Finite Group Databasebeta

← #28346917

Subgroups — #28346917

Order 13696.

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

normal subgroups only

This group has 552 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×#16812141C212yesmaximal, complemented
2C2×#16812138C212yesmaximal, complemented
3#20293176C211yesmaximal, characteristic
4C4⋊Q8×D107C211yesmaximal, characteristic, complemented
5#20293022C211yesmaximal, characteristic, complemented
6#20293021C211yesmaximal, characteristic, complemented
7#20293019C211yesmaximal, characteristic, complemented
8#20293005C212yesmaximal, complemented
9#20292998C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#497×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133817C2211yescharacteristic
14C2×Q8×D107C2212yes—
15#16812141—28no—
16#16812138—28no—
17C2×Dic428C2212yes—
18C2×#13133808C2212yes—
19C2×#13133808C2212yes—
20C2×#13133807C2212yes—
21C2×#13133806C2212yes—
22C2×#13133805C2212yes—
23C2×#13133801C2212yes—
24#16812109C2211yescharacteristic
25#16812108—22no—
26#16812107C2212yes—
27#16812107C2211yescharacteristic
28#16812105C2211yescharacteristic
29#16812097C2211yescharacteristic
30#16812082—22no—
31(C22⋊D107)⋊C4—22no—
32C4⋊C4×D107—22no—
33C42×D107C2211yescharacteristic
34#16812020C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×Q16×C107C2212yes—
37C2×SD16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39#124×C107C2211yescharacteristic
40#97×C107C2211yescharacteristic
41#13133819—24no—
42#13133817—24no—
43C2×Dic214—22no—
44C2×Dic214C2311yescharacteristic
45C2×Dic214C2312yes—
46Q8×D107—24no—
47Q8×D107—24no—
48C2×D428—21no—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—21no—
51C2×C22⋊D107—22no—
52C22×Dic107—21no—
53C2×C4×D107—22no—
54C2×C4×D107—21no—
55C2×C4×D107D412yes—
56C2×C4×D107C2311yescharacteristic
57Dic428—24no—
58#13133808—24no—
59#13133808—24no—
60#13133807—24no—
61#13133806—24no—
62#13133805—24no—
63#13133801D418yescomplemented
64C2×C107⋊C8C2312yes—
65C4⋊Dic107—22no—
66Dic107⋊C4—24no—
67Dic107⋊C4—22no—
68(C2×D107)⋊C4—22no—
69C4×Dic107D412yes—
70C4×Dic107C2311yescharacteristic
71C2×Q8×C107C2312yes—
72C2×Q8×C107C2311yescharacteristic
73C2×D4×C107C2311yescharacteristic
74Q16×C107—24no—
75SD16×C107—24no—
76C2×C8×C107C2312yes—
77C4⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C42×C107C2311yescharacteristic
80C23×C107—21no—
81C22×D107—21no—
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85Q8×C107—22no—
86D4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—42no—
92C22⋊D107—44no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—42no—
96C2×Dic107C2×D411yescharacteristic
97C2×Dic107C2×D412yes—
98D428—22no—
99D428—42no—
100C4×D107—24no—
101C4×D107—44no—
102C4×D107—42no—
103C4×D107C2×D414yes—
104Dic214—22no—
105Dic214—22no—
106Dic214—22no—
107Dic214—44no—
108C8×C107C2×D414yes—
109C107⋊C8C2×D414yes—
110C22×C107—41no—
111C22×C107—42no—
112C22×C107C22×D411yescharacteristic
113C2×D107—22no—
114C2×D107—41no—
115C2×D107—42no—
116C2×D107#12712yes—
117C4×C107—22no—
118C4×C107—42no—
119C4×C107—41no—
120C4×C107C22×D412yes—
121Dic107—42no—
122Dic107—41no—
123Dic107#12712yes—
124Dic107#13714yescomplemented
125C2×C107—81no—
