Finite Group Databasebeta

← #28335031

Subgroups — #28335031

Order 13696.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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⋊Q8×D107C211yesmaximal, characteristic
2C22⋊C8×D107C211yesmaximal, characteristic
3#20292486C211yesmaximal, characteristic
4#20292365C211yesmaximal, characteristic, complemented
5#20292364C211yesmaximal, characteristic, complemented
6#20292363C211yesmaximal, characteristic, complemented
7#328×C107C211yesmaximal, characteristic, complemented
8C2×Q8×D107C411yescharacteristic, complemented
9C22×C4×D107C2211yescharacteristic
10C2×C8×D107—21no—
11#16812089C411yescharacteristic, complemented
12#16812088C2211yescharacteristic
13#16812087—21no—
14#16812086—21no—
15#16812085C411yescharacteristic, complemented
16C2×Dic107⋊C4—21no—
17C4⋊C4×D107—21no—
18C4⋊C4×D107C411yescharacteristic, complemented
19C2×(C2×D107)⋊C4—21no—
20C22⋊C4×D107—21no—
21C22⋊(C107⋊C8)C2211yescharacteristic
22(C2×D107)⋊C8—21no—
23C107⋊#95C2211yescharacteristic
24C22⋊Q8×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28C2×Dic214—21no—
29C2×Dic214C2×C411yescharacteristic
30Q8×D107—42no—
31C22×Dic107—21no—
32C22×Dic107D411yescharacteristic
33C2×C4×D107—21no—
34C2×C4×D107—21no—
35C2×C4×D107—42no—
36C2×C4×D107C2×C411yescharacteristic
37C2×C4×D107C2×C411yescharacteristic
38C2×C4×D107D411yescharacteristic
39C2×C4×D107D411yescharacteristic
40C2×C107⋊C8—21no—
41C8×D107—42no—
42C4⋊Dic107—21no—
43C4⋊Dic107C2×C411yescharacteristic
44Dic107⋊C4—21no—
45Dic107⋊C4—21no—
46Dic107⋊C4—21no—
47Dic107⋊C4—42no—
48C22⋊Dic107—21no—
49(C2×D107)⋊C4—21no—
50(C2×D107)⋊C4—21no—
51(C2×D107)⋊C4—42no—
52C2×Q8×C107C2×C411yescharacteristic
53C22×C4×C107—21no—
54C22×C4×C107C2311yescharacteristic
55C2×C8×C107—21no—
56C4⋊C4×C107—21no—
57C4⋊C4×C107C2×C411yescharacteristic
58C22⋊C4×C107—21no—
59C23×C107C2×D411yescharacteristic
60C22×D107—41no—
61C22×D107—41no—
62C22×D107C22⋊C411yescharacteristic
63C22×D107C22⋊C411yescharacteristic
64C22×D107SD1612yes—
65C22×D107Q1612yes—
66Q8×C107—41no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×Dic107—22no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107C22⋊C411yescharacteristic
82C2×Dic107C22⋊C411yescharacteristic
83C4×D107—22no—
84C4×D107—22no—
85C4×D107—41no—
86C4×D107—41no—
87C4×D107—42no—
88C4×D107—42no—
89Dic214—42no—
90Dic214—41no—
91C8×C107—41no—
92C107⋊C8—41no—
93C22×C107—21no—
94C22×C107—21no—
95C22×C107C2×C22⋊C411yescharacteristic
96C22×C107C2×SD1611yescharacteristic
97C22×C107C2×Q1611yescharacteristic
98C2×D107—21no—
99C2×D107—21no—
100C2×D107—21no—
101C2×D107—21no—
102C2×D107—42no—
103C2×D107—42no—
104C2×D107#10312yes—
105C2×D107#10312yes—
106C4×C107—21no—
107C4×C107—21no—
108C4×C107—41no—
109C4×C107—41no—
110C4×C107—81no—
111Dic107—21no—
112Dic107—41no—
113Dic107—41no—
114Dic107—81no—
115Dic107#9912yes—
116C2×C107—21no—
117C2×C107—21no—
118C2×C107C2×#9911yescharacteristic
119C2×C107C2×#10311yescharacteristic
