Finite Group Databasebeta

← #29600346

Subgroups — #29600346

Order 15616.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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#21456273C211yesmaximal, characteristic
2#356×D61C212yesmaximal, complemented
3#21443987C212yesmaximal, complemented
4#21443986C211yesmaximal, characteristic, complemented
5C61×#1410C211yesmaximal, characteristic, complemented
6C2×#14519806—22no—
7#16840310C411yescharacteristic
8C4⋊Q8×D61C2211yescharacteristic
9#16840153—22no—
10#16840142C411yescharacteristic
11#16840141C2211yescharacteristic
12#16840134C411yescharacteristic
13#101×D61—22no—
14C4×#11758189C411yescharacteristic
15#16839561C2212yes—
16#16839560—22no—
17C61×#492C2211yescharacteristic
18C61×#356C2212yes—
19C2×#11758224—21no—
20C2×Q8×D61D412yes—
21#14519806—44no—
22#14519806—44no—
23C2×Dic244—22no—
24C2×#11758202—22no—
25C2×#11758200—22no—
26C2×#11758189—21no—
27M16×D61—42no—
28#14519777C2×C411yescharacteristic
29#14519777C2×C411yescharacteristic
30#14519776—41no—
31#14519775D412yes—
32#14519764—21no—
33#14519748—41no—
34D4×Dic61—41no—
35C4⋊C4×D61—41no—
36C42×D61C2×C411yescharacteristic
37#14519696—22no—
38#14519695—42no—
39#14519687—21no—
40C4×#92418C2×C411yescharacteristic
41C2×SD16×C61—22no—
42C4⋊Q8×C61C2311yescharacteristic
43#127×C61C2×C411yescharacteristic
44#101×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C2×Q8×C61C2×D412yes—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×C22⋊D61—41no—
50C22×Dic61—41no—
51Q8×D61—42no—
52Q8×D61—42no—
53#11758224—42no—
54#11758224—42no—
55C2×C4×D61—22no—
56C2×C4×D61—21no—
57C2×C4×D61C22⋊C411yescharacteristic
58C2×Dic122—21no—
59C2×Dic122—22no—
60C2×Dic122C22⋊C412yes—
61SD16×C61—44no—
62M16×C61—42no—
63C2×C8×C61—21no—
64C4⋊C4×C61—41no—
65C42×C61C22×C411yescharacteristic
66C22⋊Dic61—42no—
67#11758202—44no—
68#11758200—44no—
69C4⋊Dic61—41no—
70C4⋊Dic61—41no—
71#11758196—42no—
72C4×Dic61—21no—
73C4×Dic61C22⋊C411yescharacteristic
74#11758194—42no—
75C2×#92418—21no—
76Dic244—44no—
77#11758189—24no—
78#11758189—82no—
79C8×D61—82no—
80C23×C61—41no—
81C22×D61#9911yescharacteristic
82Q8×C61—42no—
83D4×C61—42no—
84D4×C61—82no—
85C2×C4×C61—21no—
86C2×C4×C61—22no—
87C2×C4×C61C2×C22⋊C411yescharacteristic
88C22⋊D61—82no—
89C2×Dic61—21no—
90C2×Dic61—22no—
91C2×Dic61—41no—
92C2×Dic61—81no—
93C2×Dic61—81no—
94C2×Dic61#9911yescharacteristic
95C4×D61—22no—
96C4×D61—42no—
97C4×D61—42no—
98C4×D61—42no—
99Dic122—42no—
100Dic122—42no—
101Dic122—82no—
102C8×C61—22no—
103C8×C61—82no—
104#92418—22no—
105#92418—82no—
106#13590—611nomaximal, Sylow
107C22×C61—82no—
108C22×C61—81no—
109C22×C61C2×#9911yescharacteristic
110C2×D61—41no—
111C2×D61#35312yes—
112C4×C61—22no—
113C4×C61—21no—
114C4×C61—42no—
115Dic61—21no—
116Dic61—41no—
117Dic61—42no—
118Dic61—81no—
119#3152—611no—
120C2×#356—612no—
121#1411—612no—
