Finite Group Databasebeta

← #22610267

Subgroups — #22610267

Order 9344.

402 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 115 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×D4×Dic73C211yesmaximal, characteristic
2C2×#14556515C211yesmaximal, characteristic, complemented
3C2×#14556512C211yesmaximal, characteristic
4#17988494C211yesmaximal, characteristic, complemented
5#17988350C211yesmaximal, characteristic, complemented
6#17988341C211yesmaximal, characteristic, complemented
7C22×SD16×C73C211yesmaximal, characteristic
8C23×Dic73—21no—
9C22×#10596831C2211yescharacteristic
10D4×Dic73—22no—
11D4×Dic73—22no—
12C2×C4⋊Dic73C2211yescharacteristic
13C2×C4⋊Dic73C2211yescharacteristic
14C2×C22⋊Dic73—21no—
15C2×C4×Dic73C2211yescharacteristic
16#14556515—22no—
17#14556512—22no—
18#14556498—21no—
19#14556498—21no—
20C22×Q8×C73C2211yescharacteristic
21C22×D4×C73C2211yescharacteristic
22C2×SD16×C73—22no—
23C2×SD16×C73—22no—
24C2×SD16×C73C414yescomplemented
25C22×C8×C73C2211yescharacteristic
26C22×Dic73—24no—
27C22×Dic73—21no—
28C22×Dic73—21no—
29C22×Dic73—42no—
30C22×Dic73—42no—
31C22×Dic73D411yescharacteristic
32C22×Dic73D411yescharacteristic
33C2×#10596831—21no—
34C2×#10596831—21no—
35C2×#10596831—21no—
36C4⋊Dic73—21no—
37C4⋊Dic73—21no—
38C4⋊Dic73—21no—
39C4⋊Dic73D412yes—
40C22⋊Dic73—24no—
41C4×Dic73—21no—
42C4×Dic73—21no—
43C24×C73—21no—
44C2×Q8×C73—21no—
45C2×Q8×C73—21no—
46C2×Q8×C73—21no—
47C2×Q8×C73—21no—
48C2×Q8×C73—21no—
49C2×Q8×C73C2×C412yescomplemented
50C2×D4×C73—21no—
51C2×D4×C73—21no—
52C2×D4×C73—21no—
53C2×D4×C73C2×C412yes—
54C2×D4×C73D412yes—
55C2×D4×C73D412yes—
56C22×C4×C73—21no—
57C22×C4×C73C2311yescharacteristic
58SD16×C73—28no—
59C2×C8×C73—21no—
60C2×C8×C73—21no—
61C2×C8×C73C2×C412yescomplemented
62C23×C73—22no—
63C23×C73—22no—
64C23×C73—22no—
65C23×C73—41no—
66C23×C73—41no—
67C23×C73—41no—
68C23×C73—41no—
69C23×C73C2×D411yescharacteristic
70Q8×C73—22no—
71Q8×C73—22no—
72Q8×C73—41no—
73Q8×C73—41no—
74D4×C73—22no—
75D4×C73—22no—
76D4×C73—22no—
77D4×C73C22⋊C414yes—
78C2×C4×C73—22no—
79C2×C4×C73—41no—
80C2×C4×C73—41no—
81C2×C4×C73C22×C411yescharacteristic
82C2×C4×C73C2×D411yescharacteristic
83C2×C4×C73C2×D411yescharacteristic
84C2×C4×C73C2×D411yescharacteristic
85C2×C4×C73#4111yescharacteristic
86C2×C4×C73#4111yescharacteristic
87C2×Dic73—21no—
88C2×Dic73—21no—
89C2×Dic73—21no—
90C2×Dic73—22no—
91C2×Dic73—21no—
92C2×Dic73—21no—
93C2×Dic73—48no—
94C2×Dic73—41no—
95C2×Dic73—41no—
96C2×Dic73—41no—
97C2×Dic73—41no—
98C2×Dic73—41no—
99C2×Dic73SD1612yescomplemented
100C8×C73—22no—
101#10596831—41no—
102C22×C73—24no—
103C22×C73—42no—
104C22×C73—42no—
105C22×C73—42no—
106C22×C73—42no—
107C22×C73—42no—
108C22×C73—42no—
109C22×C73C4×D411yescharacteristic
110C22×C73#12011yescharacteristic
111C22×C73#12111yescharacteristic
112C22×C73C2×SD1611yescharacteristic
113C22×C73#13511yescharacteristic
114C22×C73AGL(1,ℤ/8)11yescharacteristic
115C22×C73#13711yescharacteristic
116C4×C73—42no—
117C4×C73C2×C22⋊C411yescharacteristic
118C4×C73C4×D411yescharacteristic
119C4×C73#12111yescharacteristic
120C4×C73#12311yescharacteristic
121Dic73—22no—
122Dic73—42no—
123Dic73—41no—
124Dic73—81no—
125C2×C73—44no—
126C2×C73#38611yescharacteristic
127C2×C73C4×SD1611yescharacteristic
128C2×C73#44011yescharacteristic
129C2×C73#45011yescharacteristic
