Finite Group Databasebeta

← C2×#22624425 #33993820

Subgroups — C2×#22624425 #33993820

Order 18688.

6338 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624425C2132yesmaximal, direct product factor
2C2×C73⋊#564C213yesmaximal, complemented
3C2×#17989314C2112yesmaximal, complemented
4C2×#17989196C2112yesmaximal
5C2×#17989143C213yesmaximal
6C2×#564×C73C211yesmaximal, characteristic, complemented
7C73⋊#564C22148yescomplemented
8#17989314C221192yescomplemented
9#17989196C221192yes—
10#17989143C22148yes—
11C2×#14556590C22112yes—
12C2×#14556590C2216yes—
13C2×#14556586C22112yes—
14C2×#14556585C2214yes—
15C2×#14556583C2213yes—
16C2×#14556582C22112yes—
17C2×#14556581C22112yes—
18C2×#14556581C22112yes—
19C2×#14556581C22112yes—
20C2×#14556569C22112yes—
21C2×C4×Dic146C22112yes—
22C2×#14556556C22112yes—
23C2×#14556556C22112yes—
24C2×#14556549C2216yes—
25C2×C42×D73C2211yescharacteristic
26#564×C73C22116yescomplemented
27C2×#125×C73C22112yes—
28C2×#125×C73C2213yes—
29C22×Dic146C23112yes—
30C22×C4×D73C2313yes—
31#14556590C23148yes—
32#14556590C23196yes—
33#14556586C23196yes—
34#14556585C23132yes—
35#14556583C23124yes—
36#14556582C23196yes—
37#14556581C23196yes—
38#14556581C23196yes—
39#14556581C23196yes—
40#14556569C23196yes—
41C4×Dic146C23196yes—
42#14556556C23196yes—
43#14556556C23196yes—
44#14556549C23148yes—
45C2×C4⋊Dic73C2316yes—
46C2×C4⋊Dic73C23112yes—
47C2×#11924007C23112yes—
48C2×#11924007C23112yes—
49C2×#11924007C23112yes—
50C2×#11924007C23112yes—
51C2×#11924005C23112yes—
52C2×C4×Dic73C2313yes—
53C2×C4×Dic73C23112yes—
54C2×C4×Dic73C23112yes—
55C42×D73C2318yes—
56#125×C73C23124yes—
57#125×C73C23196yes—
58C2×C4⋊C4×C73C2316yes—
59C2×C4⋊C4×C73C23112yes—
60C2×C4⋊C4×C73C23112yes—
61C2×C42×C73C2311yescharacteristic
62C2×C42×C73C2314yes—
63C23×D73C2411yescharacteristic
64C2×Dic146—248no—
65C2×Dic146C24148yes—
66C22×Dic73C24112yes—
67C22×Dic73C2413yes—
68C2×C4×D73—212no—
69C2×C4×D73C24112yes—
70C4⋊Dic73C24124yes—
71C4⋊Dic73C24148yes—
72#11924007C24148yes—
73#11924007C24148yes—
74#11924007C24148yes—
75#11924007C24148yes—
76#11924005C24148yes—
77C4×Dic73C24112yes—
78C4×Dic73C24148yes—
79C4×Dic73C24148yes—
80C22×C4×C73C24112yes—
81C22×C4×C73C2413yes—
82C4⋊C4×C73C24124yes—
83C4⋊C4×C73C24148yes—
84C4⋊C4×C73C24148yes—
85C42×C73C2414yes—
86C42×C73C24116yes—
87C23×C73C2511yescharacteristic
88C22×D73—26no—
89C22×D73C2512yes—
90C2×C4×C73—26no—
91C2×C4×C73—224no—
92C2×C4×C73C25124yes—
93C2×C4×C73C2516yes—
94C2×Dic73—224no—
95C2×Dic732−1+4112yes—
96C2×Dic73C25124yes—
97C2×Dic73C2516yes—
98C4×D73—224no—
99Dic146—296no—
100C22×C73C2×2−1+416yes—
101C22×C73C2611yescharacteristic
102C2×D73—26no—
103C2×D73—44no—
104C4×C73—224no—
105C4×C73—26no—
106Dic73—224no—
107Dic73C2×2−1+4112yes—
108C2×#3556—731nomaximal, Sylow
109C2×C73#355614yescomplemented
110C2×C73C22×2−1+413yes—
111D73—42no—
112#3556—7332no—
113C2×#564—733no—
114C2×#564D7311yescharacteristic, complemented
