Finite Group Databasebeta

← #28336388

Subgroups — #28336388

Order 13696.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#13133819C211yesmaximal, characteristic
2C2×#16812028C211yesmaximal, characteristic, complemented
3C2×#16812025C212yesmaximal, complemented
4#20292590C212yesmaximal, complemented
5C2×#101×C107C211yesmaximal, characteristic, complemented
6C2×#13133819—24no—
7C2×#13133819—24no—
8C2×#13133819—22no—
9C2×#13133819—22no—
10C22×D428C412yes—
11C22×D428C2211yescharacteristic
12C22×C4×D107C412yes—
13C22×C4×D107C2211yescharacteristic
14C2×#13133802C2212yes—
15#16812028—22no—
16#16812025—24no—
17(C2×D107)⋊C8—24no—
18C22×Q8×C107C2211yescharacteristic
19C2×M16×C107C2212yes—
20#101×C107—22no—
21C23×D107C2×C412yes—
22C23×D107C2×C411yescharacteristic
23#13133819—216no—
24#13133819—216no—
25C2×D428—24no—
26C2×D428—24no—
27C2×D428—24no—
28C2×D428—22no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428D414yes—
32C22×Dic107C2×C411yescharacteristic
33C2×C4×D107—24no—
34C2×C4×D107—24no—
35C2×C4×D107—22no—
36C2×C4×D107—21no—
37C2×C4×D107—21no—
38C2×C4×D107—24no—
39C2×C4×D107D414yes—
40#13133802—22no—
41#13133802—22no—
42C2×C107⋊C8—22no—
43C2×Q8×C107—22no—
44C2×Q8×C107—22no—
45C2×Q8×C107D412yes—
46C2×Q8×C107D412yes—
47C22×C4×C107C2×C412yes—
48C22×C4×C107C2311yescharacteristic
49M16×C107—22no—
50M16×C107—22no—
51C2×C8×C107—22no—
52C23×C107C22×C411yescharacteristic
53C22×D107—21no—
54C22×D107—22no—
55C22×D107—22no—
56C22×D107—21no—
57C22×D107—42no—
58C22×D107—44no—
59C22×D107C22⋊C412yes—
60C22×D107C22⋊C414yes—
61Q8×C107—24no—
62Q8×C107—24no—
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107C22⋊C414yes—
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D412yes—
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×Dic107—21no—
71C2×Dic107—21no—
72C2×Dic107C22⋊C412yes—
73D428—216no—
74D428—24no—
75D428—24no—
76C4×D107—216no—
77C4×D107—24no—
78C4×D107—24no—
79C8×C107—42no—
80C107⋊C8—42no—
81C22×C107—22no—
82C22×C107C2×C22⋊C411yescharacteristic
83C22×C107C2×C22⋊C411yescharacteristic
84C22×C107C2×C22⋊C411yescharacteristic
85C2×D107—22no—
86C2×D107—24no—
87C2×D107—24no—
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—44no—
91C2×D107—44no—
92C2×D107—44no—
93C4×C107—24no—
94C4×C107#12012yes—
95C4×C107#12012yes—
96Dic107—22no—
97C2×C107—21no—
98C2×C107—21no—
99C2×C107#37911yescharacteristic
100C2×C107#41312yes—
101D107—42no—
102D107—44no—
103#1785—1071nomaximal, Sylow
104C107#178511yescharacteristic, complemented, Sylow
105C22×#41—1071no—
106C2×#101—1071no—
107C2×#101D10711yescharacteristic, complemented
108C2×#100—1072no—
109#407—1072no—
110C2×#41—2144no—
111C2×#41—2144no—
112C2×#41—2142no—
113C2×#41—2142no—
114C22×Q8C2×D10711yescharacteristic
115C22×D4—1072no—
116C22×D4—1071no—
117C23×C4—1071no—
118C23×C4—1072no—
119C2×M16—1072no—
120C2×M16C2×D10712yes—
121#101—22no—
122#101—2142no—
123#100—2144no—
124C22⋊C8—2144no—
125C24—1072no—
126C24—1071no—
127#41—21416no—
128#41—21416no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8D42812yes—
132C2×Q8C22⋊D10712yes—
133C2×D4—1074no—
134C2×D4—2144no—
135C2×D4—2144no—
136C2×D4—2144no—
137C2×D4—2142no—
138C2×D4—2141no—
139C2×D4—2141no—
140C22×C4—1071no—
141C22×C4—1074no—
142C22×C4—2144no—
143C22×C4—2144no—
144C22×C4—2144no—
145C22×C4—2142no—
146C22×C4—2141no—
147C22×C4—2141no—
148C22×C4C4×D10712yes—
149C22×C4C22×D10711yescharacteristic
150M16—22no—
151M16—22no—
152M16—2142no—
153M16—2142no—
154C2×C8—22no—
155C2×C8—2142no—
156C23—1074no—
157C23—1072no—
158C23—4284no—
159C23—4282no—
160C23—2141no—
161C23—2142no—
162C23—2142no—
163C23—2141no—
164C23C2×C4×D10711yescharacteristic
165Q8—24no—
166Q8—24no—
167D4—21416no—
168D4—2144no—
169D4—2144no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—1072no—
173C2×C4—21416no—
174C2×C4—2141no—
175C2×C4—2141no—
176C2×C4—2144no—
177C2×C4—2144no—
178C2×C4(C2×D107)⋊C414yes—
179C2×C4D4×D10712yes—
180C2×C4D4×D10712yes—
181C2×C4C2×C22⋊D10711yescharacteristic
182C2×C4C2×D42811yescharacteristic
183C8—42no—
184C8—4282no—
185C22—22no—
186C22—4284no—
187C22—4282no—
188C22—4284no—
189C22—4282no—
190C22—4284no—
191C22—2144no—
192C22—2144no—
193C22—2142no—
194C22C22⋊C4×D10711yescharacteristic
195C22C22⋊C4×D10711yescharacteristic
196C22C2×(C2×D107)⋊C411yescharacteristic
197C4—24no—
198C4—2142no—
199C4#1681207712yes—
200C4#1681207912yes—
201C2—21no—
202C2—21no—
203C2—4284no—
204C2—4282no—
205C2#2029248211yescharacteristic
206C2#2029261012yes—
subgroupG/H
centerC22C22⋊C4×D107
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#101×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC22×#13133819C2
cosocleC22×C4×C107C23
hypercenterC2×#101D107
hypocenterC107#1785
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16812028C2semidirect
3. C2×#16812025C2semidirect
4. #20292590C2semidirect
5. C2×#101×C107C2semidirect
104. C107#1785semidirect
107. C2×#101D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#168120282
C2×#168120252
#202925902
C2×#101×C1072
#1785107