Finite Group Databasebeta

← #14530779

Subgroups — #14530779

Order 2176.

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

normal subgroups only

This group has 246 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#11763545C211yesmaximal, characteristic, complemented
2#11762999C212yesmaximal, complemented
3#11762602C211yesmaximal, characteristic, complemented
4#11762600C212yesmaximal, complemented
5C17×#453C211yesmaximal, characteristic
6C17×H(5,𝔽2)C2211yescharacteristic
7C17×AGL(1,ℤ/8)—21no—
8C17×#127C2211yescharacteristic
9#10587622—22no—
10#10587621—21no—
11#10587621—22no—
12#10587621C2212yes—
13D4×Dic17—22no—
14C4×Dic34—21no—
15C17×#104—21no—
16C17×#100C2211yescharacteristic
17#10587505—22no—
18#10587505C2212yes—
19#41×C17—21no—
20#41×C17—41no—
21C2×D4×C17—22no—
22C2×D4×C17—22no—
23C2×D4×C17—41no—
24C2×D4×C17D411yescharacteristic
25C2×D4×C17C2311yescharacteristic
26C22×Dic17—21no—
27C22×Dic17D412yes—
28C2×Dic34—21no—
29SD16×C17—41no—
30D8×C17—41no—
31M16×C17—21no—
32C42×C17D411yescharacteristic
33C22⋊Dic17—22no—
34C22⋊Dic17—21no—
35C22⋊Dic17—22no—
36C22⋊Dic17—22no—
37C22⋊Dic17—42no—
38C22⋊Dic17D412yes—
39C4⋊Dic17—21no—
40#63225—22no—
41#63225—42no—
42C4×Dic17—22no—
43C23×C17—22no—
44C23×C17—21no—
45C23×C17C2×D412yes—
46Q8×C17—21no—
47D4×C17—21no—
48D4×C17—41no—
49D4×C17—41no—
50D4×C17—44no—
51D4×C17—42no—
52D4×C17—41no—
53C2×C4×C17—21no—
54C2×C4×C17—22no—
55C2×C4×C17—41no—
56C2×C4×C17C2×D411yescharacteristic
57C2×Dic17—22no—
58C2×Dic17—22no—
59C2×Dic17—42no—
60C2×Dic17—42no—
61C2×Dic17—81no—
62Dic34—42no—
63C8×C17—41no—
64#2198—171nomaximal, Sylow
65C22×C17—22no—
66C22×C17—42no—
67C22×C17—42no—
68C22×C17—41no—
69C22×C17—41no—
70C22×C17—82no—
71C22×C17#12011yescharacteristic
72C4×C17—22no—
73C4×C17—21no—
74C4×C17—41no—
75Dic17—42no—
76Dic17—42no—
77Dic17—82no—
78#559—171no—
79#458—172no—
80#453D1711yescharacteristic
81#353—171no—
82#352—172no—
83C2×C17—21no—
84C2×C17—42no—
85C2×C17—81no—
86C2×C17—81no—
87C2×C17B(4,𝔽2)11yescharacteristic
88H(5,𝔽2)C2×D1711yescharacteristic
89AGL(1,ℤ/8)—21no—
90#127C2×D1711yescharacteristic
91#124—342no—
92#123—172no—
93#123—341no—
94#123—342no—
95C4×Q8—341no—
96C4×D4—342no—
97#104—21no—
98#100C2×D1711yescharacteristic
99#99—172no—
100#99—342no—
101C17#219811yescharacteristic, complemented, Sylow
102#41—21no—
103#41—41no—
104C2×Q8—341no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—41no—
108C2×D4C22⋊D1711yescharacteristic
109C2×D4C22×D1711yescharacteristic
110C22×C4—172no—
111C22×C4—341no—
112SD16—41no—
113D8—41no—
114M16—21no—
115C4⋊C4—341no—
116C4⋊C4—342no—
117C4⋊C4—682no—
118C22⋊C4—172no—
119C22⋊C4—342no—
120C22⋊C4—341no—
121C22⋊C4—342no—
122C22⋊C4—342no—
123C22⋊C4—682no—
124C42—342no—
125C42C22⋊D1711yescharacteristic
126C23—22no—
127C23—21no—
128C23D4×D1712yes—
129Q8—21no—
130Q8—682no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—44no—
135D4—42no—
136D4—41no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—342no—
141C2×C4—342no—
142C2×C4—682no—
143C2×C4—682no—
144C2×C4—1361no—
145C2×C4C2×C22⋊D1711yescharacteristic
146C8—41no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—82no—
153C22#1058762311yescharacteristic
154C4—22no—
155C4—21no—
156C4—41no—
157C4—682no—
158C4—682no—
159C4—1362no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—81no—
164C2#1176299511yescharacteristic
subgroupG/H
centerC2#11762995
commutator, cosocleC2×D4×C17C23
FrattiniC2×D4C22×D17
Fitting, prime radicalC17×#453C2
socleC2×C17B(4,𝔽2)
hypercenter#453D17
hypocenterC17#2198
LCSG ▷ C2×D4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #11763545C2semidirect
2. #11762999C2semidirect
3. #11762602C2semidirect
4. #11762600C2semidirect
101. C17#2198semidirect

Maximal subgroups

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

subgroupindex
#117635452
#117629992
#117626022
#117626002
C17×#4532
#219817