Finite Group Databasebeta

← #36355586

Subgroups — #36355586

Order 22144.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012241C211yesmaximal, characteristic, complemented
2#26011676C212yesmaximal, complemented
3#26011663C211yesmaximal, characteristic
4#26011289C212yesmaximal, complemented
5#355×C173C211yesmaximal, characteristic, complemented
6#19149727C2211yescharacteristic
7#19149712—22no—
8#19149705—21no—
9#19149677C2212yes—
10#19149672—22no—
11#19149671—21no—
12#19149660—21no—
13#19149656—22no—
14#19149607—21no—
15#19149605—22no—
16#19149598C2211yescharacteristic
17#124×C173C2211yescharacteristic
18#101×C173C2212yes—
19#14583827—41no—
20#14583826—22no—
21#14583826—21no—
22C2×Dic346D412yes—
23Q8×D173—41no—
24C22×Dic173—21no—
25Dic692—42no—
26#14583815—42no—
27#14583814—41no—
28#14583811—41no—
29#14583810—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173—21no—
32#14583805—22no—
33#14583805—22no—
34#14583805—22no—
35C22⋊Dic173—22no—
36C22⋊Dic173—21no—
37C4×Dic173D412yes—
38C2×Q8×C173C2311yescharacteristic
39C2×D4×C173D411yescharacteristic
40M16×C173—22no—
41C22⋊C4×C173—22no—
42C42×C173D411yescharacteristic
43Dic346—21no—
44Dic346—22no—
45Dic346—42no—
46D692—21no—
47D692—22no—
48C22⋊D173—41no—
49C2×Dic173—22no—
50C2×Dic173—22no—
51C2×Dic173—81no—
52C4×D173—42no—
53C4×D173—41no—
54#13063436—41no—
55C23×C173—21no—
56Q8×C173—41no—
57D4×C173—41no—
58C2×C4×C173—21no—
59C2×C4×C173C2×D412yes—
60C2×C4×C173C2×D411yescharacteristic
61C8×C173—42no—
62C22×C173—42no—
63C22×C173—41no—
64C22×C173#12011yescharacteristic
65C2×D173—41no—
66C4×C173—22no—
67C4×C173—21no—
68C4×C173—41no—
69Dic173—42no—
70Dic173—41no—
71Dic173—42no—
72C2×C173—21no—
73C2×C173—81no—
74C2×C173B(4,𝔽2)11yescharacteristic
75D173—81no—
76C173#220311yescharacteristic, complemented, Sylow
77#2203—1731nomaximal, Sylow
78#563—1731no—
79#456—1732no—
80#454—1731no—
81#356—1732no—
82#355D17311yescharacteristic, complemented
832−1+4—1731no—
84#137—3462no—
85AGL(1,ℤ/8)—3461no—
86C4⋊Q8—1732no—
87#126—3462no—
88#125—3461no—
89#124C2×D17311yescharacteristic
90#123—3461no—
91C22⋊Q8—3462no—
92#104—3462no—
93#104—3461no—
94#101—1731no—
95#101C2×D17312yes—
96#41—3462no—
97#41—3461no—
98#41—6921no—
99C2×Q8—1732no—
100C2×Q8—6921no—
101C2×Q8C22×D17311yescharacteristic
102C2×D4D69211yescharacteristic
103C22×C4—3461no—
104Q16—6922no—
105SD16—6922no—
106SD16—6921no—
107D8—6921no—
108M16—22no—
109M16—3461no—
110C4⋊C4—3461no—
111C4⋊C4—3462no—
112C4⋊C4—3461no—
113C4⋊C4—3462no—
114C4⋊C4—3462no—
115C22⋊C4—22no—
116C22⋊C4—3462no—
117C22⋊C4—3461no—
118C42—1732no—
119C42D69211yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—3461no—
123Q8—3462no—
124Q8—6922no—
125D4—41no—
126D4—3461no—
127D4—3462no—
128D4—6921no—
129C2×C4—21no—
130C2×C4—3462no—
131C2×C4—3462no—
132C2×C4—13841no—
133C2×C4—6922no—
134C2×C4—6921no—
135C2×C4D4×D17312yes—
136C2×C4C2×D69211yescharacteristic
137C8—42no—
138C8—6921no—
139C22—42no—
140C22—41no—
141C22—6921no—
142C22#1914964911yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—6922no—
147C4—6922no—
148C4—6921no—
149C2—21no—
150C2—81no—
151C2—13841no—
152C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#355×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011663C2
hypercenter#355D173
hypocenterC173#2203
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #26012241C2semidirect
2. #26011676C2semidirect
4. #26011289C2semidirect
5. #355×C173C2semidirect
76. C173#2203semidirect
82. #355D173semidirect

Maximal subgroups

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

subgroupindex
#260122412
#260116762
#260116632
#260112892
#355×C1732
#2203173