Finite Group Databasebeta

← #360×D107 #28338191

Subgroups — #360×D107 #28338191

Order 13696.

232 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 67 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#135×D107C211yesmaximal, characteristic
2M32×D107C211yesmaximal, characteristic, complemented
3#106×D107C211yesmaximal, characteristic, complemented
4#20292438C211yesmaximal, characteristic, complemented
5#20292437C211yesmaximal, characteristic, complemented
6#20292436C211yesmaximal, characteristic, complemented
7#360×C107C211yesmaximal, characteristic, complemented
8#41×D107—21no—
9#16812132C411yescharacteristic, complemented
10#16812131C2211yescharacteristic
11#16812130—21no—
12#16812129—21no—
13#16812128C411yescharacteristic, complemented
14Q16×D107C411yescharacteristic, complemented
15SD16×D107—21no—
16D8×D107C411yescharacteristic, complemented
17C2×C8×D107C2211yescharacteristic
18C4⋊C4×D107—21no—
19#16812049C2211yescharacteristic, complemented
20#16812048—21no—
21C16×D107—21no—
22#16812041C2211yescharacteristic, complemented
23#16812040—21no—
24#135×C107C2211yescharacteristic
25M32×C107C2211yescharacteristic, complemented
26#106×C107C2211yescharacteristic, complemented
27#13133819—21no—
28#13133818—21no—
29#13133818—41no—
30#13133817—21no—
31#13133817—41no—
32Q8×D107—21no—
33D4×D107—21no—
34D4×D107—41no—
35C2×C4×D107—41no—
36C2×C4×D107—41no—
37C2×C4×D107D411yescharacteristic
38Dic428C2×C411yescharacteristic, complemented
39#13133808—21no—
40#13133807—21no—
41#13133806—21no—
42#13133805—21no—
43D856C2×C411yescharacteristic, complemented
44#13133803—21no—
45C2×C107⋊C8D411yescharacteristic
46C8×D107C2×C411yescharacteristic
47C8×D107C2×C411yescharacteristic
48C8×D107D411yescharacteristic
49C8×D107D411yescharacteristic
50C4⋊Dic107—21no—
51Dic107⋊C4—41no—
52C107⋊C16—21no—
53#41×C107—21no—
54Q16×C107C2×C411yescharacteristic, complemented
55SD16×C107—21no—
56D8×C107C2×C411yescharacteristic, complemented
57C2×C8×C107C2311yescharacteristic
58C4⋊C4×C107—21no—
59C16×C107—21no—
60C22×D107—41no—
61C22×D107D811yescharacteristic
62Q8×C107—21no—
63D4×C107—21no—
64D4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107C2×D411yescharacteristic
68C22⋊D107—41no—
69C22⋊D107—41no—
70C22⋊D107—41no—
71C2×Dic107—41no—
72C2×Dic107—41no—
73C2×Dic107D811yescharacteristic
74D428—21no—
75D428—41no—
76C4×D107—41no—
77C4×D107—41no—
78C4×D107—41no—
79C4×D107—82no—
80C4×D107C22⋊C411yescharacteristic
81C4×D107C22⋊C411yescharacteristic
82C4×D107SD1611yescharacteristic
83C4×D107SD1611yescharacteristic
84Dic214—21no—
85Dic214—41no—
86C8×C107C22×C411yescharacteristic
87C8×C107C2×D411yescharacteristic
88C107⋊C8C22⋊C411yescharacteristic
89C107⋊C8C22⋊C411yescharacteristic
90C22×C107—41no—
91C22×C107C2×D811yescharacteristic
92C2×D107—22no—
93C2×D107—41no—
94C2×D107—82no—
95C2×D107#10211yescharacteristic
96C2×D107#10211yescharacteristic
97C4×C107—41no—
98C4×C107—81no—
99C4×C107C2×C22⋊C411yescharacteristic
100C4×C107C2×SD1611yescharacteristic
101Dic107—41no—
102Dic107—81no—
103Dic107#10211yescharacteristic
104Dic107#10211yescharacteristic
105C2×C107—21no—
106C2×C107—81no—
107C2×C107C2×#10211yescharacteristic
108D107—21no—
109D107—81no—
110D107#36012yesdirect product factor
111C2×#360—1071nomaximal, Sylow
