Finite Group Databasebeta

← #28348340

Subgroups — #28348340

Order 13696.

388 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293337C211yesmaximal, characteristic
2#20293292C211yesmaximal, characteristic
3#20292977C211yesmaximal, characteristic, complemented
4#20292961C212yesmaximal, complemented
5#20292923C211yesmaximal, characteristic, complemented
6#20292865C212yesmaximal, complemented
7#20292781C212yesmaximal, complemented
8C4×#13133808C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic
10#103×D107C212yesmaximal, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2212yes—
13C2×#13133808C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812108—21no—
17#16812101—21no—
18#16812087—22no—
19#16812087—22no—
20#16812085C2212yes—
21#16812085C2212yes—
22C4×Dic214C2211yescharacteristic
23C4×D428—21no—
24C4×D428C2211yescharacteristic
25#16812071—21no—
26C4⋊C4×D107C2212yes—
27#16812043C2212yes—
28#16812042C2211yescharacteristic
29#16812034C2212yes—
30#16812033C2211yescharacteristic
31#16812033C2212yes—
32#16812033C2212yes—
33#16812032C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×Q16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—22no—
42C2×Dic214—21no—
43C2×Dic214C2311yescharacteristic
44Q8×D107—24no—
45Q8×D107—42no—
46C2×D428D411yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C4×D107C2312yes—
50#13133808—22no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C8×D107—42no—
54C4⋊Dic107—21no—
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—21no—
59Dic107⋊C4—22no—
60Dic107⋊C4—22no—
61Dic107⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4D412yes—
65C4×Dic107—21no—
66C2×Q8×C107C2312yes—
67C2×Q8×C107C2311yescharacteristic
68Q16×C107—42no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107—21no—
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2311yescharacteristic
74C42×C107—21no—
75C42×C107C2311yescharacteristic
76C22×D107C2×D412yes—
77Q8×C107—21no—
78Q8×C107—22no—
79Q8×C107—21no—
80Q8×C107—22no—
81C2×C4×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—22no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—22no—
90C2×Dic107C2×D412yes—
91D428Q1614yescomplemented
92C4×D107—22no—
93C4×D107—22no—
94C4×D107—44no—
95C4×D107—82no—
96Dic214—21no—
97Dic214—44no—
98Dic214—81no—
99Dic214#4112yes—
100C8×C107—22no—
101C107⋊C8—21no—
102C107⋊C8—21no—
103C22×C107C22×D411yescharacteristic
104C2×D107—22no—
105C2×D107—22no—
106C2×D107C2×Q1614yes—
107C4×C107—21no—
108C4×C107—42no—
109C4×C107—41no—
110C4×C107—41no—
111C4×C107C2×Q1612yes—
112C4×C107C2×#4111yescharacteristic
113C4×C107H(5,𝔽2)11yescharacteristic
114Dic107—22no—
115Dic107—22no—
116Dic107—42no—
117Dic107—41no—
118C2×C107#54611yescharacteristic
119C2×C107C22×Q1611yescharacteristic
120C2×C107#57711yescharacteristic
121D107—24no—
122#3299—1071nomaximal, Sylow
123C107#329911yescharacteristic, complemented, Sylow
124#554—1071no—
125D4×Q8—1071no—
126#487—1071no—
127#484—1072no—
128#477—1071no—
129#467—1072no—
130#462D10711yescharacteristic, complemented
131#451—1072no—
132C4×Q16—1071no—
133C8×D4—1071no—
134C2×#103—1072no—
135C22×Q8—1072no—
136C2×Q16—1071no—
137C2×Q16C2×D10712yes—
138C22×C8—1072no—
139C4⋊Q8—1071no—
140C4⋊Q8—2141no—
141C4⋊Q8C2×D10711yescharacteristic
142#125—2141no—
143C22⋊Q8—1072no—
144C22⋊Q8—1072no—
145C22⋊Q8—2142no—
146C22⋊Q8—2142no—
147C4×Q8—1071no—
148C4×Q8C2×D10711yescharacteristic
149C4×D4—1071no—
150C4×D4—2141no—
151C4×D4—2141no—
152C2×C4⋊C4—1072no—
153#107—1072no—
154#107—1071no—
155C4⋊C8C2×D10711yescharacteristic
156#103—1072no—
157#103—1071no—
158#103—1072no—
159#103—1072no—
160#103—1071no—
161#103C2×D10712yes—
162C22⋊C8—1072no—
163C4×C8—1071no—
164C2×Q8—1071no—
165C2×Q8—4282no—
166C2×Q8—2142no—
167C2×Q8—2144no—
168C2×Q8—2141no—
169C2×Q8C22×D10712yes—
170C2×Q8C22×D10711yescharacteristic
171C2×D4—1071no—
172C22×C4—1072no—
173C22×C4—2142no—
174C22×C4—2142no—
175Q16—42no—
176Q16—2142no—
177C2×C8—1071no—
178C2×C8—1071no—
179C2×C8—4282no—
180C2×C8C22×D10712yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—1071no—
185C4⋊C4—1072no—
186C4⋊C4—1071no—
187C4⋊C4—2142no—
188C4⋊C4—2142no—
189C4⋊C4—2142no—
190C4⋊C4—2141no—
191C4⋊C4—2141no—
192C4⋊C4C22×D10711yescharacteristic
193C22⋊C4—1072no—
194C22⋊C4—2142no—
195C22⋊C4—2142no—
196C42—21no—
197C42—2141no—
198C42C22×D10711yescharacteristic
199C23—1072no—
200Q8—21no—
201Q8—22no—
202Q8—21no—
203Q8—22no—
204Q8—8561no—
205Q8—1072no—
206Q8—4284no—
207Q8—2141no—
208D4—1074no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—8562no—
213C2×C4—1072no—
214C2×C4—4284no—
215C2×C4—2141no—
216C2×C4—2142no—
217C2×C4—2141no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4D4×D10711yescharacteristic
221C2×C4D4×D10711yescharacteristic
222C2×C4C23×D10711yescharacteristic
223C8—22no—
224C8—2141no—
225C8—2141no—
226C22—1074no—
227C22—2142no—
228C22—2142no—
229C22C2×D4×D10711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—41no—
234C4—4282no—
235C4—4281no—
236C4—2142no—
237C4—2142no—
238C4Q16×D10712yes—
239C4#41×D10711yescharacteristic
240C4#1681215411yescharacteristic
241C2—2144no—
242C2#2029327411yescharacteristic
243C2C2×Q16×D10711yescharacteristic
244C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radicalC2×D428D4
hypercenter#462D107
hypocenterC107#3299
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #20292977C2semidirect
4. #20292961C2semidirect
5. #20292923C2semidirect
6. #20292865C2semidirect
7. #20292781C2semidirect
8. C4×#13133808C2semidirect
10. #103×D107C2semidirect
11. #462×C107C2semidirect
91. D428Q16semidirect
123. C107#3299semidirect
130. #462D107semidirect

Maximal subgroups

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

subgroupindex
#202933372
#202932922
#202929772
#202929612
#202929232
#202928652
#202927812
C4×#131338082
#202926922
#103×D1072
#462×C1072
#3299107