115C2×#557—7312no—
116C2×#541—7312no—
117C2×#533—733no—
118C73C2×#355611yescharacteristic, complemented, Sylow
119#564—7348no—
120#564C2×D73116yescomplemented
121#557—73192no—
122#541—73192no—
123#533—7348no—
124C2×C4⋊Q8—736no—
125C2×C4⋊Q8—7312no—
126C2×#126—7312no—
127C2×#126—734no—
128C2×#125—7312no—
129C2×#125—7312no—
130C2×#125—7312no—
131C2×#125—7312no—
132C2×#125—733no—
133C2×#125C2×D7313yes—
134C2×#125C2×D73112yes—
135C2×C22⋊Q8—7312no—
136C2×C4×Q8—7312no—
137C2×C4×Q8—7312no—
138C2×C4×Q8—7312no—
139C2×#117—736no—
140C22×C42—731no—
141C22×Q8—7312no—
142C23×C4—733no—
143C4⋊Q8—7348no—
144C4⋊Q8—7396no—
145#126—7396no—
146#126—7332no—
147#125—7396no—
148#125—7396no—
149#125—7396no—
150#125—7396no—
151#125—7324no—
152#125C22×D73124yes—
153#125C22×D73196yes—
154C22⋊Q8—7396no—
155C4×Q8—7396no—
156C4×Q8—7396no—
157C4×Q8—7396no—
158#117—7348no—
159C2×C4⋊C4—736no—
160C2×C4⋊C4—7312no—
161C2×C4⋊C4—7312no—
162C2×C4⋊C4—7312no—
163C2×C4⋊C4—7312no—
164C2×C4⋊C4—7312no—
165C2×C4⋊C4C22×D7316yes—
166C2×C4⋊C4C22×D73112yes—
167C2×C4⋊C4C22×D73112yes—
168C2×C22⋊C4—7312no—
169C2×C42—7312no—
170C2×C42—733no—
171C2×C42—7312no—
172C2×C42—738no—
173C2×C42C22×D7314yes—
174C2×C42C22×D7311yescharacteristic
175C24—731no—
176C2×Q8—7348no—
177C2×Q8—14648no—
178C22×C4—733no—
179C22×C4—7312no—
180C22×C4—7312no—
181C22×C4—14612no—
182C22×C4C23×D73112yes—
183C22×C4C23×D7313yes—
184C4⋊C4—7348no—
185C4⋊C4—7348no—
186C4⋊C4—7348no—
187C4⋊C4—7348no—
188C4⋊C4—7324no—
189C4⋊C4—7348no—
190C4⋊C4C23×D73124yes—
191C4⋊C4C23×D73148yes—
192C4⋊C4C23×D73148yes—
193C22⋊C4—7348no—
194C42—7312no—
195C42—7348no—
196C42—7348no—
197C42C23×D7314yes—
198C42C23×D73116yes—
199C23—732no—
200C23—1466no—
201C23C24×D7311yescharacteristic
202Q8—14696no—
203C2×C4—26no—
204C2×C4—224no—
205C2×C4—7312no—
206C2×C4—736no—
207C2×C4—7324no—
208C2×C4—14624no—
209C2×C4—14624no—
210C2×C4C24×D73124yes—
211C2×C4C24×D7316yes—
212C22—2924no—
213C22—1466no—
214C222−1+4×D7316yes—
215C22C25×D7311yescharacteristic
216C4—26no—
217C4—224no—
218C4—7312no—
219C4—14624no—
220C2—2922no—
221C2#2262442514yesdirect product factor
222C2C2×2−1+4×D7313yes—
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#564×C73C2
socleC23×C73C25
prime radicalC23×D73C24
hypercenterC2×#564D73
hypocenterC73C2×#3556
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#564

Complements

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

NH ≅ G/Ntype
1. #22624425C2direct (N × H)
2. C2×C73⋊#564C2semidirect
3. C2×#17989314C2semidirect
6. C2×#564×C73C2semidirect
7. C73⋊#564C22semidirect
8. #17989314C22semidirect
26. #564×C73C22semidirect
109. C2×C73#3556semidirect
114. C2×#564D73semidirect
118. C73C2×#3556semidirect
120. #564C2×D73semidirect
221. C2#22624425direct (N × H)

Maximal subgroups

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

subgroupindex
#226244252
C2×C73⋊#5642
C2×#179893142
C2×#179891962
C2×#179891432
C2×#564×C732
C2×#355673