Finite Group Databasebeta

← #23773205

Subgroups — #23773205

Order 10112.

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

normal subgroups only

This group has 376 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
1#19099120C211yesmaximal, characteristic
2#19098870C214yesmaximal, complemented
3#19098858C212yesmaximal, complemented
4#19098848C212yesmaximal, complemented
5#19098603C212yesmaximal
6#19098587C211yesmaximal, characteristic
7#19098572C212yesmaximal
8C2×C4×Q8×C79C211yesmaximal, characteristic
9C2×#11929608C2212yes—
10#14579161C2211yescharacteristic
11#14579161C2211yescharacteristic
12#14579153C2212yes—
13#14579140—21no—
14#14579140—21no—
15C4×Dic158—21no—
16C4×Dic158—21no—
17#14579120C2212yes—
18#14579094C2214yes—
19#14579092C2214yes—
20C4⋊(C79⋊C8)C2212yes—
21C4⋊(C79⋊C8)C2212yes—
22#14579085C2214yes—
23#14579085C2214yes—
24C22×Q8×C79C2211yescharacteristic
25C4×Q8×C79—21no—
26C4×Q8×C79—21no—
27C4×Q8×C79C2212yes—
28C4×Q8×C79C2212yes—
29C2×C4⋊C4×C79—21no—
30C2×C4⋊C4×C79C2211yescharacteristic
31C2×C42×C79—21no—
32C2×C42×C79C2211yescharacteristic
33C2×Dic158—21no—
34C2×Dic158—21no—
35#11929608—24no—
36C2×C79⋊C8C2314yes—
37C4⋊Dic79—21no—
38C4⋊Dic79—21no—
39C4⋊Dic79C2312yes—
40C4⋊Dic79C2312yes—
41Dic79⋊C4—22no—
42Dic79⋊C4—22no—
43C22⋊Dic79—22no—
44C4×Dic79—22no—
45C2×Q8×C79—21no—
46C2×Q8×C79—22no—
47C2×Q8×C79Q814yescomplemented
48C2×Q8×C79C2312yes—
49C22×C4×C79—21no—
50C22×C4×C79—21no—
51C22×C4×C79D411yescharacteristic
52C22×C4×C79D411yescharacteristic
53C22×C4×C79C2311yescharacteristic
54C4⋊C4×C79—22no—
55C4⋊C4×C79—22no—
56C4⋊C4×C79—21no—
57C4⋊C4×C79C2312yes—
58C42×C79—22no—
59C42×C79—22no—
60C42×C79D411yescharacteristic
61C42×C79D411yescharacteristic
62C42×C79C2311yescharacteristic
63C42×C79C2311yescharacteristic
64C23×C79C2×D411yescharacteristic
65Q8×C79—24no—
66Q8×C79—22no—
67Q8×C79C2×Q814yes—
68C2×C4×C79—24no—
69C2×C4×C79—22no—
70C2×C4×C79—22no—
71C2×C4×C79—22no—
72C2×C4×C79—41no—
73C2×C4×C79—42no—
74C2×C4×C79C2×D411yescharacteristic
75C2×C4×C79C2×D411yescharacteristic
76C2×C4×C79C2×D411yescharacteristic
77C2×C4×C79C2×D411yescharacteristic
78C2×C4×C79C2×D411yescharacteristic
79C2×C4×C79C2×Q812yes—
80C2×C4×C79#4112yes—
81C2×C4×C79C2411yescharacteristic
82C2×Dic79—22no—
83C2×Dic79—22no—
84Dic158—42no—
85Dic158—42no—
86C79⋊C8—24no—
87C22×C79—21no—
88C22×C79—21no—
89C22×C79C22⋊Q812yes—
90C22×C79C22×D411yescharacteristic
91C4×C79—24no—
92C4×C79—21no—
93C4×C79—21no—
94C4×C79—21no—
95C4×C79—42no—
96C4×C79C22⋊Q812yes—
97C4×C79#13712yes—
98C4×C79C22×Q811yescharacteristic
99C4×C79C2×#4111yescharacteristic
100Dic79—42no—
101Dic79—42no—
102C2×C79—22no—
103C2×C79C2×C22⋊Q811yescharacteristic
104C2×C79C2×#13711yescharacteristic
105C2×C79#57511yescharacteristic
106#3078—791nomaximal, Sylow
107C79#307811yescharacteristic, complemented, Sylow
108#533—791no—
109C2×C4×Q8D7911yescharacteristic
110#479—794no—
111#477—792no—
112#475—792no—
113#428—792no—
114#423—791no—
115#419—792no—
116C22×Q8C2×D7911yescharacteristic
117C2×M16—792no—
118C4⋊Q8—791no—
119C4⋊Q8—791no—
120#125—792no—
121C22⋊Q8—1581no—
122C22⋊Q8—1581no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—1581no—
126C4×Q8—1581no—
127C4×Q8C2×D7912yes—
128C4×Q8C2×D7912yes—
129#117—792no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C2×D7911yescharacteristic
132C2×C42—21no—
133C2×C42C2×D7911yescharacteristic
134#107—794no—
135#106—794no—
136C4⋊C8—792no—
137C4⋊C8—792no—
138#103—794no—
139#103—794no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—1581no—
143C2×Q8—1581no—
144C2×Q8Dic15814yescomplemented
145C2×Q8C22×D7912yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22⋊D7911yescharacteristic
149C22×C4C22⋊D7911yescharacteristic
150C22×C4C22×D7911yescharacteristic
151M16—1584no—
152C2×C8—794no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—792no—
157C4⋊C4—792no—
158C4⋊C4—1582no—
159C4⋊C4—1582no—
160C4⋊C4—1581no—
161C4⋊C4—1581no—
162C4⋊C4C22×D7912yes—
163C22⋊C4—1582no—
164C42—22no—
165C42—22no—
166C42—1582no—
167C42C22⋊D7911yescharacteristic
168C42C22⋊D7911yescharacteristic
169C42C22×D7911yescharacteristic
170C42C22×D7911yescharacteristic
171C23C2×C22⋊D7911yescharacteristic
172Q8—22no—
173Q8—24no—
174Q8—3162no—
175Q8—3162no—
176Q8C2×Dic15814yes—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—1582no—
184C2×C4—1582no—
185C2×C4C2×C22⋊D7911yescharacteristic
186C2×C4C2×C22⋊D7911yescharacteristic
187C2×C4C2×C22⋊D7911yescharacteristic
188C2×C4C2×C22⋊D7911yescharacteristic
189C2×C4C2×C22⋊D7911yescharacteristic
190C2×C4C2×Dic15812yes—
191C2×C4#1192962412yes—
192C2×C4C23×D7911yescharacteristic
193C8—1584no—
194C22—21no—
195C22—21no—
196C22#1457914012yes—
197C22C22×C22⋊D7911yescharacteristic
198C4—24no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—3162no—
204C4—3162no—
205C4#1457914012yes—
206C4#1457919212yes—
207C4C22×Dic15811yescharacteristic
208C4C2×#1192962411yescharacteristic
209C2—22no—
210C2C2×#1457914011yescharacteristic
211C2C2×#1457919211yescharacteristic
212C2#1909936111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
FittingC2×C4×Q8×C79C2
socleC22×C79C22×D4
prime radicalC23×C79C2×D4
hypercenterC2×C4×Q8D79
hypocenterC79#3078
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #19098870C2semidirect
3. #19098858C2semidirect
4. #19098848C2semidirect
47. C2×Q8×C79Q8semidirect
107. C79#3078semidirect
144. C2×Q8Dic158semidirect