Finite Group Databasebeta

← #22621248

Subgroups — #22621248

Order 9344.

392 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 137 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#17989144C211yesmaximal, characteristic, complemented
2#17989137C211yesmaximal, characteristic
3#17988878C212yesmaximal, complemented
4#17988873C212yesmaximal, complemented
5#17988816C214yesmaximal, complemented
6#17988793C212yesmaximal, complemented
7#17988788C212yesmaximal, complemented
8C2×C4⋊C8×C73C211yesmaximal, characteristic, complemented
9C2×#11924028—21no—
10C2×#11924017C2214yes—
11#14556591C2211yescharacteristic
12#14556591C2211yescharacteristic
13#14556591C2211yescharacteristic
14#14556588C2211yescharacteristic
15#14556583—21no—
16#14556579—21no—
17#14556570—21no—
18#14556570C2212yes—
19#14556565—21no—
20#14556565C2212yes—
21C4×Dic146—21no—
22C4×Dic146C2212yes—
23C4×D292—21no—
24C4×D292C2212yes—
25#14556550—21no—
26#14556523C2214yes—
27#14556516C2214yes—
28#14556513C2214yes—
29C22×C8×C73C2212yes—
30C2×C42×C73C2211yescharacteristic
31C4⋊C8×C73C2212yes—
32C4⋊C8×C73C2212yes—
33#11924028—44no—
34C2×Dic146—21no—
35C2×Dic146—21no—
36C2×Dic146C2312yes—
37C2×D292—21no—
38C2×D292—21no—
39C2×D292C2312yes—
40C2×C22⋊D73—22no—
41C2×C4×D73—22no—
42#11924017—44no—
43C4⋊Dic73—21no—
44C4⋊Dic73—21no—
45C4⋊Dic73—21no—
46C4⋊Dic73—21no—
47C4⋊Dic73C2312yes—
48C4⋊Dic73C2312yes—
49#11924007—22no—
50#11924007—22no—
51C22⋊Dic73—22no—
52#11924005—22no—
53#11924005—22no—
54C4×Dic73—22no—
55C22×C4×C73D411yescharacteristic
56C22×C4×C73D411yescharacteristic
57C22×C4×C73C2311yescharacteristic
58C2×C8×C73—42no—
59C2×C8×C73C2314yes—
60C42×C73D411yescharacteristic
61C42×C73D411yescharacteristic
62C42×C73C2311yescharacteristic
63C42×C73C2311yescharacteristic
64C23×C73C2×D411yescharacteristic
65C22×D73—22no—
66C2×C4×C73—21no—
67C2×C4×C73—21no—
68C2×C4×C73—41no—
69C2×C4×C73SD1614yescomplemented
70C2×C4×C73C2×D411yescharacteristic
71C2×C4×C73C2×D411yescharacteristic
72C2×C4×C73C2×D411yescharacteristic
73C2×C4×C73C2×D411yescharacteristic
74C2×C4×C73C2×D411yescharacteristic
75C2×C4×C73C2411yescharacteristic
76C22⋊D73—44no—
77C2×Dic73—22no—
78C2×Dic73—22no—
79C2×Dic73—22no—
80D292—22no—
81D292—22no—
82D292—42no—
83D292—42no—
84C4×D73—44no—
85Dic146—22no—
86Dic146—22no—
87Dic146—42no—
88Dic146—42no—
89C8×C73—24no—
90C22×C73—21no—
91C22×C73—21no—
92C22×C73C2×SD1612yes—
93C22×C73C22×D411yescharacteristic
94C2×D73—42no—
95C2×D73—42no—
96C2×D73—42no—
97C4×C73—21no—
98C4×C73—21no—
99C4×C73—21no—
100C4×C73C2×SD1612yes—
101C4×C73C2×SD1612yes—
102C4×C73H(5,𝔽2)11yescharacteristic
103C4×C732−1+411yescharacteristic
104Dic73—42no—
105Dic73—42no—
106Dic73—42no—
107C2×C73—22no—
108C2×C73#53711yescharacteristic
109C2×C73C22×SD1611yescharacteristic
110C2×C73#57511yescharacteristic
111D73—82no—
