Finite Group Databasebeta

← #29600371

Subgroups — #29600371

Order 15616.

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

normal subgroups only

This group has 318 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#21456166C211yesmaximal, characteristic
2#21447503C212yesmaximal
3#21443992C211yesmaximal, characteristic, complemented
4#21443991C212yesmaximal, complemented
5C61×#1413C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic
7#16840313C2211yescharacteristic
8#16840130C2211yescharacteristic
9#16840128C411yescharacteristic
10#16840127C411yescharacteristic
11#16840123—22no—
12#16839757—22no—
13#16839745—22no—
14#97×D61C411yescharacteristic
15#16839561C2212yes—
16#16839558—22no—
17C61×#491C2211yescharacteristic
18C61×#356C2212yes—
19C2×#11758226D412yes—
20M16×D61—42no—
21C2×C8×D61—21no—
22#14519776—21no—
23#14519774C2×C411yescharacteristic
24#14519769C2×C411yescharacteristic
25#14519766D412yes—
26#14519755—41no—
27#14519752—41no—
28#14519742—41no—
29#14519739—41no—
30#14519735—21no—
31C42×D61C2×C411yescharacteristic
32#14519701—22no—
33#14519699—22no—
34#14519697—22no—
35#14519696—22no—
36#14519693—42no—
37#14519688C2×C411yescharacteristic
38#14519687—21no—
39C4⋊Q8×C61C2311yescharacteristic
40#125×C61C2×C411yescharacteristic
41#103×C61—22no—
42#101×C61—22no—
43#97×C61C2×C411yescharacteristic
44C2×Q8×C61C2×D412yes—
45#11758226—42no—
46#11758226—42no—
47C2×D244—22no—
48C2×D244—41no—
49C2×D244C22⋊C412yes—
50C2×C4×D61—22no—
51C2×C4×D61—21no—
52C2×C4×D61C22⋊C411yescharacteristic
53C2×Dic122—41no—
54M16×C61—42no—
55C2×C8×C61—21no—
56C4⋊C4×C61—21no—
57C4⋊C4×C61—41no—
58C4⋊C4×C61—41no—
59C42×C61C22×C411yescharacteristic
60#11758198—42no—
61#11758198—41no—
62C4⋊Dic61—21no—
63#11758196—41no—
64#11758196—41no—
65C4×Dic61—21no—
66C4×Dic61—41no—
67C4×Dic61C22⋊C411yescharacteristic
68#11758194—42no—
69C2×#92418—21no—
70#11758189—82no—
71C8×D61—41no—
72C8×D61—41no—
73C8×D61—82no—
74C22×D61—22no—
75C22×D61#9911yescharacteristic
76Q8×C61—42no—
77C2×C4×C61—22no—
78C2×C4×C61—21no—
79C2×C4×C61—41no—
80C2×C4×C61C2×C22⋊C411yescharacteristic
81C2×Dic61—21no—
82C2×Dic61—41no—
83C2×Dic61#9911yescharacteristic
84D244—42no—
85D244—82no—
86D244—82no—
87C4×D61—21no—
88C4×D61—21no—
89C4×D61—42no—
90C4×D61—42no—
91C4×D61—81no—
92Dic122—82no—
93C8×C61—41no—
94C8×C61—82no—
95#92418—41no—
96#92418—82no—
97#13598—611nomaximal, Sylow
98C22×C61C2×#9911yescharacteristic
99C2×D61—21no—
100C2×D61—41no—
101C2×D61—42no—
102C2×D61—82no—
103C4×C61—21no—
104C4×C61—41no—
105C4×C61—42no—
106C4×C61—81no—
107Dic61—41no—
108Dic61—81no—
109Dic61#35411yescharacteristic
110Dic61#35411yescharacteristic
111#3135—611no—
112#2132—612no—
113#1413—611no—
114#1413—612no—
115#1413D6111yescharacteristic, complemented
116C2×C61—21no—
117C2×C61C2×#35411yescharacteristic
118D61—41no—
119D61—82no—
120#533—611no—
121#532—611no—
122#491—611no—
123#491C2×D6111yescharacteristic
124#490—611no—
125#490—611no—
126#489—1222no—
127#416—1222no—
128#413—1222no—
129C2×#97—611no—
130#356—612no—
131#356C2×D6112yes—
132#355—1222no—
133C61#1359811yescharacteristic, complemented, Sylow
134C2×#41—612no—
135C2×M16—2442no—
136C22×C8—1221no—
137C4⋊Q8—1221no—
138C4⋊Q8C22×D6111yescharacteristic
139#127—611no—
140#125—611no—
141#125C4×D6111yescharacteristic
142#124—612no—
143C22⋊Q8—2441no—
144#121—2441no—
145C4×Q8—2441no—
146C4×D4—2441no—
147#117—1221no—
148C2×C42—611no—
149#103—22no—
150#103—1222no—
151#102—1222no—
152#102—1222no—
153#101—22no—
154#101—1222no—
155#100—2442no—
156#97—611no—
157#97—1221no—
158#97C4×D6111yescharacteristic
159#41—2442no—
160#41—2442no—
161C2×Q8—2441no—
162C2×Q8D4×D6112yes—
163C2×D4—612no—
164C2×D4—1222no—
165C2×D4—2441no—
166C22×C4—611no—
167C22×C4—1222no—
168C22×C4—1221no—
169M16—42no—
170M16—4882no—
171M16—2442no—
172C2×C8—21no—
173C2×C8—1221no—
174C2×C8—4882no—
175C2×C8—2441no—
176C2×C8—2441no—
177C4⋊C4—21no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—1221no—
181C4⋊C4—2441no—
182C4⋊C4—2441no—
183C22⋊C4—2442no—
184C22⋊C4—2441no—
185C42—611no—
186C42—1221no—
187C42—2441no—
188C42C2×C4×D6111yescharacteristic
189C23—611no—
190C23—1222no—
191Q8—42no—
192Q8—4882no—
193D4—4882no—
194D4—4882no—
195D4—2442no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—611no—
200C2×C4—1221no—
201C2×C4—1221no—
202C2×C4—1221no—
203C2×C4—4881no—
204C2×C4—2442no—
205C2×C4—2442no—
206C2×C4—2441no—
207C2×C4C22⋊C4×D6111yescharacteristic
208C8—41no—
209C8—82no—
210C8—4882no—
211C8—2441no—
212C22—1221no—
213C22—4882no—
214C22—2441no—
215C22—2442no—
216C22#99×D6111yescharacteristic
217C4—21no—
218C4—41no—
219C4—42no—
220C4—81no—
221C4—611no—
222C4—611no—
223C4—4881no—
224C4—2441no—
225C2—21no—
226C2—4882no—
227C2—2441no—
228C2#354×D6111yescharacteristic
subgroupG/H
centerC2#354×D61
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#1413C2
socleC2×C61C2×#354
prime radical#16840313C22
cosocleC4⋊Q8×C61C23
hypercenter#1413D61
hypocenterC61#13598
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. #21443992C2semidirect
4. #21443991C2semidirect
5. C61×#1413C2semidirect
115. #1413D61semidirect
133. C61#13598semidirect

Maximal subgroups

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

subgroupindex
#214561662
#214475032
#214439922
#214439912
C61×#14132
#1359861