126C2×C107C2×#12711yescharacteristic
127C2×C107C2×#13712yes—
128D107—22no—
129D107—81no—
130#3152—1071nomaximal, Sylow
131C107#315211yescharacteristic, complemented, Sylow
132C2×#137—1072no—
133C2×#137—1072no—
134#532—1071no—
135C2×C4⋊Q8—1071no—
136#497—1071no—
137#497—1071no—
138#497—1071no—
139#497D10711yescharacteristic, complemented
140#494—1072no—
141#492—1072no—
142C4×M16—1071no—
143C2×#41—1071no—
144C2×#41—1071no—
145C22×Q8—1072no—
146#137—2148no—
147#137—2148no—
148C2×Q16—1072no—
149C2×Q16—1072no—
150C2×Q16—1072no—
151C2×Q16C2×D10712yes—
152C2×SD16—1072no—
153C2×SD16—1072no—
154C2×SD16—1072no—
155C2×SD16C2×D10712yes—
156C2×M16—1072no—
157C4⋊Q8—1072no—
158C4⋊Q8—1071no—
159C4⋊Q8—1071no—
160C4⋊Q8—2142no—
161C4⋊Q8C2×D10711yescharacteristic
162#127—1071no—
163#124—1071no—
164#124C2×D10711yescharacteristic
165#121—2142no—
166C4×D4—2142no—
167C2×C4⋊C4—2142no—
168C2×C42—1071no—
169#97—1071no—
170#97C2×D10711yescharacteristic
171C4×C8—1072no—
172#41—2144no—
173#41—2144no—
174C2×Q8—1071no—
175C2×Q8—1072no—
176C2×Q8—2142no—
177C2×Q8—2144no—
178C2×Q8—2144no—
179C2×Q8C22×D10711yescharacteristic
180C2×Q8C22×D10712yes—
181C2×D4—1071no—
182C2×D4—2141no—
183C2×D4—2142no—
184C2×D4—2141no—
185C2×D4C22×D10711yescharacteristic
186C22×C4—1071no—
187C22×C4—1072no—
188C22×C4—2141no—
189C22×C4—2142no—
190C22×C4—2141no—
191Q16—24no—
192Q16—2144no—
193Q16—2144no—
194Q16—2144no—
195SD16—24no—
196SD16—2144no—
197SD16—2144no—
198SD16—2144no—
199M16—1078no—
200C2×C8—1072no—
201C2×C8C22×D10712yes—
202C4⋊C4—22no—
203C4⋊C4—2144no—
204C4⋊C4—2142no—
205C4⋊C4—2142no—
206C22⋊C4—22no—
207C22⋊C4—2142no—
208C42—1072no—
209C42—1071no—
210C42C22×D10711yescharacteristic
211C23—21no—
212C23—1071no—
213C23—2141no—
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8—4284no—
218Q8—2142no—
219Q8—2142no—
220Q8—2142no—
221D4—22no—
222D4—4282no—
223D4—4284no—
224D4—4282no—
225D4—2142no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—1072no—
229C2×C4—1071no—
230C2×C4—1074no—
231C2×C4—4282no—
232C2×C4—4284no—
233C2×C4—4282no—
234C2×C4—2144no—
235C2×C4—2141no—
236C2×C4—2142no—
237C2×C4D4×D10712yes—
238C2×C4C23×D10711yescharacteristic
239C8—1074no—
240C8D4×D10714yes—
241C22—42no—
242C22—41no—
243C22—1072no—
244C22—4282no—
245C22—4281no—
246C22—2142no—
247C22C2×D4×D10711yescharacteristic
248C4—22no—
249C4—41no—
250C4—42no—
251C4—1074no—
252C4—1072no—
253C4—4281no—
254C4—4282no—
255C4C2×D4×D10712yes—
256C2—81no—
257C2—8561no—
258C2—2142no—
259C2#127×D10711yescharacteristic
260C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#497×C107C2
socleC22×C107C22×D4
prime radical#20293176C2
hypercenter#497D107
hypocenterC107#3152
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. C2×#16812138C2semidirect
4. C4⋊Q8×D107C2semidirect
5. #20293022C2semidirect
6. #20293021C2semidirect
7. #20293019C2semidirect
8. #20293005C2semidirect
9. #20292998C2semidirect
10. #20292579C2semidirect
11. #497×C107C2semidirect
63. #13133801D4semidirect
124. Dic107#137semidirect
131. C107#3152semidirect
139. #497D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
C2×#168121382
#202931762
C4⋊Q8×D1072
#202930222
#202930212
#202930192
#202930052
#202929982
#202925792
#497×C1072
#3152107