Finite Group Databasebeta

← #37510343

Subgroups — #37510343

Order 22912.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#26039067C211yesmaximal, characteristic
2#26038479C211yesmaximal, characteristic
3#26038308C211yesmaximal, characteristic
4#26038207C211yesmaximal, characteristic
5#26038200C211yesmaximal, characteristic
6#26038198C211yesmaximal, characteristic
7#428×C179C211yesmaximal, characteristic
8C2×#15668243C2211yescharacteristic
9#19153113—21no—
10#19153112—21no—
11#19153105—21no—
12#19153104—21no—
13#19153090—22no—
14#19153072C2211yescharacteristic
15C2×C4⋊Dic179C2211yescharacteristic
16C2×Dic179⋊C4—22no—
17C2×C4×Dic179C2211yescharacteristic
18#19153046—21no—
19#19153044—21no—
20C2×M16×C179C2211yescharacteristic
21#117×C179C2211yescharacteristic
22C2×C4⋊C4×C179C2211yescharacteristic
23#107×C179—21no—
24#106×C179—21no—
25C2×Dic358—22no—
26C22×Dic179—21no—
27C22×Dic179D412yes—
28#15668243—21no—
29#15668243Q812yescomplemented
30C2×C179⋊C8—21no—
31C4⋊Dic179—22no—
32C4⋊Dic179—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179D411yescharacteristic
35C4⋊Dic179D411yescharacteristic
36Dic179⋊C4—22no—
37Dic179⋊C4—22no—
38Dic179⋊C4—44no—
39C22⋊Dic179—21no—
40C4×Dic179—21no—
41C4×Dic179—21no—
42C4×Dic179Q812yescomplemented
43C22×C4×C179—21no—
44C22×C4×C179C2311yescharacteristic
45M16×C179—21no—
46M16×C179Q812yescomplemented
47C2×C8×C179—21no—
48C4⋊C4×C179—21no—
49C4⋊C4×C179—21no—
50C4⋊C4×C179D411yescharacteristic
51C4⋊C4×C179D411yescharacteristic
52C22⋊C4×C179—21no—
53C42×C179—21no—
54Dic358—82no—
55C2×Dic179—22no—
56C2×Dic179—22no—
57C2×Dic179—22no—
58C2×Dic179—21no—
59C2×Dic179—21no—
60C2×Dic179—22no—
61C2×Dic179—42no—
62C179⋊C8—22no—
63C23×C179C2×D411yescharacteristic
64C2×C4×C179—21no—
65C2×C4×C179—21no—
66C2×C4×C179—42no—
67C2×C4×C179—41no—
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×Q811yescharacteristic
71C2×C4×C179C2×Q811yescharacteristic
72C2×C4×C179C2×Q811yescharacteristic
73C2×C4×C179#4111yescharacteristic
74C8×C179—22no—
75C22×C179—21no—
76C22×C179—21no—
77C22×C179#12011yescharacteristic
78C22×C179C22⋊Q811yescharacteristic
79C22×C179C4⋊Q811yescharacteristic
80C4×C179—41no—
81C4×C179—41no—
82C4×C179—81no—
83C4×C179C22⋊Q811yescharacteristic
84C4×C179C22⋊Q811yescharacteristic
85C4×C179C4⋊Q811yescharacteristic
86C4×C179C4⋊Q811yescharacteristic
87Dic179—24no—
88Dic179—42no—
89Dic179—41no—
90Dic179—41no—
91C2×C179—21no—
92C2×C179—21no—
93C2×C179#39511yescharacteristic
94C2×C179#45311yescharacteristic
95C2×C179#45611yescharacteristic
96C179#206011yescharacteristic, complemented, Sylow
97#2060—1791nomaximal, Sylow
98#544—1791no—
99#428—1791no—
100#428D17911yescharacteristic
101#384—1791no—
102#343—1791no—
103#339—1791no—
104#339—1791no—
105C2×M16—1791no—
106C2×M16C2×D17911yescharacteristic
107C4⋊Q8—3581no—
108C4⋊Q8—3581no—
109#125—3581no—
110#125—3581no—
111C22⋊Q8—3582no—
112#117—1791no—
113#117C2×D17911yescharacteristic
114C2×C4⋊C4—3582no—
115C2×C4⋊C4—1791no—
116C2×C4⋊C4C2×D17911yescharacteristic
117C2×C42—1791no—
118#107—21no—
119#107—3581no—
120#106—21no—
121#106—3581no—
122C2×Q8—3582no—
123C22×C4—21no—
124C22×C4—3581no—
125C22×C4—1792no—
126C22×C4C22×D17911yescharacteristic
127M16—21no—
128M16—3581no—
129M16—1792no—
130M16Dic35812yescomplemented
131C2×C8—21no—
132C2×C8—3581no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—3582no—
136C4⋊C4—3582no—
137C4⋊C4—3582no—
138C4⋊C4—3581no—
139C4⋊C4—3581no—
140C4⋊C4—1791no—
141C4⋊C4—1791no—
142C4⋊C4—7164no—
143C4⋊C4C22⋊D17911yescharacteristic
144C4⋊C4C22⋊D17911yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—3581no—
147C42—21no—
148C42—3581no—
149C42—3581no—
150C42—1792no—
151C23D4×D17911yescharacteristic
152Q8—14322no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—3582no—
158C2×C4—3582no—
159C2×C4—3582no—
160C2×C4—3581no—
161C2×C4—3581no—
162C2×C4—3582no—
163C2×C4—7162no—
164C2×C4D4×D17911yescharacteristic
165C2×C4C2×C22⋊D17911yescharacteristic
166C2×C4Q8×D17911yescharacteristic
167C2×C4Q8×D17911yescharacteristic
168C2×C4C2×Dic35811yescharacteristic
169C2×C4#1566825911yescharacteristic
170C8—22no—
171C8—3582no—
172C22—21no—
173C22—21no—
174C22#1915308111yescharacteristic
175C22#1915309111yescharacteristic
176C22#1915311211yescharacteristic
177C4—41no—
178C4—41no—
179C4—81no—
180C4—3584no—
181C4—7162no—
182C4—7161no—
183C4—7161no—
184C4#1915309111yescharacteristic
185C4#1915309211yescharacteristic
186C4#1915311111yescharacteristic
187C4#1915311211yescharacteristic
188C2—21no—
189C2—21no—
190C2#2603835911yescharacteristic
191C2#2603860511yescharacteristic
192C2#2603862011yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#428×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#428D179
hypocenterC179#2060
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
29. #15668243Q8semidirect
42. C4×Dic179Q8semidirect
46. M16×C179Q8semidirect
96. C179#2060semidirect
130. M16Dic358semidirect

Maximal subgroups

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

subgroupindex
#260390672
#260384792
#260383082
#260382072
#260382002
#260381982
#428×C1792
#2060179