112#3246—731nomaximal, Sylow
113C73#324611yescharacteristic, complemented, Sylow
114#533—731no—
115#532—731no—
116#476—732no—
117#475—732no—
118#465—734no—
119#461—732no—
120#460—732no—
121C2×C4⋊C8D7311yescharacteristic, complemented
122C2×#41—1461no—
123C2×SD16—734no—
124C22×C8C2×D7312yes—
125C4⋊Q8—731no—
126C4⋊Q8—731no—
127C4⋊Q8—731no—
128#127—731no—
129#125—1461no—
130#124—1461no—
131C22⋊Q8—732no—
132C22⋊Q8—1461no—
133#121—732no—
134#121—1461no—
135C4×Q8—732no—
136C4×Q8—1461no—
137C4×D4—732no—
138C4×D4—1461no—
139#117—1461no—
140C2×C42C2×D7311yescharacteristic
141#106—734no—
142C4⋊C8C2×D7312yes—
143C4⋊C8C2×D7312yes—
144#103—734no—
145#102—734no—
146#41—2924no—
147C2×Q8—732no—
148C2×Q8—1461no—
149C2×Q8—1461no—
150C2×D4—732no—
151C2×D4—1462no—
152C2×D4—1461no—
153C2×D4—1461no—
154C22×C4—1462no—
155C22×C4D29211yescharacteristic
156C22×C4D29211yescharacteristic
157C22×C4C22×D7311yescharacteristic
158SD16—2924no—
159C2×C8—42no—
160C2×C8C22×D7314yes—
161C4⋊C4—732no—
162C4⋊C4—732no—
163C4⋊C4—1462no—
164C4⋊C4—1462no—
165C4⋊C4—1461no—
166C4⋊C4—1461no—
167C4⋊C4—1461no—
168C4⋊C4—1461no—
169C22⋊C4—1462no—
170C22⋊C4—1462no—
171C22⋊C4—1462no—
172C42—1462no—
173C42D29211yescharacteristic
174C42D29211yescharacteristic
175C42C22×D7311yescharacteristic
176C42C22×D7311yescharacteristic
177C23—1462no—
178C23C2×D29211yescharacteristic
179Q8—2922no—
180Q8—2922no—
181Q8—1462no—
182Q8—1462no—
183D4—2922no—
184D4—2924no—
185D4—2922no—
186D4—1462no—
187D4—1462no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—2924no—
192C2×C4—1462no—
193C2×C4—1462no—
194C2×C4—1462no—
195C2×C4#1192401714yescomplemented
196C2×C4C2×D29211yescharacteristic
197C2×C4C2×D29211yescharacteristic
198C2×C4C2×D29211yescharacteristic
199C2×C4C2×D29211yescharacteristic
200C2×C4C2×D29211yescharacteristic
201C2×C4C23×D7311yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—2922no—
206C22—2922no—
207C22—2922no—
208C22C2×#1192401712yes—
209C22C22×D29211yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2922no—
214C4—2922no—
215C4—2922no—
216C4C2×#1192401712yes—
217C4C2×#1192401712yes—
218C4#1455663611yescharacteristic
219C4#1455663911yescharacteristic
220C2—22no—
221C2—5842no—
222C2#1798916811yescharacteristic
223C2C22×#1192401711yescharacteristic
224C2#1798938711yescharacteristic
subgroupG/H
centerC22C22×D292
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
FittingC2×C4⋊C8×C73C2
socleC22×C73C22×D4
hypercenterC2×C4⋊C8D73
hypocenterC73#3246
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #17989144C2semidirect
3. #17988878C2semidirect
4. #17988873C2semidirect
5. #17988816C2semidirect
6. #17988793C2semidirect
7. #17988788C2semidirect
8. C2×C4⋊C8×C73C2semidirect
69. C2×C4×C73SD16semidirect
113. C73#3246semidirect
121. C2×C4⋊C8D73semidirect
195. C2×C4#11924017semidirect