Finite Group Databasebeta

← #22621162

Subgroups — #22621162

Order 9344.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D73C211yesmaximal, characteristic, complemented
2#17989122C211yesmaximal, characteristic
3#17988997C211yesmaximal, characteristic, complemented
4#17988949C211yesmaximal, characteristic, complemented
5#17988947C211yesmaximal, characteristic, complemented
6#17988924C212yesmaximal, complemented
7#17988907C212yesmaximal
8#17988871C212yesmaximal, complemented
9#17988866C212yesmaximal, complemented
10#17988642C211yesmaximal, characteristic, complemented
11#501×C73C211yesmaximal, characteristic, complemented
12C2×Q8×D73—21no—
13#14556591C2211yescharacteristic
14#14556590—21no—
15#14556590C2212yes—
16#14556589—21no—
17#14556586—21no—
18#14556579C2211yescharacteristic
19#14556576—21no—
20#14556571C2211yescharacteristic
21#14556566C2211yescharacteristic
22#14556556C2211yescharacteristic
23#14556553—21no—
24#14556553C2211yescharacteristic
25#14556549—21no—
26C4⋊C4×D73—21no—
27C4⋊C4×D73C2212yes—
28C42×D73C2211yescharacteristic
29#14556524C2212yes—
30#14556522C2212yes—
31#14556520C2212yes—
32#14556516C2212yes—
33#14556514C2212yes—
34#14556513C2212yes—
35#14556511C2212yes—
36#14556503C2212yes—
37#14556502C2211yescharacteristic
38C4⋊Q8×C73C2211yescharacteristic
39#125×C73C2211yescharacteristic
40#107×C73C2212yes—
41#106×C73C2212yes—
42#97×C73C2211yescharacteristic
43C2×Dic146—21no—
44C2×Dic146—22no—
45C2×Dic146C2311yescharacteristic
46Q8×D73—44no—
47C2×D292—21no—
48C2×D292C2311yescharacteristic
49C2×C4×D73—22no—
50C2×C4×D73—21no—
51C2×C4×D73D411yescharacteristic
52C2×C4×D73D411yescharacteristic
53C2×C4×D73C2311yescharacteristic
54C2×#10596831C2312yes—
55C4⋊Dic73—21no—
56C4⋊Dic73—21no—
57C4⋊Dic73C2312yes—
58#11924007—22no—
59#11924007—22no—
60#11924007—21no—
61#11924005—21no—
62#11924005—21no—
63#11924005—21no—
64#11924005—21no—
65#11924005—21no—
66C4×Dic73—21no—
67C4×Dic73D411yescharacteristic
68C4×Dic73D411yescharacteristic
69C4×Dic73C2311yescharacteristic
70C2×Q8×C73—21no—
71C2×C8×C73C2312yes—
72C4⋊C4×C73—21no—
73C4⋊C4×C73—21no—
74C4⋊C4×C73—21no—
75C4⋊C4×C73C2311yescharacteristic
76C4⋊C4×C73C2312yes—
77C4⋊C4×C73C2311yescharacteristic
78C42×C73C2311yescharacteristic
79C22×D73—21no—
80C22×D73C2×D411yescharacteristic
81Q8×C73—42no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—22no—
85C2×C4×C73C2×D411yescharacteristic
86C2×C4×C73C2×D411yescharacteristic
87C2×C4×C73C2411yescharacteristic
88C2×Dic73—21no—
89C2×Dic73—22no—
90C2×Dic73C2×D411yescharacteristic
91C2×Dic73C2×D411yescharacteristic
92C2×Dic73C2×D411yescharacteristic
93D292—22no—
94D292—42no—
95C4×D73—22no—
96C4×D73—22no—
97C4×D73—22no—
98C4×D73—44no—
99C4×D73—42no—
100Dic146—22no—
101Dic146—42no—
102Dic146—44no—
103C8×C73—22no—
104#10596831—22no—
105C22×C73C22×D411yescharacteristic
106C2×D73—21no—
107C2×D73—21no—
108C2×D73—41no—
109C2×D73—41no—
110C2×D73—41no—
111C2×D73#13712yes—
112C4×C73—21no—
113C4×C73—21no—
114C4×C73—42no—
115C4×C73—41no—
116C4×C73—41no—
117C4×C732−1+412yes—
