Finite Group Databasebeta

← #28349079

Subgroups — #28349079

Order 13696.

342 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 115 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#20293350C211yesmaximal, characteristic
2#20293338C211yesmaximal, characteristic
3#20293023C211yesmaximal, characteristic, complemented
4#20292970C212yesmaximal, complemented
5#20292956C212yesmaximal, complemented
6#20292913C211yesmaximal, characteristic, complemented
7#20292884C212yesmaximal, complemented
8C4×#13133808C211yesmaximal, characteristic, complemented
9#20292704C211yesmaximal, characteristic
10#20292668C212yesmaximal, complemented
11#477×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2211yescharacteristic
13C2×#13133801C2212yes—
14#16812109C2211yescharacteristic
15#16812104—22no—
16#16812100—21no—
17#16812097—21no—
18#16812089C2212yes—
19#16812085—22no—
20#16812085C2212yes—
21C4×Dic214—21no—
22C4×Dic214C2211yescharacteristic
23C4×D428C2211yescharacteristic
24#16812071—21no—
25#16812067C2212yes—
26C4⋊C4×D107C2212yes—
27#16812042C2211yescharacteristic
28#16812041C2212yes—
29#16812040C2212yes—
30#16812040C2212yes—
31#16812034C2212yes—
32#16812033C2211yescharacteristic
33#16812032C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×Q8×C107C2211yescharacteristic
38#107×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214C2311yescharacteristic
43C2×D428D411yescharacteristic
44C2×C4×D107—22no—
45C2×C4×D107C2312yes—
46#13133808—22no—
47#13133801—24no—
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2311yescharacteristic
50C4⋊Dic107—21no—
51C4⋊Dic107—22no—
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2312yes—
55Dic107⋊C4—21no—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4D412yes—
62C4×Dic107—21no—
63C4×Dic107—22no—
64C2×Q8×C107—21no—
65C2×Q8×C107C2311yescharacteristic
66C2×C8×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2312yes—
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73C22×D107C2×D412yes—
74Q8×C107—21no—
75Q8×C107—21no—
76Q8×C107—81no—
77C2×C4×C107—21no—
78C2×C4×C107—21no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C2×Dic107—21no—
84C2×Dic107—22no—
85C2×Dic107—21no—
86C2×Dic107C2×D412yes—
87D428—22no—
88C4×D107—22no—
89C4×D107—22no—
90C4×D107—82no—
91Dic214—21no—
92Dic214—21no—
93Dic214—42no—
94C8×C107—22no—
95C107⋊C8—21no—
96C107⋊C8#4112yes—
97C22×C107C22×D411yescharacteristic
98C2×D107—22no—
99C2×D107—22no—
100C2×D107—22no—
101C4×C107—21no—
102C4×C107—41no—
103C4×C107—42no—
104C4×C107—41no—
105C4×C107#13712yes—
106C4×C107C2×#4111yescharacteristic
107C4×C1072−1+411yescharacteristic
108Dic107—22no—
109Dic107—41no—
110Dic107—42no—
111Dic107—41no—
112C2×C107#54711yescharacteristic
113C2×C107C2×#13711yescharacteristic
114C2×C107#57711yescharacteristic
115D107—42no—
116#3356—1071nomaximal, Sylow
117C107#335611yescharacteristic, complemented, Sylow
118#556—1071no—
119#554—1071no—
120#498—1071no—
121#485—1072no—
122#483—1072no—
123#477D10711yescharacteristic, complemented
124#475—1071no—
125#470—1072no—
126C4×Q16—1071no—
127#436—1071no—
128#428—1072no—
129C2×Q16—1071no—
130C2×M16—1072no—
131C4⋊Q8—1071no—
132C4⋊Q8C2×D10711yescharacteristic
133#126—2142no—
134#125—2141no—
135#124—2141no—
136C22⋊Q8—1072no—
137C22⋊Q8—1072no—
138C22⋊Q8—2142no—
139C4×Q8—1071no—
140C4×Q8—2141no—
141C4×Q8C2×D10711yescharacteristic
142C4×D4—1071no—
143C4×D4—2141no—
144#117—1072no—
145C2×C4⋊C4—1072no—
146#107—1071no—
147#107C2×D10712yes—
148#106—1072no—
149#106—1072no—
150#106—1072no—
151C4⋊C8C2×D10711yescharacteristic
152#103—1071no—
153#103—1072no—
154#103—1071no—
155#103C2×D10712yes—
156C22⋊C8—1072no—
157C4×C8—1071no—
158C2×Q8—21no—
159C2×Q8—1071no—
160C2×Q8—2141no—
161C2×Q8C22×D10711yescharacteristic
162C2×D4—1071no—
163C22×C4—1072no—
164C22×C4—2142no—
165Q16—2142no—
166M16—2144no—
167C2×C8—1071no—
168C2×C8—1071no—
169C2×C8C22×D10712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—1071no—
173C4⋊C4—1072no—
174C4⋊C4—1071no—
175C4⋊C4—2142no—
176C4⋊C4—2142no—
177C4⋊C4—2142no—
178C4⋊C4—2141no—
179C4⋊C4—2141no—
180C4⋊C4C22×D10711yescharacteristic
181C4⋊C4C22×D10712yes—
182C22⋊C4—1072no—
183C22⋊C4—2142no—
184C22⋊C4—2142no—
185C22⋊C4—2142no—
186C42—21no—
187C42—2142no—
188C42—2141no—
189C42C22×D10711yescharacteristic
190C23—1072no—
191Q8—21no—
192Q8—21no—
193Q8—81no—
194Q8—4282no—
195Q8—2141no—
196Q8—2141no—
197D4—2142no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—8562no—
202C2×C4—1072no—
203C2×C4—2142no—
204C2×C4—2141no—
205C2×C4—2141no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4D4×D10711yescharacteristic
209C2×C4D4×D10711yescharacteristic
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—1072no—
213C8—2141no—
214C22—2142no—
215C22—2142no—
216C22—2142no—
217C22C2×D4×D10711yescharacteristic
218C4—21no—
219C4—42no—
220C4—41no—
221C4—41no—
222C4—4282no—
223C4—4281no—
224C4—4281no—
225C4—2142no—
226C4#1681214112yes—
227C4#41×D10711yescharacteristic
228C4#1681215711yescharacteristic
229C2—4282no—
230C2#2029327911yescharacteristic
231C2C2×#1681214111yescharacteristic
232C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#477×C107C2
socleC22×C107C22×D4
prime radicalC2×D428D4
hypercenter#477D107
hypocenterC107#3356
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #20293023C2semidirect
4. #20292970C2semidirect
5. #20292956C2semidirect
6. #20292913C2semidirect
7. #20292884C2semidirect
8. C4×#13133808C2semidirect
10. #20292668C2semidirect
11. #477×C107C2semidirect
117. C107#3356semidirect
123. #477D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202933382
#202930232
#202929702
#202929562
#202929132
#202928842
C4×#131338082
#202927042
#202926682
#477×C1072
#3356107