Finite Group Databasebeta

← #28345994

Subgroups — #28345994

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×#16812139C211yesmaximal, characteristic
2C22×Dic428C211yesmaximal, characteristic, complemented
3#20293198C211yesmaximal, characteristic
4#20293104C211yesmaximal, characteristic, complemented
5#20292848C214yesmaximal, complemented
6#20292836C214yesmaximal, complemented
7#20292653C211yesmaximal, characteristic, complemented
8#20292635C211yesmaximal, characteristic, complemented
9#416×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12#16812139—24no—
13C2×Dic428—24no—
14C2×Dic428C2214yes—
15C2×#13133808C2212yes—
16C2×#13133806C2212yes—
17C2×#13133802C2211yescharacteristic
18#16812107C2212yes—
19#16812096C2212yes—
20#16812090—22no—
21#16812088—22no—
22C4×Dic214—22no—
23#16812074C2214yes—
24#16812068C2211yescharacteristic
25#16812065—22no—
26Dic107⋊C8C2214yes—
27#16812032C2214yes—
28C2×#41×C107C2211yescharacteristic
29C22×C8×C107C2211yescharacteristic
30#117×C107C2211yescharacteristic
31#103×C107C2212yes—
32#102×C107C2212yes—
33C2×Dic214—21no—
34C2×Dic214—22no—
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214—21no—
38C2×Dic214D414yes—
39C2×Dic214C2312yes—
40C2×Dic214C2312yes—
41C22×Dic107—21no—
42C22×Dic107—21no—
43Dic428—28no—
44#13133808—24no—
45#13133806—24no—
46#13133802—22no—
47C2×C107⋊C8C2312yes—
48C4⋊Dic107D412yes—
49Dic107⋊C4—24no—
50Dic107⋊C4—24no—
51C22⋊Dic107—24no—
52C22⋊Dic107D412yes—
53C4×Dic107—24no—
54C4×Dic107C2312yes—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107C2311yescharacteristic
58C2×D4×C107—21no—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62C2×C8×C107—22no—
63C2×C8×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C22⋊C4×C107—21no—
66C42×C107—21no—
67C23×C107—21no—
68C23×C107C2×D411yescharacteristic
69Q8×C107—21no—
70Q8×C107—21no—
71D4×C107—21no—
72D4×C107—21no—
73D4×C107—42no—
74C2×C4×C107—22no—
75C2×C4×C107—21no—
76C2×C4×C107—42no—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D412yes—
79C2×C4×C107#4112yes—
80C2×C4×C107C2411yescharacteristic
81C2×Dic107—24no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—42no—
85C2×Dic107C2×D414yes—
86Dic214—24no—
87Dic214—22no—
88Dic214—22no—
89Dic214—22no—
90Dic214—42no—
91Dic214C2×D414yes—
92C8×C107—22no—
93C107⋊C8—22no—
94C22×C107—22no—
95C22×C107—41no—
96C22×C107—42no—
97C22×C107#12112yes—
98C22×C107C22×D411yescharacteristic
99C4×C107—41no—
100C4×C107—42no—
101C4×C107#12112yes—
102C4×C107C22×D411yescharacteristic
103C4×C107C2×#4111yescharacteristic
104Dic107—24no—
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#536—1071no—
116#519—1071no—
117#464—1074no—
118#462—1074no—
119#424—1071no—
120#419—1071no—
121#416D10711yescharacteristic
122C2×#41C2×D10711yescharacteristic
123C22×Q8—1071no—
124C22×Q8—1071no—
125#137—2144no—
126C2×Q16—1072no—
127C2×Q16—1074no—
128C2×Q16—2144no—
129C2×SD16—1072no—
130C2×M16—1071no—
131C22×C8C2×D10711yescharacteristic
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—1074no—
143#103C2×D10712yes—
144#102C2×D10712yes—
145#41—22no—
146#41—22no—
147C2×Q8—1074no—
148C2×Q8—1072no—
149C2×Q8—1072no—
150C2×Q8—2142no—
151C2×Q8—2142no—
152C2×Q8—2141no—
153C2×Q8—2141no—
154C2×Q8—2142no—
155C2×Q8C22×D10711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D10711yescharacteristic
158C22×C4—21no—
159C22×C4—2141no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—2148no—
163Q16—2144no—
164SD16—2144no—
165M16—2142no—
166C2×C8—22no—
167C2×C8—1072no—
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—21no—
180C23D4×D10711yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—1074no—
184Q8—4282no—
185Q8—2142no—
186Q8—2142no—
187Q8—2144no—
188Q8—2142no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—1074no—
196C2×C4—4282no—
197C2×C4—2142no—
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—42no—
208C22—41no—
209C22#1681208212yes—
210C22C2×D4×D10711yescharacteristic
211C4—42no—
212C4—41no—
213C4—4282no—
214C4—2144no—
215C4—2144no—
216C4#1681208212yes—
217C4C2×D4×D10711yescharacteristic
218C4C2×#1313381811yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1681208211yescharacteristic
222C2#2029344112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×C107C22
hypercenter#416D107
hypocenterC107#3044
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic428C2semidirect
4. #20293104C2semidirect
5. #20292848C2semidirect
6. #20292836C2semidirect
7. #20292653C2semidirect
8. #20292635C2semidirect
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×Dic4282
#202931982
#202931042
#202928482
#202928362
#202926532
#202926352
#416×C1072
#3044107