Finite Group Databasebeta

← #28346299

Subgroups — #28346299

Order 13696.

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

normal subgroups only

This group has 362 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
1#20293258C211yesmaximal, characteristic
2#20293105C211yesmaximal, characteristic
3#20292936C212yesmaximal, complemented
4#20292932C212yesmaximal, complemented
5#20292920C212yesmaximal, complemented
6#20292910C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic
8#20292665C211yesmaximal, characteristic
9#20292653C211yesmaximal, characteristic
10#20292636C211yesmaximal, characteristic
11C2×#103×C107C211yesmaximal, characteristic
12C2×#13133802C2211yescharacteristic
13#16812108—21no—
14#16812108C2212yes—
15#16812100—21no—
16#16812100C2212yes—
17#16812088—22no—
18Q8×Dic107—21no—
19Q8×Dic107—21no—
20Q8×Dic107C2212yes—
21Q8×Dic107C2212yes—
22#16812068—22no—
23#16812068C2211yescharacteristic
24#16812068C2211yescharacteristic
25#16812068C2211yescharacteristic
26#16812043C2211yescharacteristic
27#16812043C2211yescharacteristic
28#16812042C2212yes—
29#16812041C2211yescharacteristic
30#16812041C2211yescharacteristic
31#16812040C2212yes—
32Dic107⋊C8C2212yes—
33Dic107⋊C8C2212yes—
34#16812033C2212yes—
35#16812033C2212yes—
36C22×Q8×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#103×C107C2212yes—
40#103×C107C2212yes—
41C2×Dic214—22no—
42#13133802—22no—
43C2×C107⋊C8C2312yes—
44C4⋊Dic107—22no—
45C4⋊Dic107—22no—
46C4⋊Dic107—22no—
47C4⋊Dic107D411yescharacteristic
48C4⋊Dic107D411yescharacteristic
49C4⋊Dic107C2311yescharacteristic
50C4⋊Dic107C2311yescharacteristic
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55C22⋊Dic107—21no—
56C22⋊Dic107—21no—
57C22⋊Dic107D412yes—
58C4×Dic107—21no—
59C4×Dic107—21no—
60C4×Dic107—22no—
61C4×Dic107—22no—
62C4×Dic107C2312yes—
63C2×Q8×C107—21no—
64C2×Q8×C107—22no—
65C2×Q8×C107Q814yescomplemented
66C2×Q8×C107C2312yes—
67C22×C4×C107—21no—
68C22×C4×C107—21no—
69C22×C4×C107C2311yescharacteristic
70C2×C8×C107—22no—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107—21no—
74C4⋊C4×C107C2312yes—
75C23×C107C2×D411yescharacteristic
76Q8×C107—24no—
77Q8×C107—22no—
78Q8×C107C2×Q814yes—
79C2×C4×C107—24no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—81no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×Q812yes—
85C2×C4×C107#4112yes—
86C2×C4×C107C2411yescharacteristic
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107C2×D412yes—
92C2×Dic107C2×D412yes—
93Dic214—82no—
94C8×C107—21no—
95C8×C107—21no—
96C107⋊C8—22no—
97C22×C107—21no—
98C22×C107—21no—
99C22×C107C22⋊Q812yes—
100C22×C107C22×D411yescharacteristic
101C4×C107—24no—
102C4×C107—42no—
103C4×C107C22⋊Q812yes—
104C4×C107C22×Q811yescharacteristic
105C4×C107C2×#4111yescharacteristic
106Dic107—22no—
107Dic107—22no—
108Dic107—41no—
109Dic107—41no—
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#428—1071no—
125#427—1071no—
126#424—1071no—
127#419—1071no—
128C2×#103D10711yescharacteristic
129C22×Q8C2×D10711yescharacteristic
130C2×M16—1071no—
131C22×C8C2×D10711yescharacteristic
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—1071no—
144#117—2142no—
145C2×C4⋊C4C2×D10711yescharacteristic
146#107—1071no—
147#107—1071no—
148#107—1072no—
149#106—1071no—
150#106—1071no—
151#106—1072no—
152C4⋊C8—1072no—
153C4⋊C8—1072no—
154#103—1072no—
155#103—1072no—
156#103C2×D10712yes—
157#103C2×D10712yes—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×Q8—2142no—
161C2×Q8Dic21414yescomplemented
162C2×Q8C22×D10712yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D10711yescharacteristic
166M16—2142no—
167C2×C8—22no—
168C2×C8—1072no—
169C2×C8C22×D10711yescharacteristic
170C2×C8C22×D10711yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—1071no—
173C4⋊C4—1071no—
174C4⋊C4—1071no—
175C4⋊C4—1071no—
176C4⋊C4—1071no—
177C4⋊C4—1071no—
178C4⋊C4—2142no—
179C4⋊C4—2142no—
180C4⋊C4—2142no—
181C4⋊C4—2142no—
182C4⋊C4—2142no—
183C4⋊C4C22×D10712yes—
184C22⋊C4—1072no—
185C22⋊C4—2141no—
186C22⋊C4—2141no—
187C42—1072no—
188C42—2142no—
189C42—2142no—
190C42—2141no—
191C42—2141no—
192C23D4×D10711yescharacteristic
193Q8—24no—
194Q8—22no—
195Q8—8562no—
196Q8C2×Dic21414yes—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—81no—
201C2×C4—1072no—
202C2×C4—1072no—
203C2×C4—2141no—
204C2×C4—2141no—
205C2×C4—2141no—
206C2×C4—2141no—
207C2×C4D4×D10711yescharacteristic
208C2×C4C2×Dic21412yes—
209C2×C4#1313381712yes—
210C2×C4C23×D10711yescharacteristic
211C8—21no—
212C8—21no—
213C8—2142no—
214C22—21no—
215C22—21no—
216C22#1681208612yes—
217C22C2×D4×D10711yescharacteristic
218C4—24no—
219C4—42no—
220C4—4281no—
221C4—4281no—
222C4—4281no—
223C4—4281no—
224C4—2142no—
225C4—2142no—
226C4#1681208612yes—
227C4C22×Dic21411yescharacteristic
228C4C2×#1313381711yescharacteristic
229C2—22no—
230C2C2×#1681208611yescharacteristic
231C2#2029343311yescharacteristic
232C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenterC2×#103D107
hypocenterC107#3082
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #20292936C2semidirect
4. #20292932C2semidirect
5. #20292920C2semidirect
6. #20292910C2semidirect
65. C2×Q8×C107Q8semidirect
117. C107#3082semidirect
161. C2×Q8Dic214semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202931052
#202929362
#202929322
#202929202
#202929102
#202926672
#202926652
#202926532
#202926362
C2×#103×C1072
#3082107