Finite Group Databasebeta

← #24913586

Subgroups — #24913586

Order 10624.

448 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14581791C211yesmaximal, characteristic
2C2×#14581782C211yesmaximal, characteristic
3#19127633C211yesmaximal, characteristic
4#19127632C211yesmaximal, characteristic
5#19127631C211yesmaximal, characteristic
6#19127622C212yesmaximal
7#19127621C212yesmaximal
8#19127570C212yesmaximal
9#19127565C212yesmaximal
10#19127546C211yesmaximal, characteristic
11#398×C83C211yesmaximal, characteristic
12C22×Dic166C2212yes—
13#14581791—24no—
14#14581782—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2212yes—
24C2×Dic83⋊C4C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C2×C4×Dic83C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C2×Dic166—24no—
38C2×Dic166—42no—
39C2×Dic166—42no—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C22×Dic83C2311yescharacteristic
46C4⋊Dic83—22no—
47C4⋊Dic83—22no—
48C4⋊Dic83—42no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—24no—
54Dic83⋊C4—42no—
55Dic83⋊C4—42no—
56C4×Dic83D414yes—
57C4×Dic83Q818yescomplemented
58C22×C4×C83C2312yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C4⋊C4×C83—42no—
66C23×C83C2411yescharacteristic
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—21no—
71C2×C4×C83—21no—
72C2×C4×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D412yes—
79C2×C4×C83C2×Q814yes—
80C2×Dic83—22no—
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—22no—
84C2×Dic83—21no—
85C2×Dic83—21no—
86C2×Dic83—22no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83C2×D412yes—
93C2×Dic83C2×D412yes—
94C2×Dic83C2×Q814yes—
95C2×Dic83C2×Q814yes—
96Dic166—48no—
97C22×C83C22×D411yescharacteristic
98C22×C83C22×Q812yes—
99C22×C83H(5,𝔽2)11yescharacteristic
100C22×C83H(5,𝔽2)11yescharacteristic
101C22×C832−1+412yes—
102C4×C83—22no—
103C4×C83—24no—
104C4×C83—42no—
105C4×C83—41no—
106C4×C83—41no—
107Dic83—22no—
108Dic83—24no—
109Dic83—42no—
110Dic83—41no—
111Dic83—41no—
112Dic83C4⋊Q814yes—
113C2×C83C2×C4⋊Q811yescharacteristic
114C2×C83#53511yescharacteristic
115C2×C83#53611yescharacteristic
116C2×C83#54412yes—
117C2×C83#54412yes—
118#2660—831nomaximal, Sylow
119C83#266011yescharacteristic, complemented, Sylow
120C2×C4⋊Q8—831no—
121C2×#125—831no—
122#398—831no—
123#398—831no—
124#398—831no—
125#398D8311yescharacteristic
126#395—832no—
127#395—832no—
128#384—832no—
129#382—832no—
130#376—831no—
131C22×Q8—832no—
132C4⋊Q8—1664no—
133#125—1664no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—831no—
138C2×C4⋊C4—831no—
139C2×C4⋊C4—831no—
140C2×C4⋊C4—831no—
141C2×C4⋊C4—831no—
142C2×C4⋊C4—832no—
143C2×C4⋊C4—831no—
144C2×C4⋊C4C2×D8311yescharacteristic
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C4⋊C4C2×D8312yes—
147C2×C42—831no—
148C2×C42—832no—
149#95—832no—
150#95—831no—
151#95—832no—
152#95—832no—
153#95—832no—
154#95C2×D8311yescharacteristic
155#95C2×D8312yes—
156C2×Q8—3322no—
157C2×Q8—3322no—
158C2×Q8—1664no—
159C22×C4—831no—
160C22×C4—832no—
161C22×C4—831no—
162C22×C4—831no—
163C22×C4—832no—
164C22×C4—831no—
165C22×C4C22×D8311yescharacteristic
166C22×C4C22×D8312yes—
167C22×C4C22×D8311yescharacteristic
168C22×C4C22×D8311yescharacteristic
169C22×C4C22×D8312yes—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—3322no—
174C4⋊C4—3322no—
175C4⋊C4—3322no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1664no—
181C4⋊C4—1662no—
182C4⋊C4—1662no—
183C42—834no—
184C42—838no—
185C23C23×D8311yescharacteristic
186Q8—3328no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—832no—
199C2×C4—834no—
200C2×C4—832no—
201C2×C4—834no—
202C2×C4—1661no—
203C2×C4—1661no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4—1662no—
207C2×C4—1661no—
208C2×C4—1662no—
209C2×C4—1662no—
210C2×C4—1661no—
211C2×C4—1662no—
212C2×C4—1661no—
213C2×C4—1662no—
214C2×C4D4×D8312yes—
215C2×C4Q8×D8314yes—
216C22C2×D4×D8311yescharacteristic
217C22C2×Q8×D8312yes—
218C22#1458183711yescharacteristic
219C22#1458183811yescharacteristic
220C22#1458183912yes—
221C4—24no—
222C4—22no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—3322no—
227C4—3321no—
228C4—3321no—
229C4—834no—
230C4—1664no—
231C4—1662no—
232C2C4⋊Q8×D8311yescharacteristic
233C2#1912826711yescharacteristic
234C2#1912827311yescharacteristic
235C2#1912833012yes—
236C2#1912833112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#398×C83C2
hypercenter#398D83
hypocenterC83#2660
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
57. C4×Dic83Q8semidirect
119. C83#2660semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817822
#191276332
#191276322
#191276312
#191276222
#191276212
#191275702
#191275652
#191275462
#398×C832
#266083