130C2×C73#45211yescharacteristic
131C2×C73#46111yescharacteristic
132C2×C73#46311yescharacteristic
133#1877—731nomaximal, Sylow
134C73#187711yescharacteristic, complemented, Sylow
135C22×SD16D7311yescharacteristic
136C2×C4×D4—731no—
137C2×#103—731no—
138C2×#102—731no—
139#391—731no—
140#340—731no—
141#336—731no—
142C22×Q8C2×D7311yescharacteristic
143C22×D4C2×D7311yescharacteristic
144C23×C4—1461no—
145C2×SD16—22no—
146C2×SD16—22no—
147C2×SD16Dic7314yescomplemented
148C22×C8—731no—
149C22×C8C2×D7311yescharacteristic
150C4×D4—1462no—
151C4×D4—1462no—
152C2×C4⋊C4—731no—
153C2×C4⋊C4—731no—
154C2×C22⋊C4—1461no—
155C2×C42—731no—
156#103—1462no—
157#102—1462no—
158#95—1461no—
159#95—1461no—
160C24—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8C2×Dic7312yescomplemented
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4C2×Dic7312yes—
171C2×D4C22⋊D7312yes—
172C2×D4C22⋊D7312yes—
173C22×C4—21no—
174C22×C4—2922no—
175C22×C4—2922no—
176C22×C4—731no—
177C22×C4—731no—
178C22×C4—1461no—
179C22×C4—1464no—
180C22×C4—1461no—
181C22×C4C22×D7311yescharacteristic
182SD16—28no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—1461no—
186C2×C8—1461no—
187C2×C8—1461no—
188C2×C8C2×Dic7312yescomplemented
189C4⋊C4—732no—
190C4⋊C4—1461no—
191C4⋊C4—1461no—
192C4⋊C4—1461no—
193C22⋊C4—1464no—
194C42—1461no—
195C42—1461no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23D4×D7311yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208D4—22no—
209D4—22no—
210D4—22no—
211D4C22⋊Dic7314yes—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—2921no—
216C2×C4—2921no—
217C2×C4—2928no—
218C2×C4—2921no—
219C2×C4—2921no—
220C2×C4—2921no—
221C2×C4—732no—
222C2×C4—1461no—
223C2×C4—1462no—
224C2×C4—1461no—
225C2×C4—1461no—
226C2×C4—1461no—
227C2×C4—1461no—
228C2×C4C22×Dic7311yescharacteristic
229C2×C4D4×D7311yescharacteristic
230C2×C4C2×C22⋊D7311yescharacteristic
231C2×C4C2×C22⋊D7311yescharacteristic
232C2×C4#1192402711yescharacteristic
233C2×C4#1192402711yescharacteristic
234C8—22no—
235C8—2921no—
236C22—24no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22D4×Dic7311yescharacteristic
244C22#1455655911yescharacteristic
245C22#1455656311yescharacteristic
246C22SD16×D7311yescharacteristic
247C22#1455661211yescharacteristic
248C22#1455661811yescharacteristic
249C22#1455662011yescharacteristic
250C4—42no—
251C4—2922no—
252C4—2921no—
253C4—5841no—
254C4—1462no—
255C4C2×C22⋊Dic7311yescharacteristic
256C4D4×Dic7311yescharacteristic
257C4#1455656311yescharacteristic
258C4#1455657211yescharacteristic
259C2—44no—
260C2#1798846711yescharacteristic
261C2SD16×Dic7311yescharacteristic
262C2#1798868811yescharacteristic
263C2#1798873611yescharacteristic
264C2#1798874611yescharacteristic
265C2#1798879011yescharacteristic
266C2#1798880411yescharacteristic
subgroupG/H
centerC23D4×D73
commutatorC2×C4×C73C22×C4
FrattiniC22×C4C22×D73
FittingC22×SD16×C73C2
socleC23×C73C2×D4
hypercenterC22×SD16D73
hypocenterC73#1877
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#14556515C2semidirect
4. #17988494C2semidirect
5. #17988350C2semidirect
6. #17988341C2semidirect
24. C2×SD16×C73C4semidirect
49. C2×Q8×C73C2×C4semidirect
61. C2×C8×C73C2×C4semidirect
99. C2×Dic73SD16semidirect
134. C73#1877semidirect
147. C2×SD16Dic73semidirect
166. C2×Q8C2×Dic73semidirect
188. C2×C8C2×Dic73semidirect