Finite Group Databasebeta

← #28345987

Subgroups — #28345987

Order 13696.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812139C211yesmaximal, characteristic
2C22×#13133808C211yesmaximal, characteristic, complemented
3#20293104C211yesmaximal, characteristic, complemented
4#20292846C214yesmaximal, complemented
5#20292834C214yesmaximal, complemented
6#20292654C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292623C211yesmaximal, characteristic
9#536×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11#16812139—24no—
12C2×#13133808—24no—
13C2×#13133808C2214yes—
14C2×#13133808C2212yes—
15C2×#13133806C2212yes—
16C2×#13133802C2211yescharacteristic
17C22×C107⋊C8C2211yescharacteristic
18C4×Dic214C2214yes—
19#16812074—22no—
20#16812068C2211yescharacteristic
21#16812065—22no—
22C4⋊(C107⋊C8)C2214yes—
23#16812033C2212yes—
24#16812033C2214yes—
25#16812030C2212yes—
26C2×#41×C107C2211yescharacteristic
27C22×Q8×C107C2211yescharacteristic
28C4⋊Q8×C107C2212yes—
29#124×C107C2212yes—
30#123×C107—22no—
31C22⋊Q8×C107—22no—
32#117×C107C2211yescharacteristic
33C2×Dic214—21no—
34C2×Dic214—22no—
35C2×Dic214D414yes—
36C2×Dic214C2312yes—
37C22×Dic107—21no—
38#13133808—28no—
39#13133808—24no—
40#13133806—24no—
41#13133802—22no—
42C2×C107⋊C8—22no—
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—24no—
47C22⋊Dic107—21no—
48C4×Dic107—24no—
49C4×Dic107—21no—
50#41×C107—22no—
51#41×C107—22no—
52C2×Q8×C107—21no—
53C2×Q8×C107—22no—
54C2×Q8×C107—22no—
55C2×Q8×C107C2312yes—
56C2×Q8×C107C2311yescharacteristic
57C2×D4×C107—21no—
58C2×D4×C107C2311yescharacteristic
59C22×C4×C107—21no—
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62C4⋊C4×C107—24no—
63C4⋊C4×C107D412yes—
64C22⋊C4×C107—24no—
65C22⋊C4×C107D412yes—
66C42×C107C2312yes—
67C23×C107—21no—
68C23×C107C2×D411yescharacteristic
69Q8×C107—22no—
70Q8×C107—21no—
71Q8×C107—22no—
72Q8×C107—21no—
73Q8×C107—42no—
74D4×C107—21no—
75D4×C107—21no—
76D4×C107—42no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—42no—
80C2×C4×C107—42no—
81C2×C4×C107C2×D412yes—
82C2×C4×C107C2×D414yes—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107#4112yes—
85C2×C4×C107C2411yescharacteristic
86C2×Dic107—24no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89Dic214—24no—
90Dic214—22no—
91Dic214C2×D414yes—
92C107⋊C8—22no—
93C107⋊C8—22no—
94C22×C107—22no—
95C22×C107—41no—
96C22×C107—42no—
97C22×C107#12112yes—
98C22×C107C22×D411yescharacteristic
99C4×C107—24no—
100C4×C107—41no—
101C4×C107—42no—
102C4×C107#12112yes—
103C4×C107C22×D411yescharacteristic
104C4×C107C2×#4111yescharacteristic
105Dic107—24no—
106Dic107—42no—
107C2×C107—22no—
108C2×C107—81no—
109C2×C107C2×#12111yescharacteristic
110C2×C107#57812yes—
111#3044—1071nomaximal, Sylow
112C107#304411yescharacteristic, complemented, Sylow
113C2×#137—1071no—
114C22×Q16—1071no—
115#536D10711yescharacteristic
116#519—1071no—
117#464—1074no—
118#462—1074no—
119#424—1071no—
120#419—1071no—
121#416—1071no—
122C2×#41C2×D10711yescharacteristic
123C22×Q8—1071no—
124C22×Q8C2×D10711yescharacteristic
125#137—2144no—
126C2×Q16—1074no—
127C2×Q16—1072no—
128C2×Q16—2144no—
129C2×SD16—1072no—
130C2×M16—1071no—
131C22×C8—1071no—
132C4⋊Q8C2×D10712yes—
133#124C2×D10712yes—
134#123—22no—
135C22⋊Q8—22no—
136C4×Q8—1074no—
137C4×Q8—2142no—
138#117—1071no—
139#117—2142no—
140#117C2×D10711yescharacteristic
141C4⋊C8—1074no—
142#103—1072no—
143#103—1074no—
144#102—1072no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—1074no—
151C2×Q8—1072no—
152C2×Q8—2141no—
153C2×Q8—2142no—
154C2×Q8C22×D10712yes—
155C2×Q8C22×D10711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D10711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—2148no—
163Q16—2144no—
164SD16—2144no—
165M16—2142no—
166C2×C8—1072no—
167C2×C8—1072no—
168C2×C8—2142no—
169C4⋊C4—24no—
170C4⋊C4—1072no—
171C4⋊C4—2144no—
172C4⋊C4C22⋊D10712yes—
173C22⋊C4—24no—
174C22⋊C4—2141no—
175C22⋊C4C22⋊D10712yes—
176C42—2141no—
177C42—2144no—
178C42C22×D10712yes—
179C23—21no—
180C23C2×C22⋊D10711yescharacteristic
181Q8—22no—
182Q8—21no—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—1074no—
187Q8—2144no—
188Q8—2142no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2142no—
197C2×C4—2144no—
198C2×C4—2142no—
199C2×C4D4×D10712yes—
200C2×C4C2×C22⋊D10714yes—
201C2×C4C2×C22⋊D10711yescharacteristic
202C2×C4#1313381712yes—
203C2×C4C23×D10711yescharacteristic
204C8—2142no—
205C8—2142no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1681208012yes—
210C22C22×C22⋊D10711yescharacteristic
211C4—24no—
212C4—41no—
213C4—42no—
214C4—4282no—
215C4—2144no—
216C4#1681208012yes—
217C4C2×D4×D10711yescharacteristic
218C4C2×#1313381711yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1681208011yescharacteristic
222C2#2029344212yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×C107C22
hypercenter#536D107
hypocenterC107#3044
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#13133808C2semidirect
3. #20293104C2semidirect
4. #20292846C2semidirect
5. #20292834C2semidirect
6. #20292654C2semidirect
7. #20292636C2semidirect
112. C107#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C22×#131338082
#202931042
#202928462
#202928342
#202926542
#202926362
#202926232
#536×C1072
#3044107