Finite Group Databasebeta

← #28346301

Subgroups — #28346301

Order 13696.

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

normal subgroups only

This group has 362 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
1#20293258C211yesmaximal, characteristic
2#20293104C211yesmaximal, characteristic
3#20292937C212yesmaximal, complemented
4#20292933C212yesmaximal, complemented
5#20292921C212yesmaximal, complemented
6#20292911C212yesmaximal, complemented
7#20292663C211yesmaximal, characteristic
8#20292655C211yesmaximal, characteristic
9#20292637C211yesmaximal, characteristic
10C2×#16812032C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic
12C22×Dic214C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812108—21no—
15#16812108C2212yes—
16#16812100—21no—
17#16812100C2212yes—
18#16812088—22no—
19C4×Dic214—21no—
20C4×Dic214—21no—
21#16812074C2212yes—
22#16812074C2212yes—
23#16812068C2211yescharacteristic
24#16812068C2211yescharacteristic
25#16812065—22no—
26#16812042C2211yescharacteristic
27#16812042C2211yescharacteristic
28#16812040C2211yescharacteristic
29#16812040C2211yescharacteristic
30Dic107⋊C8C2212yes—
31Dic107⋊C8C2212yes—
32#16812034C2212yes—
33#16812034C2212yes—
34#16812032C2212yes—
35#16812032C2212yes—
36C2×M16×C107C2211yescharacteristic
37#117×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#107×C107C2212yes—
40#106×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214—22no—
44C2×Dic214Q814yescomplemented
45C2×Dic214C2312yes—
46C22×Dic107—21no—
47C2×C107⋊C8—22no—
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2311yescharacteristic
50C4⋊Dic107—22no—
51C4⋊Dic107D411yescharacteristic
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—21no—
60C22⋊Dic107D412yes—
61C4×Dic107—21no—
62C4×Dic107—22no—
63C4×Dic107—22no—
64C4×Dic107C2312yes—
65C22×C4×C107—21no—
66C22×C4×C107C2311yescharacteristic
67M16×C107—22no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2312yes—
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C42×C107—21no—
75C23×C107C2×D411yescharacteristic
76C2×C4×C107—21no—
77C2×C4×C107—21no—
78C2×C4×C107—22no—
79C2×C4×C107—81no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×Q812yes—
82C2×C4×C107#4112yes—
83C2×C4×C107C2411yescharacteristic
84C2×Dic107—24no—
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107C2×D412yes—
89C2×Dic107C2×D412yes—
90Dic214—24no—
91Dic214—22no—
92Dic214—82no—
93Dic214C2×Q814yes—
94C8×C107—22no—
95C107⋊C8—21no—
96C107⋊C8—21no—
97C22×C107—21no—
98C22×C107—21no—
99C22×C107C22⋊Q812yes—
100C22×C107C22×D411yescharacteristic
101C4×C107—42no—
102C4×C107—41no—
103C4×C107—41no—
104C4×C107C22⋊Q812yes—
105C4×C107C22×Q811yescharacteristic
106C4×C107C2×#4111yescharacteristic
107Dic107—24no—
108Dic107—22no—
109Dic107—22no—
110Dic107—41no—
111Dic107—41no—
112C2×C107—22no—
113C2×C107C2×C22⋊Q811yescharacteristic
114C2×C107#57711yescharacteristic
115C2×C107#57811yescharacteristic
116#3082—1071nomaximal, Sylow
117C107#308211yescharacteristic, complemented, Sylow
118#544—1071no—
119#519—1071no—
120#479—1072no—
121#479—1072no—
122#477—1072no—
123#475—1072no—
124#428D10711yescharacteristic
125#427—1071no—
126#424—1071no—
127#419—1071no—
128C2×#103—1071no—
129C22×Q8—1071no—
130C2×M16C2×D10711yescharacteristic
131C22×C8—1071no—
132C4⋊Q8—1072no—
133C4⋊Q8—2141no—
134#125—1072no—
135#125—2141no—
136C22⋊Q8—2142no—
137C4×Q8—1072no—
138C4×Q8—1072no—
139C4×Q8—2141no—
140C4×Q8—2141no—
141#117—1071no—
142#117—1071no—
143#117—2142no—
144#117C2×D10711yescharacteristic
145C2×C4⋊C4C2×D10711yescharacteristic
146#107—1071no—
147#107—1071no—
148#107C2×D10712yes—
149#106—1071no—
150#106—1071no—
151#106C2×D10712yes—
152C4⋊C8—1072no—
153C4⋊C8—1072no—
154#103—1072no—
155#103—1072no—
156#103—1072no—
157#103—1072no—
158C2×Q8—1074no—
159C2×Q8—1072no—
160C2×Q8—2142no—
161C2×Q8—2141no—
162C2×Q8—2142no—
163C22×C4—21no—
164C22×C4—2141no—
165C22×C4C22×D10711yescharacteristic
166M16—22no—
167C2×C8—1071no—
168C2×C8—1071no—
169C2×C8—2142no—
170C2×C8C22×D10712yes—
171C4⋊C4—21no—
172C4⋊C4—1071no—
173C4⋊C4—1071no—
174C4⋊C4—1071no—
175C4⋊C4—1071no—
176C4⋊C4—2142no—
177C4⋊C4—2142no—
178C4⋊C4—2142no—
179C4⋊C4—2142no—
180C4⋊C4—2142no—
181C4⋊C4C22×D10712yes—
182C4⋊C4C22×D10711yescharacteristic
183C4⋊C4C22×D10711yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—1072no—
186C22⋊C4—2141no—
187C42—21no—
188C42—1072no—
189C42—2142no—
190C42—2142no—
191C42—2141no—
192C23D4×D10711yescharacteristic
193Q8—8562no—
194Q8—1074no—
195Q8—2144no—
196Q8—2142no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—81no—
201C2×C4—1072no—
202C2×C4—1072no—
203C2×C4—2141no—
204C2×C4—2141no—
205C2×C4—2144no—
206C2×C4—2142no—
207C2×C4D4×D10711yescharacteristic
208C2×C4Q8×D10712yes—
209C2×C4#1313381812yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2141no—
213C8—2141no—
214C22—21no—
215C22—21no—
216C22#1681208712yes—
217C22C2×D4×D10711yescharacteristic
218C4—42no—
219C4—41no—
220C4—41no—
221C4—4281no—
222C4—4281no—
223C4—2144no—
224C4—2142no—
225C4—2142no—
226C4#1681208712yes—
227C4C2×Q8×D10711yescharacteristic
228C4C2×#1313381811yescharacteristic
229C2—22no—
230C2C2×#1681208711yescharacteristic
231C2#2029343711yescharacteristic
232C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#428D107
hypocenterC107#3082
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20292937C2semidirect
4. #20292933C2semidirect
5. #20292921C2semidirect
6. #20292911C2semidirect
44. C2×Dic214Q8semidirect
117. C107#3082semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202931042
#202929372
#202929332
#202929212
#202929112
#202926632
#202926552
#202926372
C2×#168120322
#428×C1072
#3082107