112C107C2×#36011yescharacteristic, complemented, Sylow
113C2×#135—1071no—
114C2×M32—1071no—
115C2×#106—1071no—
116#360—1071no—
117#360—1071no—
118#360—1071no—
119#360D10711yescharacteristic, direct product factor
120C2×#41—2141no—
121#135—1071no—
122#135—1071no—
123#135—1071no—
124#135—2141no—
125#135—2141no—
126#135C2×D10711yescharacteristic
127C2×Q16—1071no—
128C2×SD16—2141no—
129C2×D8—1071no—
130C22×C8—1071no—
131C2×C4⋊C4—2141no—
132M32—1071no—
133M32—2141no—
134M32C2×D10711yescharacteristic, complemented
135C2×C16—2141no—
136#106—1071no—
137#106—2141no—
138#106C2×D10711yescharacteristic, complemented
139#41—21no—
140#41—4281no—
141#41—4281no—
142#41—2141no—
143#41—2141no—
144#41—2141no—
145C2×Q8—2141no—
146C2×D4—4281no—
147C2×D4—2141no—
148C22×C4—1071no—
149C22×C4—4281no—
150C22×C4—4281no—
151Q16—1071no—
152Q16—2141no—
153Q16C4×D10711yescharacteristic, complemented
154SD16—21no—
155SD16—2141no—
156SD16—2141no—
157SD16—2141no—
158D8—1071no—
159D8—2141no—
160D8C4×D10711yescharacteristic, complemented
161C2×C8—1071no—
162C2×C8—1071no—
163C2×C8—1071no—
164C2×C8—1071no—
165C2×C8—1071no—
166C2×C8C22×D10711yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—4281no—
169C4⋊C4—2141no—
170C16—21no—
171C16—2141no—
172C23—1071no—
173C23—4281no—
174Q8—21no—
175Q8—4281no—
176Q8—2141no—
177D4—21no—
178D4—41no—
179D4—4281no—
180D4—4281no—
181D4—4281no—
182D4—4281no—
183D4—2141no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—8562no—
187C2×C4—1071no—
188C2×C4—1071no—
189C2×C4—1071no—
190C2×C4—1071no—
191C2×C4—1071no—
192C2×C4—4281no—
193C2×C4—4281no—
194C2×C4—4281no—
195C2×C4—4281no—
196C2×C4—4281no—
197C2×C4D4×D10711yescharacteristic
198C8—1071no—
199C8—1071no—
200C8C2×C4×D10711yescharacteristic
201C8D4×D10711yescharacteristic
202C22—41no—
203C22—8562no—
204C22—1071no—
205C22—1071no—
206C22—4281no—
207C22—2142no—
208C22D8×D10711yescharacteristic
209C4—41no—
210C4—81no—
211C4—8561no—
212C4—1071no—
213C4—1071no—
214C4—4281no—
215C4C22⋊C4×D10711yescharacteristic
216C4SD16×D10711yescharacteristic
217C2—21no—
218C2—81no—
219C2—8561no—
220C2—1072no—
221C2—2141no—
222C2#102×D10711yescharacteristic
subgroupG/H
centerC2#102×D107
commutatorC8×C107C22×C4
FrattiniC2×C8C22×D107
Fitting#360×C107C2
socleC2×C107C2×#102
prime radical#135×D107C2
cosocleC2×C8×C107C23
hypercenter#360D107
hypocenterC107C2×#360
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. M32×D107C2semidirect
3. #106×D107C2semidirect
4. #20292438C2semidirect
5. #20292437C2semidirect
6. #20292436C2semidirect
7. #360×C107C2semidirect
9. #16812132C4semidirect
13. #16812128C4semidirect
14. Q16×D107C4semidirect
16. D8×D107C4semidirect
19. #16812049C22semidirect
22. #16812041C22semidirect
25. M32×C107C22semidirect
26. #106×C107C22semidirect
38. Dic428C2×C4semidirect
43. D856C2×C4semidirect
54. Q16×C107C2×C4semidirect
56. D8×C107C2×C4semidirect
110. D107#360direct (N × H)
112. C107C2×#360semidirect
119. #360D107direct (N × H)
134. M32C2×D107semidirect
138. #106C2×D107semidirect
153. Q16C4×D107semidirect
160. D8C4×D107semidirect

Maximal subgroups

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

subgroupindex
#135×D1072
M32×D1072
#106×D1072
#202924382
#202924372
#202924362
#360×C1072
C2×#360107