Finite Group Databasebeta

← #28345990

Subgroups — #28345990

Order 13696.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16812142C211yesmaximal, characteristic
2C22×#13133808C211yesmaximal, characteristic, complemented
3#20293201C211yesmaximal, characteristic
4#20293104C211yesmaximal, characteristic, complemented
5#20292852C214yesmaximal, complemented
6#20292836C214yesmaximal, complemented
7#20292655C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic
9#419×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12#16812142—24no—
13C2×Dic428C2212yes—
14C2×#13133808—24no—
15C2×#13133808C2214yes—
16C2×#13133807C2212yes—
17C22×C107⋊C8C2211yescharacteristic
18#16812107C2212yes—
19#16812098C2212yes—
20#16812092—22no—
21#16812085—22no—
22C4×Dic214—22no—
23#16812074C2214yes—
24#16812068C2211yescharacteristic
25#16812065—22no—
26Dic107⋊C8C2214yes—
27#16812032C2212yes—
28#16812029C2212yes—
29C22×Q8×C107C2211yescharacteristic
30C2×M16×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32#103×C107C2214yes—
33#13133818—22no—
34#13133818—22no—
35C2×Dic214—21no—
36C2×Dic214—22no—
37C2×Dic214D414yes—
38C2×Dic214C2311yescharacteristic
39C2×Dic214C2312yes—
40C2×D428C2311yescharacteristic
41C2×C22⋊D107—21no—
42C22×Dic107—21no—
43C2×C4×D107—21no—
44Dic428—24no—
45#13133808—28no—
46#13133807—24no—
47C2×C107⋊C8—22no—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107D412yes—
50Dic107⋊C4—24no—
51Dic107⋊C4—24no—
52C22⋊Dic107D412yes—
53(C2×D107)⋊C4—24no—
54C4×Dic107—24no—
55C4×Dic107C2312yes—
56C2×Q8×C107—21no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107C2312yes—
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—21no—
66C42×C107—21no—
67C23×C107C2×D411yescharacteristic
68C22×D107—21no—
69Q8×C107—22no—
70Q8×C107—22no—
71Q8×C107—42no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—42no—
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D412yes—
77C2×C4×C107#4112yes—
78C2×C4×C107C2411yescharacteristic
79C22⋊D107—42no—
80C2×Dic107—24no—
81C2×Dic107—22no—
82C2×Dic107—21no—
83C2×Dic107C2×D414yes—
84D428—21no—
85D428—21no—
86C4×D107—42no—
87Dic214—24no—
88Dic214—22no—
89Dic214—21no—
90Dic214—21no—
91Dic214C2×D414yes—
92C8×C107—22no—
93C107⋊C8—22no—
94C22×C107—22no—
95C22×C107#12112yes—
96C22×C107C22×D411yescharacteristic
97C2×D107—41no—
98C2×D107—42no—
99C4×C107—42no—
100C4×C107—42no—
101C4×C107#12112yes—
102C4×C107C22×D411yescharacteristic
103C4×C107C2×#4111yescharacteristic
104Dic107—24no—
105Dic107—24no—
106Dic107—41no—
107C2×C107—22no—
108C2×C107C2×#12111yescharacteristic
109C2×C107#57812yes—
110D107—81no—
111#3044—1071nomaximal, Sylow
112C107#304411yescharacteristic, complemented, Sylow
113C2×#137—1071no—
114C22×Q16—1071no—
115#536—1071no—
116#519—1071no—
117#464—1074no—
118#462—1074no—
119#424—1071no—
120#419D10711yescharacteristic, complemented
121#416—1071no—
122C2×#41—1071no—
123C22×Q8—1071no—
124C22×Q8C2×D10711yescharacteristic
125#137—2144no—
126C2×Q16—1074no—
127C2×Q16—1072no—
128C2×Q16—2144no—
129C2×SD16—1072no—
130C2×M16C2×D10711yescharacteristic
131C22×C8—1071no—
132C4⋊Q8—1072no—
133#124—1072no—
134#123—2142no—
135C22⋊Q8—2142no—
136C4×Q8—1074no—
137C4×Q8—2142no—
138#117—1071no—
139#117—2142no—
140#117C2×D10711yescharacteristic
141C4⋊C8—1074no—
142#103—1072no—
143#103C2×D10714yes—
144#102—1072no—
145#41—2142no—
146#41—2142no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—1074no—
151C2×Q8—1072no—
152C2×Q8—1071no—
153C2×Q8—2141no—
154C2×Q8—2142no—
155C2×Q8C22×D10712yes—
156C2×D4—1071no—
157C2×D4—2141no—
158C22×C4—21no—
159C22×C4—2141no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—2144no—
163Q16—2148no—
164SD16—2144no—
165M16—22no—
166C2×C8—1072no—
167C2×C8—2142no—
168C2×C8C22×D10712yes—
169C4⋊C4—1072no—
170C4⋊C4—2144no—
171C4⋊C4—2144no—
172C4⋊C4C22×D10712yes—
173C22⋊C4—21no—
174C22⋊C4—1072no—
175C22⋊C4—2144no—
176C42—21no—
177C42—1072no—
178C42—2144no—
179C23—2141no—
180C23D4×D10711yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—1074no—
185Q8—2141no—
186Q8—2141no—
187Q8—2144no—
188Q8—2142no—
189D4—4282no—
190D4—2141no—
191D4—2141no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—1074no—
196C2×C4—4282no—
197C2×C4—2141no—
198C2×C4—2144no—
199C2×C4—2142no—
200C2×C4D4×D10711yescharacteristic
201C2×C4D4×D10712yes—
202C2×C4#1313381812yes—
203C2×C4C23×D10711yescharacteristic
204C8—22no—
205C8—2142no—
206C22—22no—
207C22—4282no—
208C22—4281no—
209C22#1681208212yes—
210C22C2×D4×D10711yescharacteristic
211C4—42no—
212C4—42no—
213C4—4281no—
214C4—2144no—
215C4—2144no—
216C4#1681208212yes—
217C4C2×D4×D10711yescharacteristic
218C4C2×#1313381811yescharacteristic
219C2—22no—
220C2—8561no—
221C2C2×#1681208211yescharacteristic
222C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#419D107
hypocenterC107#3044
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
2. C22×#13133808C2semidirect
4. #20293104C2semidirect
5. #20292852C2semidirect
6. #20292836C2semidirect
7. #20292655C2semidirect
9. #419×C107C2semidirect
112. C107#3044semidirect
120. #419D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C22×#131338082
#202932012
#202931042
#202928522
#202928362
#202926552
#202926212
#419×C1072
#3044107