Finite Group Databasebeta

← #21429082

Subgroups — #21429082

Order 7552.

540 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#14273725C211yesmaximal, characteristic
2C2×#14273725C211yesmaximal, characteristic
3C2×Q8×Dic59C211yesmaximal, characteristic
4#16837335C214yesmaximal
5#16837309C211yesmaximal, characteristic
6#16837308C211yesmaximal, characteristic
7#16837297C214yesmaximal
8C4×Dic59⋊C4C211yesmaximal, characteristic
9C2×C4⋊Q8×C59C211yesmaximal, characteristic
10C22×Dic118C2211yescharacteristic
11#14273725—24no—
12#14273725—24no—
13Q8×Dic59—24no—
14C2×C4⋊Dic59C2211yescharacteristic
15C2×C4⋊Dic59C2212yes—
16C2×Dic59⋊C4C2212yes—
17C2×Dic59⋊C4C2214yes—
18C2×Dic59⋊C4C2212yes—
19C2×Dic59⋊C4C2212yes—
20C2×C4×Dic59C2211yescharacteristic
21C2×C4×Dic59C2212yes—
22C2×C4×Dic59C2211yescharacteristic
23C59⋊#95C2212yes—
24C59⋊#95C2214yes—
25C59⋊#95C2214yes—
26C22×Q8×C59C2211yescharacteristic
27C22×Q8×C59C2211yescharacteristic
28C4⋊Q8×C59—24no—
29C2×C4⋊C4×C59C2214yes—
30C2×C42×C59C2211yescharacteristic
31C2×Q8×C59—22no—
32C2×Q8×C59—22no—
33C2×Q8×C59—22no—
34C2×Q8×C59—22no—
35C2×Q8×C59—22no—
36C2×Q8×C59D414yes—
37C22×C4×C59C2312yes—
38C22×C4×C59C2312yes—
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×Dic59C2312yes—
42C22×Dic59C2312yes—
43C22×Dic59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2311yescharacteristic
46C2×Dic118—22no—
47C2×Dic118—22no—
48C2×Dic118—22no—
49C4⋊C4×C59—28no—
50C42×C59—22no—
51C4⋊Dic59—22no—
52C4⋊Dic59—24no—
53Dic59⋊C4—28no—
54Dic59⋊C4—24no—
55Dic59⋊C4—24no—
56Dic59⋊C4Q818yescomplemented
57C4×Dic59—22no—
58C4×Dic59—24no—
59C4×Dic59D414yes—
60C23×C59C2411yescharacteristic
61Q8×C59—28no—
62Q8×C59—44no—
63C2×C4×C59—22no—
64C2×C4×C59—22no—
65C2×C4×C59—22no—
66C2×C4×C59—22no—
67C2×C4×C59—22no—
68C2×C4×C59—22no—
69C2×C4×C59—22no—
70C2×C4×C59C2×D412yes—
71C2×C4×C59C2×D414yes—
72C2×C4×C59C2×D412yes—
73C2×C4×C59C2×Q814yes—
74C2×C4×C59#4112yes—
75C2×Dic59—22no—
76C2×Dic59—22no—
77C2×Dic59—22no—
78C2×Dic59—22no—
79C2×Dic59—22no—
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—24no—
83C2×Dic59C2×D412yes—
84C2×Dic59C2×D412yes—
85C2×Dic59C2×Q814yes—
86C2×Dic59C2×Q814yes—
87Dic118—44no—
88C22×C59C22×D411yescharacteristic
89C22×C59C22×D411yescharacteristic
90C22×C59C22×Q812yes—
91C22×C59C2×#4111yescharacteristic
92C22×C592−1+412yes—
93C4×C59—24no—
94C4×C59—24no—
95C4×C59—42no—
96C4×C59#12114yes—
97Dic59—22no—
98Dic59—24no—
99Dic59—42no—
100Dic59—42no—
101Dic59C4⋊Q814yescomplemented
102#2546—591nomaximal, Sylow
103C2×C59C2×#12111yescharacteristic
104C2×C59C2×C4⋊Q811yescharacteristic
105C2×C59#53611yescharacteristic
106C2×C59D4×Q812yes—
107C2×C59#55712yes—
108C2×C4⋊Q8—591no—
109C2×C4⋊Q8—591no—
110C2×C4⋊Q8D5911yescharacteristic
111C2×C4×Q8—591no—
112#398—594no—
113#391—591no—
114#391—591no—
115#389—594no—
116C4×C4⋊C4—591no—
117C59#254611yescharacteristic, complemented, Sylow
118C22×Q8—591no—
119C22×Q8C2×D5911yescharacteristic
120C22×Q8C2×D5911yescharacteristic
121C4⋊Q8—24no—
122C4⋊Q8—1184no—
123C4⋊Q8—1184no—
124C4×Q8—1184no—
125C2×C4⋊C4—592no—
126C2×C4⋊C4—591no—
127C2×C4⋊C4—592no—
128C2×C4⋊C4—592no—
129C2×C4⋊C4—594no—
130C2×C4⋊C4—592no—
131C2×C4⋊C4C2×D5914yes—
132C2×C42—591no—
133C2×C42—592no—
134C2×C42—591no—
135C2×C42C2×D5911yescharacteristic
136#95—592no—
137#95—594no—
138#95—594no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—1182no—
145C2×Q8—1182no—
146C2×Q8—1182no—
147C2×Q8C22⋊D5914yes—
148C22×C4—591no—
149C22×C4—592no—
150C22×C4—591no—
151C22×C4—592no—
152C22×C4—592no—
153C22×C4C22×D5912yes—
154C22×C4C22×D5912yes—
155C22×C4C22×D5912yes—
156C22×C4C22×D5911yescharacteristic
157C4⋊C4—28no—
158C4⋊C4—598no—
159C4⋊C4—1184no—
160C4⋊C4—1184no—
161C4⋊C4—1184no—
162C4⋊C4—1182no—
163C4⋊C4—1188no—
164C42—22no—
165C42—594no—
166C42—1184no—
167C42—1182no—
168C23C23×D5911yescharacteristic
169Q8—28no—
170Q8—44no—
171Q8—2364no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—592no—
180C2×C4—594no—
181C2×C4—592no—
182C2×C4—594no—
183C2×C4—1182no—
184C2×C4—1182no—
185C2×C4—1182no—
186C2×C4—1182no—
187C2×C4—1182no—
188C2×C4—1182no—
189C2×C4—1182no—
190C2×C4—1184no—
191C2×C4D4×D5912yes—
192C2×C4#1174568712yes—
193C2×C4Q8×D5914yes—
194C2×C4C2×C22⋊D5914yes—
195C2×C4C2×C22⋊D5912yes—
196C22C2×D4×D5911yescharacteristic
197C22C22×C22⋊D5911yescharacteristic
198C22C2×Q8×D5912yes—
199C22C2×#1174568711yescharacteristic
200C22#1427377512yes—
201C4—24no—
202C4—24no—
203C4—42no—
204C4—594no—
205C4—1182no—
206C4—1184no—
207C4—2362no—
208C4—2362no—
209C4#1427369914yes—
210C2C2×#1427369911yescharacteristic
211C2C4⋊Q8×D5911yescharacteristic
212C2#1683797711yescharacteristic
213C2Q8×C22⋊D5912yes—
214C2#1683812712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×C4⋊Q8×C59C2
hypercenterC2×C4⋊Q8D59
hypocenterC59#2546
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
56. Dic59⋊C4Q8semidirect
101. Dic59C4⋊Q8semidirect
117. C59#2546semidirect