Finite Group Databasebeta

← #28336803

Subgroups — #28336803

Order 13696.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×Dic428C211yesmaximal, characteristic
2C2×#16812074C211yesmaximal, characteristic
3C2×#16812032C211yesmaximal, characteristic
4C2×#16812032C211yesmaximal, characteristic
5#20292532C211yesmaximal, characteristic
6#20292381C211yesmaximal, characteristic
7#340×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9C22×Dic214C2211yescharacteristic
10C2×Dic428—22no—
11C2×Dic428—22no—
12C2×Dic428C414yescomplemented
13C22×C107⋊C8C2211yescharacteristic
14#16812074—22no—
15#16812074—22no—
16C2×Dic107⋊C4—21no—
17C2×C4×Dic107—21no—
18C2×C4×Dic107C2211yescharacteristic
19#16812032—22no—
20#16812032—22no—
21C107⋊#95—21no—
22C107⋊#95—21no—
23C22×C8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25C2×C4⋊C4×C107C2211yescharacteristic
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214—21no—
34C2×Dic214C2×C412yes—
35C2×Dic214C2×C412yes—
36C2×Dic214D412yes—
37C2×Dic214D412yes—
38C22×Dic107—21no—
39C22×Dic107—21no—
40C22×Dic107D411yescharacteristic
41C22×Dic107D411yescharacteristic
42Dic428—28no—
43C2×C107⋊C8—21no—
44C2×C107⋊C8—21no—
45C2×C107⋊C8—21no—
46Dic107⋊C4—24no—
47C4×Dic107—24no—
48C4×Dic107—21no—
49C4×Dic107—21no—
50C22×C4×C107—21no—
51C22×C4×C107—21no—
52C22×C4×C107C2311yescharacteristic
53C2×C8×C107—21no—
54C2×C8×C107—21no—
55C2×C8×C107C2×C412yes—
56C4⋊C4×C107—21no—
57C4⋊C4×C107—21no—
58C4⋊C4×C107—21no—
59C4⋊C4×C107D412yes—
60C23×C107C2×D411yescharacteristic
61C2×C4×C107—22no—
62C2×C4×C107—41no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107C22×C411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107#4111yescharacteristic
72C2×C4×C107#4111yescharacteristic
73C2×Dic107—22no—
74C2×Dic107—21no—
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—21no—
78C2×Dic107—22no—
79C2×Dic107—21no—
80C2×Dic107—21no—
81C2×Dic107—21no—
82C2×Dic107—41no—
83C2×Dic107—41no—
84C2×Dic107Q1612yescomplemented
85Dic214—22no—
86Dic214—22no—
87Dic214—22no—
88Dic214—22no—
89Dic214—22no—
90Dic214—41no—
91Dic214—41no—
92Dic214C22⋊C414yes—
93C8×C107—22no—
94C107⋊C8—41no—
95C22×C107C4×D411yescharacteristic
96C22×C107#12011yescharacteristic
97C22×C107#12111yescharacteristic
98C22×C107C2×Q1611yescharacteristic
99C22×C107#13511yescharacteristic
100C22×C107#13711yescharacteristic
101C22×C107#13711yescharacteristic
102C4×C107—42no—
103C4×C107—81no—
104C4×C107C2×C22⋊C411yescharacteristic
105C4×C107C4×D411yescharacteristic
106C4×C107#12111yescharacteristic
107C4×C107#12311yescharacteristic
108Dic107—24no—
109Dic107—22no—
110Dic107—42no—
111Dic107—41no—
112C2×C107#38611yescharacteristic
113C2×C107C4×Q1611yescharacteristic
114C2×C107#44111yescharacteristic
115C2×C107#45111yescharacteristic
116C2×C107#45211yescharacteristic
117C2×C107#46211yescharacteristic
118C2×C107#46411yescharacteristic
119#1878—1071nomaximal, Sylow
120C107#187811yescharacteristic, complemented, Sylow
121C22×Q16—1071no—
122C2×C4×Q8—1071no—
123C2×#103—1071no—
124C2×#103—1071no—
125#391—1071no—
126#340D10711yescharacteristic
127#336—1071no—
128C22×Q8—1071no—
129C22×Q8—1071no—
130C2×Q16—1074no—
131C2×Q16—2142no—
132C2×Q16—2142no—
133C22×C8—1071no—
134C22×C8C2×D10711yescharacteristic
135C4×Q8—2142no—
136C4×Q8—2142no—
137C2×C4⋊C4—2141no—
138C2×C4⋊C4C2×D10711yescharacteristic
139C2×C4⋊C4C2×D10711yescharacteristic
140C2×C42—1071no—
141C2×C42—2141no—
142#103—2142no—
143#103—2142no—
144#95—2141no—
145#95—2141no—
146C2×Q8—1072no—
147C2×Q8—1072no—
148C2×Q8—1072no—
149C2×Q8—1072no—
150C2×Q8—2141no—
151C2×Q8—2141no—
152C2×Q8—2141no—
153C2×Q8—2141no—
154C2×Q8—2141no—
155C2×Q8—2141no—
156C2×Q8—2141no—
157C2×Q8—2141no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1071no—
161C22×C4—1071no—
162C22×C4—2141no—
163C22×C4—2141no—
164C22×C4C22×D10711yescharacteristic
165Q16—2148no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—2141no—
169C2×C8—2141no—
170C2×C8—2141no—
171C2×C8C4×D10712yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—2144no—
176C4⋊C4D42812yes—
177C42—2144no—
178C42—2141no—
179C42—2141no—
180C23D4×D10711yescharacteristic
181Q8—1074no—
182Q8—4281no—
183Q8—4281no—
184Q8—2142no—
185Q8—2142no—
186Q8—2142no—
187Q8—2142no—
188Q8—2142no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—1072no—
196C2×C4—4281no—
197C2×C4—4281no—
198C2×C4—2142no—
199C2×C4—2141no—
200C2×C4—2141no—
201C2×C4—2141no—
202C2×C4—2142no—
203C2×C4—2141no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2141no—
207C2×C4C2×C4×D10711yescharacteristic
208C2×C4D4×D10711yescharacteristic
209C2×C4D4×D10711yescharacteristic
210C2×C4C2×D42811yescharacteristic
211C2×C4#1313381811yescharacteristic
212C2×C4#1313381911yescharacteristic
213C8—22no—
214C8—4281no—
215C22#1681207111yescharacteristic
216C22#1681207911yescharacteristic
217C22#1681208211yescharacteristic
218C22Q16×D10711yescharacteristic
219C22#1681212811yescharacteristic
220C22#1681213811yescharacteristic
221C22#1681213811yescharacteristic
222C4—42no—
223C4—81no—
224C4—4282no—
225C4—4281no—
226C4—2144no—
227C4—2142no—
228C4C22⋊C4×D10711yescharacteristic
229C4C4×D42811yescharacteristic
230C4#1681208411yescharacteristic
231C4#1681209411yescharacteristic
232C2#2029250511yescharacteristic
233C2#2029272111yescharacteristic
234C2#2029273311yescharacteristic
235C2#2029278411yescharacteristic
236C2#2029279011yescharacteristic
237C2#2029283611yescharacteristic
238C2#2029284811yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#340×C107C2
socle, prime radicalC23×C107C2×D4
cosocleC22×C4×C107C23
hypercenter#340D107
hypocenterC107#1878
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
12. C2×Dic428C4semidirect
84. C2×Dic107Q16semidirect
120. C107#1878semidirect

Maximal subgroups

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

subgroupindex
C22×Dic4282
C2×#168120742
C2×#168120322
#202925322
#202923812
#340×C1072
#1878107