Finite Group Databasebeta

← #14541694

Subgroups — #14541694

Order 2176.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic34C211yesmaximal, characteristic, complemented
2#11763219C212yesmaximal, complemented
3#11763212C211yesmaximal, characteristic, complemented
4#11763113C212yesmaximal, complemented
5#11763101C211yesmaximal, characteristic, complemented
6#11763099C211yesmaximal, characteristic
7#11763089C212yesmaximal
8#11762940C211yesmaximal, characteristic, complemented
9#11762909C212yesmaximal
10C4×#63229C211yesmaximal, characteristic
11C17×#554C211yesmaximal, characteristic
12C17×C4⋊Q8C2211yescharacteristic
13C17×#125C2212yes—
14C17×C22⋊Q8—21no—
15C17×C22⋊Q8—22no—
16C4×Q8×C17C2211yescharacteristic
17C4×D4×C17C2212yes—
18C4×D4×C17C2211yescharacteristic
19C2×C4⋊C4×C17—21no—
20Q8×Dic17—21no—
21C2×#63229C2211yescharacteristic
22#10587574—22no—
23#10587574—21no—
24#10587555C2211yescharacteristic
25#10587555C2212yes—
26C4×Dic34C2211yescharacteristic
27C4×Dic34C2212yes—
28#10587503C2211yescharacteristic
29#10587503C2212yes—
30#10587481C2212yes—
31#10587481C2211yescharacteristic
32#10587479C2211yescharacteristic
33#10587479C2212yes—
34#10587478C2212yes—
35#10587478C2212yes—
36#10587478C2212yes—
37C4⋊#3622C2212yes—
38C4⋊#3622C2211yescharacteristic
39#10587474C2212yes—
40C4×#3622C2211yescharacteristic
41C2×Q8×C17—21no—
42C2×Q8×C17D411yescharacteristic
43C2×D4×C17C2311yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17—22no—
46C2×Dic34—21no—
47C2×Dic34—22no—
48C2×Dic34C2311yescharacteristic
49C4⋊C4×C17—22no—
50C4⋊C4×C17—22no—
51C4⋊C4×C17—21no—
52C4⋊C4×C17D412yes—
53C4⋊C4×C17D411yescharacteristic
54C4⋊C4×C17C2311yescharacteristic
55C4⋊C4×C17C2312yes—
56C22⋊C4×C17—21no—
57C22⋊C4×C17—22no—
58C42×C17C2312yes—
59C42×C17C2311yescharacteristic
60#63229Q814yescomplemented
61C4⋊Dic17—22no—
62C4⋊Dic17—21no—
63C4⋊Dic17C2312yes—
64C4⋊Dic17C2311yescharacteristic
65#63225—21no—
66#63225—22no—
67C4×Dic17—22no—
68C4×Dic17—21no—
69C2×#3622C2311yescharacteristic
70C2×#3622C2312yes—
71C2×#3622C2311yescharacteristic
72C23×C17—21no—
73Q8×C17—22no—
74Q8×C17—81no—
75D4×C17—21no—
76D4×C17C2×Q812yes—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—42no—
80C2×C4×C17—82no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2×D411yescharacteristic
85C2×C4×C17C2411yescharacteristic
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17—21no—
89Dic34—21no—
90Dic34—82no—
91Dic34—81no—
92Dic34C2×Q812yes—
93#3622—21no—
94#3622—22no—
95#3622C2×Q812yes—
96#3385—171nomaximal, Sylow
97C22×C17—22no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17C22×D411yescharacteristic
101C4×C17—22no—
102C4×C17—22no—
103C4×C17—21no—
104C4×C17—41no—
105C4×C17—42no—
106C4×C17#13712yes—
107C4×C17C22×Q811yescharacteristic
108C4×C172−1+411yescharacteristic
109Dic17—22no—
110Dic17—41no—
111Dic17—42no—
112Q82—171no—
113#554D1711yescharacteristic
114#501—172no—
115#500—171no—
116#479—172no—
117#477—171no—
118#476—171no—
119#474—172no—
120#446—171no—
121#440—172no—
122C4×SD16—171no—
123C2×C17—42no—
124C2×C17D4×Q811yescharacteristic
125C2×C17C2×#13711yescharacteristic
126C2×C17#57611yescharacteristic
127C2×SD16—171no—
128C4⋊Q8—172no—
129C4⋊Q8—171no—
130C4⋊Q8—341no—
131C4⋊Q8—342no—
132C4⋊Q8C2×D1711yescharacteristic
133#125C2×D1712yes—
134C22⋊Q8—21no—
135C22⋊Q8—22no—
136C4×Q8—171no—
137C4×Q8—172no—
138C4×Q8—341no—
139C4×Q8C2×D1711yescharacteristic
140C4×D4C2×D1711yescharacteristic
141C4×D4C2×D1712yes—
142C2×C4⋊C4—21no—
143#107—172no—
144#107—172no—
145#107—172no—
146#106—171no—
147#106—172no—
148C4⋊C8—172no—
149C4⋊C8—171no—
150#103—171no—
151#103—172no—
152#102—171no—
153#102—172no—
154#97—172no—
155C4×C8—171no—
156C17#338511yescharacteristic, complemented, Sylow
157C2×Q8—21no—
158C2×Q8—171no—
159C2×Q8—341no—
160C2×Q8—342no—
161C2×Q8C22⋊D1711yescharacteristic
162C2×D4C22×D1711yescharacteristic
163C22×C4—21no—
164C22×C4—22no—
165SD16—174no—
166C2×C8—171no—
167C2×C8—171no—
168C2×C8—172no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—171no—
173C4⋊C4—172no—
174C4⋊C4—341no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—341no—
178C4⋊C4C22⋊D1711yescharacteristic
179C4⋊C4C22⋊D1712yes—
180C4⋊C4C22×D1711yescharacteristic
181C4⋊C4C22×D1712yes—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C42—342no—
185C42—341no—
186C42C22×D1711yescharacteristic
187C42C22×D1712yes—
188C23—21no—
189Q8—22no—
190Q8—81no—
191Q8—172no—
192Q8—341no—
193Q8—1362no—
194Q8—1361no—
195D4—21no—
196D4Q8×D1712yes—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—341no—
202C2×C4—342no—
203C2×C4—341no—
204C2×C4C2×C22⋊D1712yes—
205C2×C4C2×C22⋊D1711yescharacteristic
206C2×C4C2×C22⋊D1712yes—
207C2×C4C2×C22⋊D1711yescharacteristic
208C2×C4C23×D1711yescharacteristic
209C8—172no—
210C8—341no—
211C8—342no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22C22×C22⋊D1711yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—342no—
222C4—681no—
223C4—682no—
224C4#1058763712yes—
225C4C2×Q8×D1711yescharacteristic
226C4#1058769411yescharacteristic
227C2—42no—
228C2Q8×C22⋊D1711yescharacteristic
229C2C2×#1058763711yescharacteristic
230C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#554C2
socleC22×C17C22×D4
prime radicalC2×D4×C17C23
hypercenter#554D17
hypocenterC17#3385
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. Q8×Dic34C2semidirect
2. #11763219C2semidirect
3. #11763212C2semidirect
4. #11763113C2semidirect
5. #11763101C2semidirect
8. #11762940C2semidirect
60. #63229Q8semidirect
156. C17#3385semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic342
#117632192
#117632122
#117631132
#117631012
#117630992
#117630892
#117629402
#117629092
C4×#632292
C17×#5542
#338517