Finite Group Databasebeta

← #22566405

Subgroups — #22566405

Order 8576.

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

normal subgroups only

This group has 408 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#16863332C212yesmaximal
2#16863330C212yesmaximal, complemented
3#16863328C212yesmaximal, complemented
4#16863319C212yesmaximal
5#16863293C212yesmaximal, complemented
6#16863283C212yesmaximal, complemented
7#16863256C211yesmaximal, characteristic
8C4×C22⋊Dic67C211yesmaximal, characteristic
9#381×C67C211yesmaximal, characteristic
10C2×C4⋊Dic67C2212yes—
11C2×C4⋊Dic67C2212yes—
12C2×C4⋊Dic67C2212yes—
13C2×Dic67⋊C4C2212yes—
14C2×Dic67⋊C4C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×Dic67⋊C4C2212yes—
17C2×C22⋊Dic67C2212yes—
18C2×C22⋊Dic67C2211yescharacteristic
19C2×C22⋊Dic67C2211yescharacteristic
20C2×C4×Dic67C2212yes—
21C67⋊#95C2211yescharacteristic
22C67⋊#95C2211yescharacteristic
23C67⋊#95C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2212yes—
26C23×C4×C67C2211yescharacteristic
27C2×C22⋊C4×C67C2212yes—
28#95×C67C2211yescharacteristic
29#95×C67C2211yescharacteristic
30#95×C67C2212yes—
31C22×Dic67C2312yes—
32C22×Dic67C2312yes—
33C22×Dic67C2312yes—
34C22×Dic67C2312yes—
35C4⋊Dic67—42no—
36C4⋊Dic67—42no—
37C4⋊Dic67—42no—
38Dic67⋊C4—24no—
39Dic67⋊C4—24no—
40Dic67⋊C4—42no—
41Dic67⋊C4—42no—
42C22⋊Dic67—24no—
43C22⋊Dic67—41no—
44C22⋊Dic67—41no—
45C4×Dic67—42no—
46C24×C67C2311yescharacteristic
47C22×C4×C67—42no—
48C22×C4×C67Q814yes—
49C22×C4×C67C2312yes—
50C22×C4×C67C2312yes—
51C22×C4×C67C2311yescharacteristic
52C22×C4×C67C2311yescharacteristic
53C22⋊C4×C67—42no—
54C23×C67—22no—
55C23×C67—22no—
56C23×C67—21no—
57C23×C67—21no—
58C23×C67C2×Q812yes—
59C23×C67C2411yescharacteristic
60C2×C4×C67—22no—
61C2×C4×C67—22no—
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—44no—
69C2×C4×C67C2×Q812yes—
70C2×C4×C67C2×Q812yes—
71C2×Dic67—22no—
72C2×Dic67—22no—
73C2×Dic67—22no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67#4114yes—
83C22×C67—22no—
84C22×C67—22no—
85C22×C67—22no—
86C22×C67—42no—
87C22×C67—42no—
88C22×C67C22×Q811yescharacteristic
89C22×C67C2×#4112yes—
90C22×C67H(5,𝔽2)12yes—
91C22×C67H(5,𝔽2)11yescharacteristic
92C22×C672−1+411yescharacteristic
93C4×C67—22no—
94C4×C67—22no—
95C4×C67—42no—
96C4×C67—42no—
97Dic67—24no—
98Dic67—42no—
99Dic67—42no—
100Dic67—42no—
101C2×C67—42no—
102C2×C67#53311yescharacteristic
103C2×C67#53912yes—
104C2×C67#54012yes—
105C2×C67#54311yescharacteristic
106C2×C67#54411yescharacteristic
107#2667—671nomaximal, Sylow
108C67#266711yescharacteristic, complemented, Sylow
109#399—672no—
110#398—672no—
111#398—672no—
112#396—672no—
113#389—672no—
114#387—672no—
115#381D6711yescharacteristic
116#380—671no—
117C4×C22⋊C4—671no—
118C23×C4C2×D6711yescharacteristic
119C2×C4⋊C4—672no—
120C2×C4⋊C4—672no—
121C2×C4⋊C4—672no—
122C2×C4⋊C4—672no—
123C2×C4⋊C4—672no—
124C2×C4⋊C4—672no—
125C2×C4⋊C4—672no—
126C2×C22⋊C4—671no—
127C2×C22⋊C4—671no—
128C2×C22⋊C4—672no—
129C2×C22⋊C4C2×D6712yes—
130C2×C42—672no—
131#95—671no—
132#95—671no—
133#95—672no—
134#95—672no—
135#95—672no—
136#95C2×D6711yescharacteristic
137#95C2×D6711yescharacteristic
138#95C2×D6712yes—
139C24C22×D6711yescharacteristic
140C22×C4—42no—
141C22×C4—672no—
142C22×C4—672no—
143C22×C4—672no—
144C22×C4—672no—
145C22×C4Dic13414yes—
146C22×C4C22×D6712yes—
147C22×C4C22×D6712yes—
148C22×C4C22×D6711yescharacteristic
149C22×C4C22×D6711yescharacteristic
150C4⋊C4—2682no—
151C4⋊C4—2682no—
152C4⋊C4—2682no—
153C4⋊C4—2682no—
154C4⋊C4—2682no—
155C4⋊C4—1344no—
156C4⋊C4—1344no—
157C22⋊C4—42no—
158C22⋊C4—2681no—
159C22⋊C4—2681no—
160C22⋊C4—1344no—
161C42—2682no—
162C23—22no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23C2×Dic13412yes—
167C23C23×D6711yescharacteristic
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—674no—
178C2×C4—1342no—
179C2×C4—1342no—
180C2×C4—1342no—
181C2×C4—1342no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4—1342no—
185C2×C4—1342no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1342no—
189C2×C4C2×Dic13412yes—
190C2×C4C2×Dic13412yes—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22C22×Dic13411yescharacteristic
197C22#41×D6712yes—
198C22#1452408812yes—
199C22#1452408911yescharacteristic
200C22#1452409011yescharacteristic
201C4—22no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—2682no—
206C4—2682no—
207C4—2682no—
208C4—1344no—
209C2—42no—
210C2#1686394911yescharacteristic
211C2#1686398912yes—
212C2#1686399812yes—
213C2#1686402211yescharacteristic
214C2#1686402411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#381×C67C2
hypercenter#381D67
hypocenterC67#2667
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #16863330C2semidirect
3. #16863328C2semidirect
5. #16863293C2semidirect
6. #16863283C2semidirect
108. C67#2667semidirect