Finite Group Databasebeta

← #37510673

Subgroups — #37510673

Order 22912.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039009C211yesmaximal, characteristic
2#26038411C212yesmaximal
3#26038234C211yesmaximal, characteristic, complemented
4#26038233C212yesmaximal, complemented
5#354×C179C211yesmaximal, characteristic, complemented
6C2×#15668258C2211yescharacteristic
7C2×Q8×D179C411yescharacteristic, complemented
8#19153113C411yescharacteristic, complemented
9#19153111C411yescharacteristic, complemented
10#19153100C2211yescharacteristic
11#19153095—22no—
12#19153090—22no—
13#19153070C411yescharacteristic, complemented
14#19153069—22no—
15#19153028C2212yes—
16#19153027—22no—
17#124×C179C2211yescharacteristic
18#99×C179C2212yes—
19#15668258—42no—
20C2×Dic358—21no—
21C2×Dic358C2×C411yescharacteristic
22C2×Dic358C2×C411yescharacteristic
23Q8×D179—42no—
24C2×C22⋊D179D412yes—
25C22×Dic179D412yes—
26C2×C4×D179—21no—
27C2×C4×D179C2×C411yescharacteristic
28C4⋊Dic179—22no—
29Dic179⋊C4—21no—
30Dic179⋊C4—22no—
31Dic179⋊C4—42no—
32C22⋊Dic179—22no—
33C22⋊Dic179—22no—
34(C2×D179)⋊C4—21no—
35C4×Dic179—42no—
36C4×Dic179C2×C411yescharacteristic
37C2×Q8×C179C2×C411yescharacteristic
38C2×D4×C179C2311yescharacteristic
39C22⋊C4×C179—22no—
40C22⋊C4×C179—22no—
41C42×C179C2×C411yescharacteristic
42C22×D179C22⋊C411yescharacteristic
43Dic358—41no—
44Dic358—42no—
45Dic358—41no—
46C22⋊D179—42no—
47C2×Dic179—22no—
48C2×Dic179—22no—
49C2×Dic179—21no—
50C2×Dic179—21no—
51C2×Dic179—42no—
52C2×Dic179C22⋊C412yes—
53C2×Dic179C22⋊C411yescharacteristic
54C4×D179—22no—
55C4×D179—42no—
56C23×C179C2×D412yes—
57Q8×C179—41no—
58D4×C179—41no—
59C2×C4×C179—21no—
60C2×C4×C179—21no—
61C2×C4×C179—42no—
62C2×C4×C179C22×C411yescharacteristic
63C22×C179—22no—
64C22×C179—42no—
65C22×C179C2×C22⋊C411yescharacteristic
66C2×D179—21no—
67C2×D179—41no—
68C4×C179—21no—
69C4×C179—41no—
70C4×C179—41no—
71C4×C179—82no—
72Dic179—22no—
73Dic179—41no—
74Dic179—41no—
75Dic179—82no—
76Dic179#9912yes—
77C2×C179—21no—
78C2×C179—42no—
79C2×C179C2×#9911yescharacteristic
80D179—41no—
81C179#212911yescharacteristic, complemented, Sylow
82#2129—1791nomaximal, Sylow
83#536—1791no—
84#410—1792no—
85#354—1791no—
86#354—1792no—
87#354D17911yescharacteristic, complemented
88C2×#41—1791no—
89C22×Q8—1791no—
90C4⋊Q8—1791no—
91C4⋊Q8—1791no—
92#124—1791no—
93#124C2×D17911yescharacteristic
94#123—3582no—
95C22⋊Q8—3582no—
96#117—3582no—
97#117—1791no—
98#99—3582no—
99#99—1792no—
100#99C2×D17912yes—
101#41—7162no—
102C2×Q8—3581no—
103C2×Q8—1791no—
104C2×Q8—1791no—
105C2×Q8—7162no—
106C2×Q8C4×D17911yescharacteristic
107C2×D4—1792no—
108C2×D4C22×D17911yescharacteristic
109C22×C4—3581no—
110C22×C4—1792no—
111C22×C4—1791no—
112C4⋊C4—3581no—
113C4⋊C4—3582no—
114C4⋊C4—3582no—
115C4⋊C4—7162no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—3581no—
119C22⋊C4—3582no—
120C22⋊C4—3582no—
121C42—1791no—
122C42—7162no—
123C42C4×D17911yescharacteristic
124C23—1791no—
125C23D4×D17912yes—
126Q8—41no—
127Q8—7161no—
128Q8—7161no—
129Q8—7162no—
130D4—41no—
131D4—7162no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—3582no—
136C2×C4—3582no—
137C2×C4—3582no—
138C2×C4—3581no—
139C2×C4—3581no—
140C2×C4—1792no—
141C2×C4—1791no—
142C2×C4—7162no—
143C2×C4—7162no—
144C2×C4C2×C4×D17911yescharacteristic
145C22—22no—
146C22—42no—
147C22—3581no—
148C22—7161no—
149C22C22⋊C4×D17911yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—3582no—
155C4—14322no—
156C4—1792no—
157C4—7161no—
158C4—7161no—
159C2—21no—
160C2—42no—
161C2—7161no—
162C2#99×D17911yescharacteristic
subgroupG/H
centerC2#99×D179
commutatorC2×C4×C179C22×C4
FrattiniC2×D4C22×D179
Fitting#354×C179C2
socleC2×C179C2×#99
prime radicalC2×#15668258C22
cosocleC2×D4×C179C23
hypercenter#354D179
hypocenterC179#2129
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #26038234C2semidirect
4. #26038233C2semidirect
5. #354×C179C2semidirect
7. C2×Q8×D179C4semidirect
8. #19153113C4semidirect
9. #19153111C4semidirect
13. #19153070C4semidirect
81. C179#2129semidirect
87. #354D179semidirect

Maximal subgroups

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

subgroupindex
#260390092
#260384112
#260382342
#260382332
#354×C1792
#2129179