120C2×C107#42111yescharacteristic
121D107—22no—
122D107—22no—
123#1520—1071nomaximal, Sylow
124C107#152011yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—1071no—
126C2×C22⋊C8—1071no—
127#380—1071no—
128#328—1071no—
129#328—1071no—
130#328—1071no—
131#328D10711yescharacteristic, complemented
132C22×Q8—1071no—
133C23×C4—1071no—
134C22×C8—2141no—
135C22⋊Q8—1071no—
136C22⋊Q8—1071no—
137C22⋊Q8—1071no—
138C22⋊Q8—2141no—
139C22⋊Q8—2141no—
140C22⋊Q8C2×D10711yescharacteristic
141C2×C4⋊C4—1071no—
142C2×C4⋊C4—2141no—
143C2×C4⋊C4—2141no—
144C2×C22⋊C4—2141no—
145C2×C22⋊C4—2141no—
146C22⋊C8—1071no—
147C22⋊C8—2141no—
148C22⋊C8C2×D10711yescharacteristic
149#95—1071no—
150#95C2×D10711yescharacteristic
151C24—1071no—
152C2×Q8—1071no—
153C2×Q8—4282no—
154C2×Q8—2141no—
155C2×Q8C4×D10711yescharacteristic
156C22×C4—21no—
157C22×C4—1071no—
158C22×C4—1071no—
159C22×C4—1071no—
160C22×C4—1071no—
161C22×C4—1071no—
162C22×C4—4282no—
163C22×C4—2141no—
164C22×C4—2141no—
165C22×C4—2141no—
166C22×C4C22×D10711yescharacteristic
167C2×C8—21no—
168C2×C8—4282no—
169C2×C8—2141no—
170C4⋊C4—21no—
171C4⋊C4—1071no—
172C4⋊C4—4282no—
173C4⋊C4—2141no—
174C4⋊C4—2141no—
175C4⋊C4—2141no—
176C4⋊C4—2141no—
177C4⋊C4C4×D10711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4282no—
180C22⋊C4—2141no—
181C22⋊C4—2141no—
182C22⋊C4—2141no—
183C23—1072no—
184C23—1072no—
185C23—1071no—
186C23—1071no—
187C23—4281no—
188C23—4281no—
189C23D4×D10711yescharacteristic
190Q8—41no—
191Q8—4281no—
192Q8—4282no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1071no—
200C2×C4—1071no—
201C2×C4—4281no—
202C2×C4—4281no—
203C2×C4—4282no—
204C2×C4—4282no—
205C2×C4—4281no—
206C2×C4—4281no—
207C2×C4—4281no—
208C2×C4—2142no—
209C2×C4—2142no—
210C2×C4—2142no—
211C2×C4—2141no—
212C2×C4—2141no—
213C2×C4C2×C4×D10711yescharacteristic
214C2×C4D4×D10711yescharacteristic
215C8—41no—
216C8—4281no—
217C22—21no—
218C22—21no—
219C22—1072no—
220C22—1072no—
221C22—4282no—
222C22—4282no—
223C22—2141no—
224C22—2141no—
225C22—2141no—
226C22—2141no—
227C22C22⋊C4×D10711yescharacteristic
228C22SD16×D10711yescharacteristic
229C22Q16×D10711yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—8561no—
236C4—1072no—
237C4—4281no—
238C4—4281no—
239C4—2141no—
240C2—21no—
241C2—21no—
242C2—2142no—
243C2—2142no—
244C2#99×D10711yescharacteristic
245C2#103×D10711yescharacteristic
246C2#2029264211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#328×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC23×D107D4
cosocleC22×C4×C107C23
hypercenter#328D107
hypocenterC107#1520
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #20292365C2semidirect
5. #20292364C2semidirect
6. #20292363C2semidirect
7. #328×C107C2semidirect
8. C2×Q8×D107C4semidirect
11. #16812089C4semidirect
15. #16812085C4semidirect
18. C4⋊C4×D107C4semidirect
124. C107#1520semidirect
131. #328D107semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1072
C22⋊C8×D1072
#202924862
#202923652
#202923642
#202923632
#328×C1072
#1520107