Finite Group Databasebeta

← #28349134

Subgroups — #28349134

Order 13696.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic214C211yesmaximal, characteristic
2#20293352C211yesmaximal, characteristic
3#20292977C211yesmaximal, characteristic
4#20292976C212yesmaximal
5#20292923C211yesmaximal, characteristic
6#20292920C212yesmaximal
7#20292835C212yesmaximal
8#20292751C211yesmaximal, characteristic
9Q16×Dic107C212yesmaximal
10C4×#13133808C211yesmaximal, characteristic
11#462×C107C211yesmaximal, characteristic
12C2×#13133808C2211yescharacteristic
13#16812109—21no—
14#16812109C2211yescharacteristic
15#16812108—22no—
16#16812108C2212yes—
17#16812107C2212yes—
18#16812100—21no—
19#16812099—22no—
20Q8×Dic107C2212yes—
21Q8×Dic107C2212yes—
22C4×Dic214—21no—
23C4×Dic214C2211yescharacteristic
24C4×Dic214C2211yescharacteristic
25#16812074—21no—
26#16812043C2212yes—
27#16812042C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812034C2212yes—
30#16812033C2212yes—
31#16812033C2211yescharacteristic
32#16812033C2212yes—
33#16812032C2211yescharacteristic
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×Q16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214D411yescharacteristic
44C2×Dic214C2311yescharacteristic
45#13133808—22no—
46C2×C107⋊C8C2311yescharacteristic
47C2×C107⋊C8C2311yescharacteristic
48C4⋊Dic107—21no—
49C4⋊Dic107—22no—
50C4⋊Dic107—22no—
51C4⋊Dic107D411yescharacteristic
52C4⋊Dic107C2312yes—
53C4⋊Dic107C2311yescharacteristic
54Dic107⋊C4—21no—
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4D412yes—
59C4×Dic107—21no—
60C4×Dic107—22no—
61C4×Dic107—22no—
62C4×Dic107C2312yes—
63C2×Q8×C107C2312yes—
64C2×Q8×C107C2311yescharacteristic
65Q16×C107—24no—
66C2×C8×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2311yescharacteristic
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73Q8×C107—21no—
74Q8×C107—22no—
75Q8×C107—22no—
76Q8×C107#4112yes—
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—21no—
85C2×Dic107—22no—
86C2×Dic107C2×D412yes—
87C2×Dic107C2×D412yes—
88Dic214—21no—
89Dic214—21no—
90Dic214—82no—
91Dic214—81no—
92Dic214Q1614yescomplemented
93C8×C107—22no—
94C107⋊C8—21no—
95C107⋊C8—21no—
96C22×C107C22×D411yescharacteristic
97C4×C107—22no—
98C4×C107—21no—
99C4×C107—42no—
100C4×C107—41no—
101C4×C107C2×Q1612yes—
102C4×C107C2×#4111yescharacteristic
103C4×C1072−1+411yescharacteristic
104Dic107—22no—
105Dic107—41no—
106Dic107—42no—
107Dic107—41no—
108Dic107C2×Q1614yes—
109C2×C107#54811yescharacteristic
110C2×C107C22×Q1611yescharacteristic
111C2×C107#57711yescharacteristic
112#3363—1071nomaximal, Sylow
113C107#336311yescharacteristic, complemented, Sylow
114Q82—1071no—
115#557—1071no—
116#487—1071no—
117#487—1072no—
118#477—1071no—
119#477—1072no—
120#462—1072no—
121#462D10711yescharacteristic
122C8×Q8—1071no—
123C4×Q16—1072no—
124C4×Q16—1071no—
125C2×Q16—1071no—
126C2×Q16C2×D10712yes—
127C4⋊Q8—1072no—
128C4⋊Q8—1072no—
129C4⋊Q8—1071no—
130C4⋊Q8—2141no—
131C4⋊Q8—2142no—
132C4⋊Q8C2×D10711yescharacteristic
133#125—2142no—
134#125—2141no—
135C4×Q8—1071no—
136C4×Q8—1072no—
137C4×Q8—1072no—
138C4×Q8—1071no—
139C4×Q8—2141no—
140C4×Q8—2141no—
141C4×Q8C2×D10711yescharacteristic
142#107—1072no—
143#107—1071no—
144C4⋊C8—1072no—
145C4⋊C8C2×D10711yescharacteristic
146#103—1071no—
147#103—1072no—
148#103—1072no—
149#103—1072no—
150#103—1071no—
151#103C2×D10712yes—
152C4×C8—1072no—
153C4×C8—1071no—
154C2×Q8—1071no—
155C2×Q8—1071no—
156C2×Q8—2142no—
157C2×Q8—2141no—
158C2×Q8C22×D10712yes—
159C2×Q8C22×D10711yescharacteristic
160Q16—24no—
161Q16—2142no—
162C2×C8—1071no—
163C2×C8—1071no—
164C2×C8C22×D10712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—1072no—
169C4⋊C4—1071no—
170C4⋊C4—1072no—
171C4⋊C4—1071no—
172C4⋊C4—2142no—
173C4⋊C4—2142no—
174C4⋊C4—2142no—
175C4⋊C4—2142no—
176C4⋊C4—2142no—
177C4⋊C4—2141no—
178C4⋊C4—2141no—
179C4⋊C4C22×D10711yescharacteristic
180C42—21no—
181C42—1072no—
182C42—2142no—
183C42—2142no—
184C42—2141no—
185C42C22×D10711yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—8561no—
190Q8—8562no—
191Q8—1074no—
192Q8—2141no—
193Q8—2141no—
194Q8#1313381712yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—1072no—
199C2×C4—1072no—
200C2×C4—2141no—
201C2×C4—2142no—
202C2×C4—2141no—
203C2×C4D4×D10711yescharacteristic
204C2×C4D4×D10711yescharacteristic
205C2×C4C23×D10711yescharacteristic
206C8—22no—
207C8—2141no—
208C8—2141no—
209C22C2×D4×D10711yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—42no—
214C4—1074no—
215C4—4282no—
216C4—4281no—
217C4—4281no—
218C4—2142no—
219C4Q16×D10712yes—
220C4C2×#1313381711yescharacteristic
221C4#1681215611yescharacteristic
222C2#2029328311yescharacteristic
223C2C2×Q16×D10711yescharacteristic
224C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socle, prime radicalC22×C107C22×D4
hypercenter#462D107
hypocenterC107#3363
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
92. Dic214Q16semidirect
113. C107#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2142
#202933522
#202929772
#202929762
#202929232
#202929202
#202928352
#202927512
Q16×Dic1072
C4×#131338082
#462×C1072
#3363107