122#1410—611no—
123#1410D6111yescharacteristic, complemented
124C2×C61—21no—
125C2×C61—161no—
126C2×C61C2×#35311yescharacteristic
127D61—22no—
128C2×#137—1222no—
129#532—611no—
130C2×C4⋊Q8—611no—
131#497—1222no—
132#494—611no—
133#494—611no—
134#492—611no—
135#492C2×D6111yescharacteristic
136C2×#101—1222no—
137C4×M16—611no—
138#356—612no—
139#356—1222no—
140#356C2×D6112yes—
141C61#1359011yescharacteristic, complemented, Sylow
142C2×#41—1221no—
143C22×Q8—612no—
144#137—2444no—
145#137—2444no—
146C2×Q16—1222no—
147C2×Q16—1222no—
148C2×SD16—22no—
149C2×SD16—1222no—
150C2×M16—1221no—
151C2×M16—2442no—
152C4⋊Q8—612no—
153C4⋊Q8—611no—
154C4⋊Q8—611no—
155C4⋊Q8—2441no—
156C4⋊Q8C22×D6111yescharacteristic
157#127C4×D6111yescharacteristic
158#124—1221no—
159#121—2441no—
160C4×D4—2441no—
161C2×C4⋊C4—2441no—
162C2×C42—611no—
163#101—22no—
164#101—1222no—
165#101—2442no—
166#97—1221no—
167C4×C8—611no—
168C4×C8C4×D6111yescharacteristic
169#41—2442no—
170#41—2442no—
171C2×Q8—612no—
172C2×Q8—1222no—
173C2×Q8—1221no—
174C2×Q8—2442no—
175C2×Q8—2442no—
176C2×Q8D4×D6112yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—2441no—
180C22×C4—611no—
181C22×C4—1221no—
182C22×C4—1222no—
183C22×C4—2441no—
184Q16—2444no—
185Q16—2444no—
186SD16—44no—
187SD16—2444no—
188M16—42no—
189M16—1224no—
190M16—4882no—
191M16—2442no—
192C2×C8—21no—
193C2×C8—1221no—
194C2×C8—4882no—
195C4⋊C4—41no—
196C4⋊C4—2442no—
197C4⋊C4—2441no—
198C4⋊C4—2441no—
199C22⋊C4—2442no—
200C42—611no—
201C42—1221no—
202C42C2×C4×D6111yescharacteristic
203C23—41no—
204C23—611no—
205Q8—42no—
206Q8—4882no—
207Q8—2442no—
208Q8—2442no—
209D4—42no—
210D4—82no—
211D4—4882no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—611no—
215C2×C4—1222no—
216C2×C4—1221no—
217C2×C4—1222no—
218C2×C4—4881no—
219C2×C4—4881no—
220C2×C4—2442no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4—2441no—
224C2×C4C22⋊C4×D6111yescharacteristic
225C8—22no—
226C8—82no—
227C8—1222no—
228C8—4882no—
229C22—82no—
230C22—81no—
231C22—612no—
232C22—2441no—
233C22#99×D6111yescharacteristic
234C4—22no—
235C4—21no—
236C4—42no—
237C4—1221no—
238C4—4881no—
239C4—2441no—
240C4—2442no—
241C2—21no—
242C2—161no—
243C2—1222no—
244C2#353×D6111yescharacteristic
subgroupG/H
centerC2#353×D61
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#1410C2
socleC2×C61C2×#353
prime radical#16840310C4
cosocleC4⋊Q8×C61C23
hypercenter#1410D61
hypocenterC61#13590
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1410

Complements

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

NH ≅ G/Ntype
2. #356×D61C2semidirect
3. #21443987C2semidirect
4. #21443986C2semidirect
5. C61×#1410C2semidirect
123. #1410D61semidirect
141. C61#13590semidirect

Maximal subgroups

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

subgroupindex
#214562732
#356×D612
#214439872
#214439862
C61×#14102
#1359061