#1455656211yescharacteristic
256C2×C4#1455656312yes—
257C2×C4#1455657212yes—
258C2×C4#1455657811yescharacteristic
259C8—2922no—
260C8—5842no—
261C22—21no—
262C22—21no—
263C22#1798850111yescharacteristic
264C22#1798877411yescharacteristic
265C22#1798877711yescharacteristic
266C4—21no—
267C4—22no—
268C4—22no—
269C4—21no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—81no—
274C4—5841no—
275C4—5841no—
276C2—21no—
277C2—21no—
278C2#2261117911yescharacteristic
279C2#2261195211yescharacteristic
280C2#2261196111yescharacteristic
subgroupG/H
centerC22#17988774
commutator, cosocleC22×Q8×C73C23
FrattiniC22×Q8C22×D73
FittingC73×#2734C2
socleC22×C73B(4,𝔽2)
prime radicalC23×C73#120
hypercenter#2734D73
hypocenterC73#13428
LCSG ▷ C22×Q8×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2734

Complements

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

NH ≅ G/Ntype
146. C73#13428semidirect

Maximal subgroups

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

subgroupindex
C2×#179883862
#226103252
#226101452
#226080582
C73×#27342
#1342873