Finite Group Databasebeta

← #28346839

Subgroups — #28346839

Order 13696.

380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293180C211yesmaximal, characteristic
2C2×#16812107C211yesmaximal, characteristic
3#20292994C214yesmaximal, complemented
4#20292976C214yesmaximal, complemented
5#20292637C211yesmaximal, characteristic
6#20292636C211yesmaximal, characteristic
7#20292635C211yesmaximal, characteristic
8M16×Dic107C211yesmaximal, characteristic
9#419×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C2×#13133802C2211yescharacteristic
12#16812108C2212yes—
13#16812107—22no—
14#16812107C2212yes—
15#16812107C2212yes—
16#16812100C2212yes—
17#16812086—22no—
18C4×Dic214—22no—
19#16812068C2211yescharacteristic
20C2×Dic107⋊C4—22no—
21C2×C4×Dic107C2211yescharacteristic
22#16812034C2214yes—
23#16812033C2214yes—
24#16812032C2214yes—
25#16812019C2212yes—
26C8×Dic107C2212yes—
27C22×Q8×C107C2211yescharacteristic
28C2×M16×C107C2211yescharacteristic
29#117×C107C2211yescharacteristic
30#103×C107C2214yes—
31C2×Dic214—22no—
32C2×Dic214—22no—
33C2×Dic214—21no—
34C2×Dic214—22no—
35C2×Dic214C2312yes—
36C22×Dic107—21no—
37C22×Dic107D412yes—
38#13133802—22no—
39C2×C107⋊C8C2312yes—
40C4⋊Dic107—22no—
41C4⋊Dic107C2312yes—
42Dic107⋊C4—24no—
43Dic107⋊C4—24no—
44Dic107⋊C4—22no—
45Dic107⋊C4—22no—
46C22⋊Dic107—21no—
47C4×Dic107—21no—
48C4×Dic107D412yes—
49C4×Dic107C2311yescharacteristic
50C4×Dic107C2311yescharacteristic
51C2×Q8×C107—21no—
52C2×Q8×C107—22no—
53C2×Q8×C107—22no—
54C2×Q8×C107C2312yes—
55C22×C4×C107—21no—
56C22×C4×C107C2311yescharacteristic
57M16×C107—22no—
58C2×C8×C107C2312yes—
59C4⋊C4×C107C2312yes—
60C22⋊C4×C107—21no—
61C42×C107—21no—
62C23×C107C2×D411yescharacteristic
63Q8×C107—22no—
64Q8×C107—22no—
65Q8×C107—42no—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107—42no—
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107#4112yes—
71C2×C4×C107#4112yes—
72C2×C4×C107C2411yescharacteristic
73C2×Dic107—24no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—42no—
77C2×Dic107C2×D412yes—
78C2×Dic107C2×D412yes—
79Dic214—22no—
80Dic214—22no—
81Dic214—42no—
82Dic214—44no—
83C8×C107—22no—
84C107⋊C8—22no—
85C22×C107—22no—
86C22×C107#12412yes—
87C22×C107C22×D411yescharacteristic
88C4×C107—42no—
89C4×C107—42no—
90C4×C107#12412yes—
91C4×C107C2×#4111yescharacteristic
92C4×C107C2×#4111yescharacteristic
93Dic107—22no—
94Dic107—42no—
95Dic107—42no—
96Dic107#13714yes—
97C2×C107—22no—
98C2×C107C2×#12411yescharacteristic
99C2×C107C2×#13712yes—
100#3139—1071nomaximal, Sylow
101C107#313911yescharacteristic, complemented, Sylow
102#533—1071no—
103C2×C4⋊Q8—1071no—
104#491—1074no—
105#487—1074no—
106#419—1071no—
107#419—1071no—
108#419—1071no—
109#419D10711yescharacteristic
110C4×M16—1071no—
111C22×Q8—1071no—
112C22×Q8C2×D10711yescharacteristic
113C2×M16—1071no—
114C2×M16C2×D10711yescharacteristic
115C4⋊Q8—1072no—
116C4⋊Q8—1072no—
117C4⋊Q8—1072no—
118C4⋊Q8—2142no—
119#125—1072no—
120C22⋊Q8—2142no—
121C4×Q8—2142no—
122#117—1071no—
123#117C2×D10711yescharacteristic
124C2×C4⋊C4—2142no—
125C2×C42—1071no—
126#103—1074no—
127#103—1074no—
128#103—1074no—
129#103C2×D10714yes—
130#97—1072no—
131C4×C8—1072no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—1072no—
136C2×Q8—2142no—
137C2×Q8—2142no—
138C2×Q8—2142no—
139C2×Q8—2141no—
140C2×Q8C22×D10712yes—
141C22×C4—21no—
142C22×C4—1072no—
143C22×C4—2141no—
144C22×C4C22×D10711yescharacteristic
145M16—22no—
146M16—2142no—
147C2×C8—1072no—
148C2×C8C22×D10712yes—
149C4⋊C4—1072no—
150C4⋊C4—2144no—
151C4⋊C4—2144no—
152C4⋊C4—2142no—
153C4⋊C4—2142no—
154C4⋊C4—2142no—
155C4⋊C4C22×D10712yes—
156C22⋊C4—21no—
157C22⋊C4—2141no—
158C42—21no—
159C42—1072no—
160C42—1071no—
161C42—1071no—
162C42—2141no—
163C23D4×D10711yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—4284no—
168Q8—4282no—
169Q8—2142no—
170Q8—2142no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—1072no—
175C2×C4—1072no—
176C2×C4—4282no—
177C2×C4—2144no—
178C2×C4—2142no—
179C2×C4—2142no—
180C2×C4D4×D10711yescharacteristic
181C2×C4#1313381712yes—
182C2×C4#1313381812yes—
183C2×C4C23×D10711yescharacteristic
184C8—22no—
185C8—2142no—
186C22—22no—
187C22#1681209512yes—
188C22C2×D4×D10711yescharacteristic
189C4—42no—
190C4—42no—
191C4—1074no—
192C4—4282no—
193C4—4282no—
194C4—2142no—
195C4#1681209512yes—
196C4C2×#1313381711yescharacteristic
197C4C2×#1313381811yescharacteristic
198C2—22no—
199C2C2×#1681209511yescharacteristic
200C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#419D107
hypocenterC107#3139
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20292994C2semidirect
4. #20292976C2semidirect
101. C107#3139semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×#168121072
#202929942
#202929762
#202926372
#202926362
#202926352
M16×Dic1072
#419×C1072
#3139107