118Dic73—21no—
119Dic73—21no—
120Dic73—41no—
121Dic73—42no—
122Dic73AGL(1,ℤ/8)12yes—
123C2×C73#53611yescharacteristic
124C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C73C2×#13711yescharacteristic
126D73—22no—
127D73—81no—
128#3239—731nomaximal, Sylow
129C73#323911yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—731no—
131#529—731no—
132#501—731no—
133#501D7311yescharacteristic, complemented
134#489—731no—
135#489—731no—
136#484—732no—
137#481—732no—
138#475—732no—
139#474—732no—
140#432—731no—
141C22×Q8—1461no—
142C4⋊Q8—732no—
143C4⋊Q8—731no—
144C4⋊Q8—1461no—
145C4⋊Q8—1461no—
146C4⋊Q8C2×D7311yescharacteristic
147#126—1461no—
148#125C2×D7311yescharacteristic
149#124—731no—
150#123—1461no—
151C22⋊Q8—731no—
152#121—731no—
153C4×Q8—731no—
154C4×D4—731no—
155C4×D4—1461no—
156#117—1461no—
157C2×C4⋊C4—732no—
158C2×C4⋊C4—1461no—
159C2×C42—731no—
160#107—732no—
161#107C2×D7312yes—
162#106—732no—
163#106C2×D7312yes—
164C4⋊C8—732no—
165#103—732no—
166#103—732no—
167#102—732no—
168#102—732no—
169C22⋊C8—732no—
170#97—731no—
171#97C2×D7311yescharacteristic
172C2×Q8—21no—
173C2×Q8—2924no—
174C2×Q8—731no—
175C2×Q8—1462no—
176C2×Q8—1461no—
177C2×D4—731no—
178C2×D4—1461no—
179C22×C4—731no—
180C22×C4—731no—
181C22×C4—731no—
182C22×C4—1461no—
183C22×C4—1462no—
184C2×C8—732no—
185C2×C8C22×D7312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—732no—
190C4⋊C4—1462no—
191C4⋊C4—1462no—
192C4⋊C4—1461no—
193C4⋊C4—1461no—
194C4⋊C4—1461no—
195C4⋊C4C22×D7312yes—
196C4⋊C4C22×D7311yescharacteristic
197C4⋊C4C22×D7311yescharacteristic
198C22⋊C4—1461no—
199C22⋊C4—1461no—
200C22⋊C4—1461no—
201C22⋊C4—1461no—
202C22⋊C4—1461no—
203C42—731no—
204C42—731no—
205C42—731no—
206C42—1461no—
207C42C22×D7311yescharacteristic
208C23—731no—
209C23—1461no—
210Q8—42no—
211Q8—2922no—
212Q8—2924no—
213Q8—1462no—
214D4—2922no—
215D4—1462no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—2924no—
220C2×C4—2922no—
221C2×C4—731no—
222C2×C4—731no—
223C2×C4—731no—
224C2×C4—1462no—
225C2×C4—1462no—
226C2×C4—1461no—
227C2×C4—1462no—
228C2×C4—1462no—
229C2×C4D4×D7311yescharacteristic
230C2×C4D4×D7311yescharacteristic
231C2×C4C23×D7311yescharacteristic
232C8—22no—
233C8—1462no—
234C22—2921no—
235C22—2921no—
236C22—2921no—
237C22—732no—
238C22—1461no—
239C22—1461no—
240C22C2×D4×D7311yescharacteristic
241C4—21no—
242C4—21no—
243C4—41no—
244C4—42no—
245C4—41no—
246C4—2921no—
247C4—2922no—
248C4—732no—
249C4—1461no—
250C4—1461no—
251C4#1455663912yes—
252C2—5841no—
253C2—1462no—
254C2#1798916211yescharacteristic
255C2AGL(1,ℤ/8)×D7311yescharacteristic
256C2#137×D7311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#501×C73C2
socleC22×C73C22×D4
hypercenter#501D73
hypocenterC73#3239
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D73C2semidirect
3. #17988997C2semidirect
4. #17988949C2semidirect
5. #17988947C2semidirect
6. #17988924C2semidirect
8. #17988871C2semidirect
9. #17988866C2semidirect
10. #17988642C2semidirect
11. #501×C73C2semidirect
129. C73#3239semidirect
133. #501D73semidirect