Finite Group Databasebeta

← #14527229

Subgroups — #14527229

Order 2176.

252 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×Q8×D17C211yesmaximal, characteristic
2#11762865C211yesmaximal, characteristic
3C4×#63218C211yesmaximal, characteristic
4#11762541C211yesmaximal, characteristic, complemented
5Q8⋊#3622C211yesmaximal, characteristic, complemented
6#11762539C211yesmaximal, characteristic, complemented
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D17C411yescharacteristic
9C4×Q8×C17C2211yescharacteristic
10Q8×Dic17C411yescharacteristic
11C2×#63218—21no—
12C4⋊C4×D17—21no—
13C4⋊C4×D17C411yescharacteristic
14#10587573—21no—
15#10587573C411yescharacteristic
16C42×D17—21no—
17C42×D17C2211yescharacteristic
18C4×Dic34—21no—
19C4×Dic34C2211yescharacteristic
20C17×C4⋊C8C2211yescharacteristic
21C4×C8×C17C2211yescharacteristic
22#10587491—21no—
23#10587486—21no—
24#10587486—21no—
25C4⋊#3622C2211yescharacteristic
26C4×#3622C2211yescharacteristic
27C2×Q8×C17C2×C411yescharacteristic
28Q8×D17—22no—
29Q8×D17—22no—
30C2×C4×D17—21no—
31C2×C4×D17—21no—
32C2×C4×D17C2×C411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×Dic34—21no—
36C2×Dic34C2×C411yescharacteristic
37C2×C8×C17—21no—
38C2×C8×C17—21no—
39C4⋊C4×C17—21no—
40C4⋊C4×C17C2×C411yescharacteristic
41C42×C17—21no—
42C42×C17C2311yescharacteristic
43C4⋊Dic17—21no—
44C4⋊Dic17C2×C411yescharacteristic
45#63225—21no—
46#63225—21no—
47#63225—21no—
48C4×Dic17—21no—
49C4×Dic17—21no—
50C4×Dic17—21no—
51C4×Dic17C2×C411yescharacteristic
52C4×Dic17D411yescharacteristic
53C4×Dic17D411yescharacteristic
54C2×#3622—21no—
55C2×#3622—21no—
56#63218—24no—
57C22×D17C22⋊C411yescharacteristic
58Q8×C17—21no—
59Q8×C17M1612yescomplemented
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17C22×C411yescharacteristic
63C2×C4×C17C2×D411yescharacteristic
64C2×C4×C17C2×D411yescharacteristic
65C2×Dic17—21no—
66C2×Dic17—21no—
67C2×Dic17C22⋊C411yescharacteristic
68C2×Dic17C22⋊C411yescharacteristic
69C2×Dic17C22⋊C411yescharacteristic
70C4×D17—24no—
71C4×D17—22no—
72C4×D17—21no—
73C4×D17—21no—
74C4×D17—21no—
75C4×D17—21no—
76C4×D17—42no—
77Dic34—24no—
78Dic34—21no—
79Dic34M1612yescomplemented
80C8×C17—22no—
81C8×C17—41no—
82#3622—22no—
83#3622—41no—
84#1491—171nomaximal, Sylow
85C22×C17C2×C22⋊C411yescharacteristic
86C2×D17—21no—
87C2×D17—21no—
88C2×D17#10412yes—
89C4×C17—22no—
90C4×C17—21no—
91C4×C17—41no—
92C4×C17C2×M1611yescharacteristic
93C4×C17C2×M1611yescharacteristic
94C4×C17#13711yescharacteristic
95C4×C17#13711yescharacteristic
96Dic17—21no—
97Dic17—22no—
98Dic17—21no—
99Dic17—21no—
100Dic17—41no—
101C2×C4×Q8—171no—
102#432—171no—
103C4×M16—171no—
104Q8⋊C8—171no—
105Q8⋊C8—171no—
106Q8⋊C8—171no—
107Q8⋊C8D1711yescharacteristic, complemented
108C2×C17#40711yescharacteristic
109C2×C17#41911yescharacteristic
110C2×C17C2×#10411yescharacteristic
111D17—22no—
112C22×Q8—171no—
113C2×M16—341no—
114C4×Q8—171no—
115C4×Q8—171no—
116C4×Q8—171no—
117C4×Q8—341no—
118C4×Q8—341no—
119C4×Q8C2×D1711yescharacteristic
120C2×C4⋊C4—171no—
121C2×C4⋊C4—341no—
122C2×C42—171no—
123C2×C42—341no—
124C4⋊C8—171no—
125C4⋊C8C2×D1711yescharacteristic
126C22⋊C8—341no—
127#97—341no—
128#97—341no—
129C4×C8—171no—
130C4×C8C2×D1711yescharacteristic
131C17#149111yescharacteristic, complemented, Sylow
132C2×Q8—171no—
133C2×Q8—341no—
134C2×Q8—342no—
135C2×Q8—342no—
136C2×Q8C4×D1711yescharacteristic
137C22×C4—171no—
138C22×C4—171no—
139C22×C4—171no—
140C22×C4—341no—
141C22×C4—341no—
142M16—344no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—341no—
146C2×C8—341no—
147C4⋊C4—21no—
148C4⋊C4—171no—
149C4⋊C4—341no—
150C4⋊C4—341no—
151C4⋊C4—341no—
152C4⋊C4—341no—
153C4⋊C4C4×D1711yescharacteristic
154C42—21no—
155C42—171no—
156C42—171no—
157C42—171no—
158C42—341no—
159C42—341no—
160C42—341no—
161C42C22×D1711yescharacteristic
162C23—171no—
163Q8—21no—
164Q8—172no—
165Q8—344no—
166Q8—341no—
167Q8#6321812yescomplemented
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—171no—
171C2×C4—171no—
172C2×C4—171no—
173C2×C4—342no—
174C2×C4—344no—
175C2×C4—341no—
176C2×C4—341no—
177C2×C4—341no—
178C2×C4—341no—
179C2×C4—341no—
180C2×C4—341no—
181C2×C4—682no—
182C2×C4C2×C4×D1711yescharacteristic
183C2×C4D4×D1711yescharacteristic
184C2×C4D4×D1711yescharacteristic
185C8—22no—
186C8—41no—
187C8—342no—
188C8—681no—
189C22—172no—
190C22—341no—
191C22—341no—
192C22C22⋊C4×D1711yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—341no—
197C4—342no—
198C4—341no—
199C4—341no—
200C4—681no—
201C4C2×#6321811yescharacteristic
202C4M16×D1711yescharacteristic
203C4#1058760111yescharacteristic
204C4#1058760411yescharacteristic
205C2—342no—
206C2#1176277111yescharacteristic
207C2#1176281611yescharacteristic
208C2#104×D1711yescharacteristic
subgroupG/H
centerC2×C4D4×D17
commutatorC2×C4×C17C22×C4
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C22⋊C4
cosocleC42×C17C23
hypercenterQ8⋊C8D17
hypocenterC17#1491
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #11762541C2semidirect
5. Q8⋊#3622C2semidirect
6. #11762539C2semidirect
7. C17×Q8⋊C8C2semidirect
59. Q8×C17M16semidirect
79. Dic34M16semidirect
107. Q8⋊C8D17semidirect
131. C17#1491semidirect
167. Q8#63218semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D172
#117628652
C4×#632182
#117625412
Q8⋊#36222
#117625392
C17×Q8⋊C82
#149117