Finite Group Databasebeta

← #37509213

Subgroups — #37509213

Order 22912.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#26038916C211yesmaximal, characteristic
2#26038479C211yesmaximal, characteristic
3C2×#19153031C211yesmaximal, characteristic
4#26038344C211yesmaximal, characteristic
5#26038200C211yesmaximal, characteristic
6#26038198C211yesmaximal, characteristic
7#428×C179C211yesmaximal, characteristic
8C22×Dic358C2211yescharacteristic
9C2×#15668243C2211yescharacteristic
10C4×Dic358—21no—
11C4×Dic358—21no—
12#19153078—21no—
13#19153078—21no—
14#19153072C2211yescharacteristic
15#19153069—22no—
16C2×C4×Dic179C2211yescharacteristic
17#19153046—21no—
18#19153044—21no—
19#19153031C414yescomplemented
20C179⋊#95—22no—
21C2×M16×C179C2211yescharacteristic
22#117×C179C2211yescharacteristic
23C2×C4⋊C4×C179C2211yescharacteristic
24#107×C179—21no—
25#106×C179—21no—
26C2×Dic358—21no—
27C2×Dic358—21no—
28C2×Dic358—21no—
29C2×Dic358—21no—
30C2×Dic358C2×C412yes—
31C2×Dic358Q812yes—
32C22×Dic179—21no—
33C22×Dic179D412yes—
34#15668243—21no—
35#15668243C2×C412yes—
36C2×C179⋊C8—21no—
37C4⋊Dic179D411yescharacteristic
38C4⋊Dic179D411yescharacteristic
39Dic179⋊C4—22no—
40Dic179⋊C4—22no—
41C22⋊Dic179—21no—
42C4×Dic179—22no—
43C4×Dic179—22no—
44C4×Dic179—21no—
45C4×Dic179—21no—
46C4×Dic179—21no—
47C22×C4×C179—21no—
48C22×C4×C179C2311yescharacteristic
49M16×C179—21no—
50M16×C179C2×C412yes—
51C2×C8×C179—21no—
52C4⋊C4×C179—21no—
53C4⋊C4×C179—21no—
54C4⋊C4×C179D411yescharacteristic
55C4⋊C4×C179D411yescharacteristic
56C22⋊C4×C179—21no—
57C42×C179—21no—
58Dic358—22no—
59Dic358—22no—
60Dic358—22no—
61Dic358—22no—
62C2×Dic179—22no—
63C2×Dic179—22no—
64C2×Dic179—22no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—41no—
68C2×Dic179—41no—
69C2×Dic179C4⋊C414yes—
70C179⋊C8—22no—
71C23×C179C2×D411yescharacteristic
72C2×C4×C179—21no—
73C2×C4×C179—21no—
74C2×C4×C179—42no—
75C2×C4×C179—41no—
76C2×C4×C179C22×C411yescharacteristic
77C2×C4×C179C2×D411yescharacteristic
78C2×C4×C179C2×D411yescharacteristic
79C2×C4×C179C2×Q811yescharacteristic
80C2×C4×C179#4111yescharacteristic
81C2×C4×C179#4111yescharacteristic
82C8×C179—22no—
83C22×C179—21no—
84C22×C179—21no—
85C22×C179C2×C4⋊C411yescharacteristic
86C22×C179#12011yescharacteristic
87C22×C179#12311yescharacteristic
88C4×C179—41no—
89C4×C179—41no—
90C4×C179—81no—
91C4×C179C4×D411yescharacteristic
92C4×C179C4×D411yescharacteristic
93C4×C179C22⋊Q811yescharacteristic
94C4×C179C22⋊Q811yescharacteristic
95Dic179—24no—
96Dic179—42no—
97Dic179—41no—
98Dic179—41no—
99C2×C179—21no—
100C2×C179—21no—
101C2×C179#38511yescharacteristic
102C2×C179#45411yescharacteristic
103C2×C179#45611yescharacteristic
104C179#185711yescharacteristic, complemented, Sylow
105#1857—1791nomaximal, Sylow
106#519—1791no—
107#428—1791no—
108#428D17911yescharacteristic
109C2×#101—1791no—
110#391—1791no—
111#339—1791no—
112#339—1791no—
113C22×Q8—1791no—
114C2×M16—1791no—
115C2×M16C2×D17911yescharacteristic
116C4×Q8—3581no—
117C4×Q8—3581no—
118C4×Q8—3581no—
119C4×Q8—3581no—
120#117—3582no—
121#117—1791no—
122#117C2×D17911yescharacteristic
123C2×C4⋊C4C2×D17911yescharacteristic
124C2×C42—1791no—
125#107—21no—
126#107—3581no—
127#106—21no—
128#106—3581no—
129#101—1794no—
130#95—3582no—
131C2×Q8—3581no—
132C2×Q8—3581no—
133C2×Q8—3581no—
134C2×Q8—3581no—
135C2×Q8—1792no—
136C2×Q8—1792no—
137C22×C4—21no—
138C22×C4—3581no—
139C22×C4—1792no—
140C22×C4C22×D17911yescharacteristic
141M16—21no—
142M16—3581no—
143M16—1792no—
144M16C4×D17912yes—
145C2×C8—21no—
146C2×C8—3581no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—3582no—
150C4⋊C4—3582no—
151C4⋊C4—1791no—
152C4⋊C4—1791no—
153C4⋊C4C22⋊D17911yescharacteristic
154C4⋊C4C22⋊D17911yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3581no—
157C42—21no—
158C42—3582no—
159C42—3582no—
160C42—3581no—
161C42—3581no—
162C42—3581no—
163C23D4×D17911yescharacteristic
164Q8—3582no—
165Q8—3582no—
166Q8—3582no—
167Q8—3582no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—3582no—
173C2×C4—3582no—
174C2×C4—3582no—
175C2×C4—3581no—
176C2×C4—3581no—
177C2×C4—1794no—
178C2×C4—7161no—
179C2×C4—7161no—
180C2×C4C2×C4×D17911yescharacteristic
181C2×C4D4×D17911yescharacteristic
182C2×C4C2×C22⋊D17911yescharacteristic
183C2×C4Q8×D17911yescharacteristic
184C2×C4#1566825911yescharacteristic
185C2×C4#1566826011yescharacteristic
186C8—22no—
187C8—3582no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D17911yescharacteristic
191C22#1915308111yescharacteristic
192C22#1915309811yescharacteristic
193C4—41no—
194C4—41no—
195C4—81no—
196C4—3584no—
197C4—7162no—
198C4—7161no—
199C4—7161no—
200C4C4×C22⋊D17911yescharacteristic
201C4#1915307511yescharacteristic
202C4#1915308911yescharacteristic
203C4#1915309111yescharacteristic
204C2—21no—
205C2—21no—
206C2#2603831011yescharacteristic
207C2#2603861011yescharacteristic
208C2#2603862011yescharacteristic
subgroupG/H
centerC22#19153081
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#428×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
cosocleC22×C4×C179C23
hypercenter#428D179
hypocenterC179#1857
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
19. #19153031C4semidirect
104. C179#1857semidirect

Maximal subgroups

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

subgroupindex
#260389162
#260384792
C2×#191530312
#260383442
#260382002
#260381982
#428×C1792
#1857179