Finite Group Databasebeta

← #28344423

Subgroups — #28344423

Order 13696.

392 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 149 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#20292560C212yesmaximal
2#20292559C212yesmaximal
3#20292547C212yesmaximal
4#20292533C211yesmaximal, characteristic
5#20292533C211yesmaximal, characteristic
6#20292531C212yesmaximal
7#20292526C212yesmaximal
8#20292513C212yesmaximal
9#389×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12C2×C4⋊Dic107C2212yes—
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C4⋊Dic107C2212yes—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×C4×Dic107C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C2×C4⋊C4×C107C2212yes—
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#95×C107C2212yes—
31C2×Dic214—22no—
32C2×Dic214—41no—
33C2×Dic214—41no—
34C2×Dic214—41no—
35C2×Dic214—41no—
36C2×Dic214Q814yescomplemented
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C4⋊Dic107—42no—
42C4⋊Dic107—42no—
43C4⋊Dic107—41no—
44C4⋊Dic107—41no—
45Dic107⋊C4—42no—
46Dic107⋊C4—42no—
47C4×Dic107—42no—
48C22×C4×C107C2312yes—
49C22×C4×C107C2312yes—
50C22×C4×C107C2311yescharacteristic
51C22×C4×C107C2311yescharacteristic
52C22×C4×C107C2311yescharacteristic
53C4⋊C4×C107—24no—
54C4⋊C4×C107—41no—
55C4⋊C4×C107Q814yescomplemented
56C42×C107—41no—
57C23×C107C2411yescharacteristic
58C2×C4×C107—22no—
59C2×C4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—21no—
62C2×C4×C107—22no—
63C2×C4×C107—21no—
64C2×C4×C107—21no—
65C2×C4×C107—21no—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107C2×Q812yes—
70C2×C4×C107C2×Q812yes—
71C2×C4×C107C2×Q814yes—
72C2×C4×C107#4112yes—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107C2×Q814yes—
85Dic214—44no—
86Dic214—44no—
87C22×C107C22×Q811yescharacteristic
88C22×C107C22×Q811yescharacteristic
89C22×C107C2×#4111yescharacteristic
90C22×C107H(5,𝔽2)11yescharacteristic
91C22×C107H(5,𝔽2)11yescharacteristic
92C22×C107H(5,𝔽2)11yescharacteristic
93C22×C1072−1+411yescharacteristic
94C4×C107—22no—
95C4×C107—22no—
96C4×C107—22no—
97C4×C107—24no—
98C4×C107—42no—
99Dic107—24no—
100Dic107—42no—
101Dic107—42no—
102Dic107—42no—
103C2×C107#53811yescharacteristic
104C2×C107#54311yescharacteristic
105C2×C107#54411yescharacteristic
106C2×C107#55411yescharacteristic
107C2×C107#55711yescharacteristic
108C2×C107Q8211yescharacteristic
109C2×C107#56211yescharacteristic
110#2806—1071nomaximal, Sylow
111C107#280611yescharacteristic, complemented, Sylow
112#398—1072no—
113#398—1072no—
114#395—1072no—
115#391—1071no—
116#391—1071no—
117#391—1072no—
118#389—1072no—
119#389D10711yescharacteristic
120#387—1072no—
121C22×Q8—1071no—
122C22×Q8—1071no—
123C2×C4⋊C4—1072no—
124C2×C4⋊C4—1072no—
125C2×C4⋊C4—1072no—
126C2×C4⋊C4—1071no—
127C2×C4⋊C4—1072no—
128C2×C4⋊C4—1071no—
129C2×C4⋊C4C2×D10712yes—
130C2×C4⋊C4C2×D10711yescharacteristic
131C2×C4⋊C4C2×D10711yescharacteristic
132C2×C42—1072no—
133C2×C42C2×D10711yescharacteristic
134#95—1072no—
135#95—1072no—
136#95—1072no—
137#95—1072no—
138#95—1072no—
139#95—1072no—
140#95—1072no—
141#95C2×D10712yes—
142C2×Q8—1074no—
143C2×Q8—4281no—
144C2×Q8—4281no—
145C2×Q8—4281no—
146C2×Q8—4281no—
147C2×Q8—2142no—
148C22×C4—1072no—
149C22×C4—1072no—
150C22×C4—1072no—
151C22×C4—1072no—
152C22×C4C22×D10712yes—
153C22×C4C22×D10711yescharacteristic
154C22×C4C22×D10711yescharacteristic
155C22×C4C22×D10712yes—
156C22×C4C22×D10711yescharacteristic
157C4⋊C4—24no—
158C4⋊C4—41no—
159C4⋊C4—4282no—
160C4⋊C4—4282no—
161C4⋊C4—4282no—
162C4⋊C4—4281no—
163C4⋊C4—4281no—
164C4⋊C4—4282no—
165C4⋊C4Dic21414yescomplemented
166C42—41no—
167C42—4282no—
168C23C23×D10711yescharacteristic
169Q8—4284no—
170Q8—4284no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—1074no—
183C2×C4—2142no—
184C2×C4—2142no—
185C2×C4—2142no—
186C2×C4—2142no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4—2142no—
192C2×C4—2142no—
193C2×C4—2142no—
194C2×C4Q8×D10712yes—
195C2×C4C2×Dic21412yes—
196C2×C4C2×Dic21414yes—
197C2×C4#1313381712yes—
198C22C2×Q8×D10711yescharacteristic
199C22C22×Dic21411yescharacteristic
200C22C2×#1313381711yescharacteristic
201C22#1681215411yescharacteristic
202C22#1681215511yescharacteristic
203C22#1681215511yescharacteristic
204C22#1681215711yescharacteristic
205C4—22no—
206C4—22no—
207C4—22no—
208C4—24no—
209C4—42no—
210C4—4282no—
211C4—4282no—
212C4—4282no—
213C4—2144no—
214C2#2029321411yescharacteristic
215C2#2029325211yescharacteristic
216C2#2029325811yescharacteristic
217C2#2029333411yescharacteristic
218C2#2029335211yescharacteristic
219C2Q8×Dic21411yescharacteristic
220C2#2029337511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, prime radical, cosocleC23×C107C24
Fitting#389×C107C2
hypercenter#389D107
hypocenterC107#2806
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
36. C2×Dic214Q8semidirect
55. C4⋊C4×C107Q8semidirect
111. C107#2806semidirect
165. C4⋊C4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202925602
#202925592
#202925472
#202925332
#202925312
#202925262
#202925132
#389×C1072
#2806107