Finite Group Databasebeta

← #14540225

Subgroups — #14540225

Order 2176.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#11763414C211yesmaximal, characteristic
2#11763116C212yesmaximal, complemented
3#11763106C212yesmaximal, complemented
4#11763057C212yesmaximal, complemented
5#11763040C212yesmaximal, complemented
6#11763011C212yesmaximal, complemented
7#11763009C212yesmaximal, complemented
8#11762836C211yesmaximal, characteristic, complemented
9C17×#541C211yesmaximal, characteristic, complemented
10C17×C4⋊Q8C2211yescharacteristic
11C17×#126—22no—
12C17×#125—22no—
13C17×#125C2211yescharacteristic
14C17×C22⋊Q8C2212yes—
15C4×Q8×C17C2212yes—
16C17×#117C2211yescharacteristic
17C2×#63230C2212yes—
18C2×#63230C2212yes—
19#10587616C2212yes—
20C2×#63223C2211yescharacteristic
21C22×#3622C2211yescharacteristic
22#10587580—22no—
23#10587570—22no—
24#10587560C2211yescharacteristic
25C4×D68C2212yes—
26#10587556C2211yescharacteristic
27#10587506C2212yes—
28#10587506C2212yes—
29#10587480C2212yes—
30#10587480C2212yes—
31#10587479C2212yes—
32#10587478C2212yes—
33C4⋊#3622C2212yes—
34C4⋊#3622C2212yes—
35C2×Q8×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C2×C22⋊D17—22no—
38C2×D68—22no—
39C2×D68C2312yes—
40C2×C4×D17—22no—
41C4⋊C4×C17—22no—
42C4⋊C4×C17—22no—
43C4⋊C4×C17—22no—
44C4⋊C4×C17—22no—
45C4⋊C4×C17D412yes—
46C4⋊C4×C17C2312yes—
47C22⋊C4×C17—22no—
48C22⋊C4×C17D412yes—
49C42×C17—22no—
50C42×C17C2311yescharacteristic
51C42×C17C2311yescharacteristic
52#63230—42no—
53#63230—42no—
54#63227—22no—
55#63227—22no—
56C4⋊Dic17C2312yes—
57#63225—22no—
58#63223—41no—
59C2×#3622—41no—
60C2×#3622C2312yes—
61C2×#3622C2312yes—
62C23×C17C2×D411yescharacteristic
63C22×D17—22no—
64Q8×C17—22no—
65Q8×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17C2×D412yes—
71C2×C4×C17C2×D412yes—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2411yescharacteristic
74C22⋊D17—82no—
75C2×Dic17—22no—
76D68—22no—
77D68—22no—
78D68—82no—
79C4×D17—82no—
80#3622—22no—
81#3622—22no—
82#3260—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—22no—
85C22×C17C22×D411yescharacteristic
86C2×D17—42no—
87C2×D17—42no—
88C2×D17—42no—
89C4×C17—21no—
90C4×C17—22no—
91C4×C17—22no—
92C4×C17—42no—
93C4×C17—42no—
94C4×C17H(5,𝔽2)11yescharacteristic
95C4×C172−1+411yescharacteristic
96Dic17—42no—
97#541D1711yescharacteristic, complemented
98#540—171no—
99#479—172no—
100#478—172no—
101#468—172no—
102#465—172no—
103#461—172no—
104#460—172no—
105#424—171no—
106C2×C17—41no—
107C2×C17#53711yescharacteristic
108C2×C17#57712yes—
109D17—82no—
110C2×SD16—172no—
111C2×SD16—172no—
112C2×M16—171no—
113C22×C8—171no—
114C4⋊Q8C2×D1711yescharacteristic
115#127—171no—
116#126—22no—
117#125—22no—
118#125—171no—
119#125C2×D1711yescharacteristic
120#123—342no—
121C22⋊Q8C2×D1712yes—
122#121—172no—
123#121—342no—
124C4×Q8C2×D1712yes—
125C4×D4—172no—
126#117C2×D1711yescharacteristic
127#107—172no—
128#106—172no—
129C4⋊C8—172no—
130C4⋊C8—172no—
131#103—172no—
132#103—172no—
133#102—172no—
134#102—172no—
135C17#326011yescharacteristic, complemented, Sylow
136C2×Q8C22×D1712yes—
137C2×D4—172no—
138C2×D4—342no—
139C2×D4—342no—
140C22×C4—342no—
141C22×C4C22×D1711yescharacteristic
142SD16—682no—
143SD16—682no—
144M16—681no—
145C2×C8—172no—
146C2×C8—172no—
147C2×C8—681no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—172no—
153C4⋊C4—342no—
154C4⋊C4C22⋊D1712yes—
155C4⋊C4C22×D1712yes—
156C22⋊C4—22no—
157C22⋊C4—342no—
158C22⋊C4—342no—
159C22⋊C4C22⋊D1712yes—
160C42—22no—
161C42C22×D1711yescharacteristic
162C42C22×D1711yescharacteristic
163C23—342no—
164C23C2×C22⋊D1711yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—342no—
168D4—342no—
169D4—1362no—
170D4—1362no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—342no—
176C2×C4—1362no—
177C2×C4C2×C22⋊D1712yes—
178C2×C4C2×C22⋊D1712yes—
179C2×C4C2×C22⋊D1711yescharacteristic
180C2×C4C23×D1711yescharacteristic
181C8—342no—
182C8—342no—
183C22—22no—
184C22—21no—
185C22—682no—
186C22—682no—
187C22—682no—
188C22C22×C22⋊D1711yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—682no—
195C4#1058769611yescharacteristic
196C4#1058769711yescharacteristic
197C2—41no—
198C2—1362no—
199C2#1176338711yescharacteristic
200C2#1176361712yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#541C2
socleC22×C17C22×D4
prime radical#11763414C2
hypercenter#541D17
hypocenterC17#3260
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #11763116C2semidirect
3. #11763106C2semidirect
4. #11763057C2semidirect
5. #11763040C2semidirect
6. #11763011C2semidirect
7. #11763009C2semidirect
8. #11762836C2semidirect
9. C17×#541C2semidirect
97. #541D17semidirect
135. C17#3260semidirect

Maximal subgroups

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

subgroupindex
#117634142
#117631162
#117631062
#117630572
#117630402
#117630112
#117630092
#117628362
C17×#5412
#326017