Finite Group Databasebeta

← #22568397

Subgroups — #22568397

Order 8576.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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#16863338C212yesmaximal, complemented
2#16863337C212yesmaximal, complemented
3#16863334C212yesmaximal, complemented
4#16863324C212yesmaximal
5#16863321C211yesmaximal, characteristic
6#16863316C212yesmaximal, complemented
7#16863308C211yesmaximal, characteristic
8#16863269C212yesmaximal
9#385×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic
11C2×C4⋊Dic67C2211yescharacteristic
12C2×Dic67⋊C4C2212yes—
13C2×Dic67⋊C4C2212yes—
14C2×Dic67⋊C4C2212yes—
15C2×C22⋊Dic67C2212yes—
16C2×C22⋊Dic67C2212yes—
17C67⋊#95C2212yes—
18C67⋊#95C2212yes—
19C67⋊#95C2211yescharacteristic
20C67⋊#95C2212yes—
21C67⋊#95C2212yes—
22C67⋊#95C2212yes—
23C67⋊#95C2212yes—
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2212yes—
26C23×C4×C67C2211yescharacteristic
27C2×C4⋊C4×C67C2212yes—
28C2×C22⋊C4×C67C2211yescharacteristic
29C2×C22⋊C4×C67C2211yescharacteristic
30#95×C67C2212yes—
31C2×Dic134—42no—
32C2×Dic134—41no—
33C22×Dic67C2312yes—
34C22×Dic67C2312yes—
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C4⋊Dic67—41no—
38Dic67⋊C4—42no—
39Dic67⋊C4—42no—
40Dic67⋊C4—42no—
41C22⋊Dic67—24no—
42C22⋊Dic67—42no—
43C24×C67C2311yescharacteristic
44C22×C4×C67—42no—
45C22×C4×C67—41no—
46C22×C4×C67C2311yescharacteristic
47C22×C4×C67C2312yes—
48C22×C4×C67C2311yescharacteristic
49C22×C4×C67C2312yes—
50C4⋊C4×C67—42no—
51C22⋊C4×C67—22no—
52C22⋊C4×C67—41no—
53C23×C67—21no—
54C23×C67—22no—
55C23×C67—22no—
56C23×C67—21no—
57C23×C67#4112yes—
58C23×C67C2411yescharacteristic
59C2×C4×C67—22no—
60C2×C4×C67—21no—
61C2×C4×C67—21no—
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—22no—
69C2×C4×C67—44no—
70C2×Dic67—22no—
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—
82Dic134—44no—
83C22×C67—24no—
84C22×C67—22no—
85C22×C67—41no—
86C22×C67—42no—
87C22×C67—41no—
88C22×C67C2×#4111yescharacteristic
89C22×C67H(5,𝔽2)12yes—
90C22×C67H(5,𝔽2)11yescharacteristic
91C22×C672−1+412yes—
92C22×C672−1+411yescharacteristic
93C4×C67—42no—
94C4×C67—41no—
95C4×C67—42no—
96C4×C67—41no—
97Dic67—42no—
98Dic67—42no—
99Dic67—42no—
100Dic67—42no—
101C2×C67—42no—
102C2×C67#53912yes—
103C2×C67#53911yescharacteristic
104C2×C67H(3,𝔽4)11yescharacteristic
105C2×C67#56312yes—
106C2×C67#56311yescharacteristic
107#2835—671nomaximal, Sylow
108C67#283511yescharacteristic, complemented, Sylow
109#400—672no—
110#400—672no—
111#400—672no—
112#397—671no—
113#397—672no—
114#395—672no—
115#393—671no—
116#385—672no—
117#385D6711yescharacteristic
118C22×Q8—671no—
119C23×C4C2×D6711yescharacteristic
120C2×C4⋊C4—672no—
121C2×C4⋊C4—672no—
122C2×C4⋊C4—672no—
123C2×C4⋊C4—671no—
124C2×C4⋊C4C2×D6712yes—
125C2×C22⋊C4—672no—
126C2×C22⋊C4—672no—
127C2×C22⋊C4C2×D6711yescharacteristic
128C2×C22⋊C4C2×D6711yescharacteristic
129#95—672no—
130#95—672no—
131#95—671no—
132#95—672no—
133#95—672no—
134#95—672no—
135#95—672no—
136#95—672no—
137#95—671no—
138#95C2×D6712yes—
139C24C22×D6711yescharacteristic
140C2×Q8—2681no—
141C2×Q8—2682no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—672no—
145C22×C4—672no—
146C22×C4—672no—
147C22×C4—672no—
148C22×C4C22×D6711yescharacteristic
149C22×C4C22×D6712yes—
150C22×C4C22×D6711yescharacteristic
151C22×C4C22×D6712yes—
152C4⋊C4—42no—
153C4⋊C4—2682no—
154C4⋊C4—2682no—
155C4⋊C4—2682no—
156C4⋊C4—2681no—
157C22⋊C4—22no—
158C22⋊C4—41no—
159C22⋊C4—2682no—
160C22⋊C4—1344no—
161C23—21no—
162C23—22no—
163C23—22no—
164C23—21no—
165C23#1176162812yes—
166C23C23×D6711yescharacteristic
167Q8—2684no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—44no—
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×C4—1342no—
190C2×C4—1342no—
191C22—24no—
192C22—22no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22C2×#1176162811yescharacteristic
197C22#1452408812yes—
198C22#1452408811yescharacteristic
199C22#1452409012yes—
200C22#1452409111yescharacteristic
201C4—42no—
202C4—41no—
203C4—41no—
204C4—42no—
205C4—2682no—
206C4—2682no—
207C4—2682no—
208C4—2682no—
209C2—42no—
210C2#1686398812yes—
211C2#1686399311yescharacteristic
212C2#1686413911yescharacteristic
213C2#1686414712yes—
214C2#1686414911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#385×C67C2
hypercenter#385D67
hypocenterC67#2835
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #16863338C2semidirect
2. #16863337C2semidirect
3. #16863334C2semidirect
6. #16863316C2semidirect
108. C67#2835semidirect