Finite Group Databasebeta

← #31808090

Subgroups — #31808090

Order 17792.

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

normal subgroups only

This group has 230 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#22581523C211yesmaximal, characteristic
2#101×D139C212yesmaximal, complemented
3#22580751C212yesmaximal, complemented
4#22580750C211yesmaximal, characteristic, complemented
5#355×C139C211yesmaximal, characteristic, complemented
6C2×#14555784C411yescharacteristic
7C2×Q8×D139C2211yescharacteristic
8M16×D139—22no—
9#17984842C411yescharacteristic
10#17984840C2211yescharacteristic
11#17984829C411yescharacteristic
12#17984824—22no—
13#17984819—22no—
14#17984799C411yescharacteristic
15#17984761C2212yes—
16#17984760—22no—
17#124×C139C2211yescharacteristic
18#101×C139C2212yes—
19#14555784—42no—
20C2×Dic278C2×C411yescharacteristic
21C2×Dic278C2×C411yescharacteristic
22C2×Dic278D412yes—
23Q8×D139—42no—
24C2×C22⋊D139—21no—
25C22×Dic139—21no—
26C2×C4×D139C2×C411yescharacteristic
27C2×C4×D139D412yes—
28#14555769—22no—
29#14555768—42no—
30C8×D139—42no—
31C4⋊Dic139—22no—
32Dic139⋊C4—21no—
33Dic139⋊C4—22no—
34C22⋊Dic139—22no—
35(C2×D139)⋊C4—21no—
36C4×Dic139C2×C411yescharacteristic
37C2×Q8×C139C2311yescharacteristic
38C2×D4×C139C2×C411yescharacteristic
39M16×C139—22no—
40C22⋊C4×C139—22no—
41C42×C139C2×C411yescharacteristic
42C22×D139C22⋊C411yescharacteristic
43Dic278—41no—
44Dic278—42no—
45Dic278—41no—
46C22⋊D139—42no—
47C2×Dic139—21no—
48C2×Dic139—21no—
49C2×Dic139—42no—
50C2×Dic139C22⋊C412yes—
51C2×Dic139C22⋊C411yescharacteristic
52C4×D139—22no—
53C4×D139—22no—
54C4×D139—22no—
55C139⋊C8—42no—
56C23×C139—21no—
57Q8×C139—41no—
58D4×C139—41no—
59C2×C4×C139—21no—
60C2×C4×C139C22×C411yescharacteristic
61C2×C4×C139C2×D412yes—
62C8×C139—42no—
63C22×C139—42no—
64C22×C139—41no—
65C22×C139C2×C22⋊C411yescharacteristic
66C2×D139—41no—
67C2×D139#9912yes—
68C4×C139—22no—
69C4×C139—21no—
70C4×C139—41no—
71Dic139—21no—
72Dic139—22no—
73Dic139—41no—
74Dic139—41no—
75C2×C139—21no—
76C2×C139—81no—
77C2×C139C2×#9911yescharacteristic
78D139—22no—
79C139#213311yescharacteristic, complemented, Sylow
80#2133—1391nomaximal, Sylow
81#536—1391no—
82C2×#101—1392no—
83#356—1392no—
84#355—1391no—
85#355D13911yescharacteristic, complemented
86C2×#41—1391no—
87C22×Q8—1391no—
88C2×M16—2782no—
89C4⋊Q8—1391no—
90C4⋊Q8—1391no—
91#124—1391no—
92#124C2×D13911yescharacteristic
93#123—2782no—
94C22⋊Q8—2782no—
95#117—1391no—
96#101—2782no—
97#101—1392no—
98#101C2×D13912yes—
99#41—5562no—
100C2×Q8—5562no—
101C2×Q8—1391no—
102C2×Q8—1391no—
103C2×Q8—1392no—
104C2×Q8C22×D13911yescharacteristic
105C2×D4—2781no—
106C2×D4C4×D13911yescharacteristic
107C22×C4—2781no—
108C22×C4—1391no—
109C22×C4—1392no—
110M16—22no—
111M16—2782no—
112M16—5562no—
113C2×C8—5562no—
114C4⋊C4—2781no—
115C4⋊C4—2782no—
116C4⋊C4—2782no—
117C22⋊C4—22no—
118C22⋊C4—2781no—
119C22⋊C4—2782no—
120C42—1391no—
121C42C4×D13911yescharacteristic
122C23—21no—
123C23—1391no—
124Q8—41no—
125Q8—5562no—
126Q8—5561no—
127Q8—5561no—
128D4—41no—
129D4—5562no—
130C2×C4—21no—
131C2×C4—2782no—
132C2×C4—2782no—
133C2×C4—2782no—
134C2×C4—2781no—
135C2×C4—2781no—
136C2×C4—5562no—
137C2×C4—1392no—
138C2×C4—1391no—
139C2×C4C2×C4×D13911yescharacteristic
140C2×C4D4×D13912yes—
141C8—42no—
142C8—5562no—
143C22—42no—
144C22—41no—
145C22—5561no—
146C22—1392no—
147C22C22⋊C4×D13911yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—2781no—
152C4—2782no—
153C4—5561no—
154C4—5561no—
155C2—21no—
156C2—81no—
157C2—2782no—
158C2#99×D13911yescharacteristic
subgroupG/H
centerC2#99×D139
commutatorC2×C4×C139C22×C4
FrattiniC2×Q8C22×D139
Fitting#355×C139C2
socleC2×C139C2×#99
prime radicalC2×#14555784C4
cosocleC2×Q8×C139C23
hypercenter#355D139
hypocenterC139#2133
LCSG ▷ C2×C4×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D139C2semidirect
3. #22580751C2semidirect
4. #22580750C2semidirect
5. #355×C139C2semidirect
79. C139#2133semidirect
85. #355D139semidirect

Maximal subgroups

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

subgroupindex
#225815232
#101×D1392
#225807512
#225807502
#355×C1392
#2133139