Finite Group Databasebeta

← #36353538

Subgroups — #36353538

Order 22144.

536 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 121 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#26011508C212yesmaximal, complemented
2#26011500C212yesmaximal, complemented
3#26011475C211yesmaximal, characteristic, complemented
4#26011465C211yesmaximal, characteristic, complemented
5C2×2−1+4×C173C211yesmaximal, characteristic
6C2×#14583810C2212yes—
7#19149638C2212yes—
8#19149606—24no—
9#19149603—24no—
10#19149598—21no—
11#19149596—21no—
12#19149594—22no—
132−1+4×C173—24no—
142−1+4×C173—24no—
15C2×#41×C173—21no—
16C2×#41×C173—22no—
17C2×#41×C173C412yescomplemented
18C2×#41×C173C2211yescharacteristic
19C2×#41×C173C2211yescharacteristic
20C22×Q8×C173—21no—
21C22×Q8×C173C412yescomplemented
22C22×Q8×C173C2211yescharacteristic
23#14583810—22no—
24#14583810—22no—
25C2×#13063436—22no—
26C4⋊Dic173—22no—
27C22⋊Dic173—22no—
28C4×Dic173—22no—
29#41×C173—216no—
30#41×C173—28no—
31#41×C173—24no—
32#41×C173—24no—
33#41×C173—22no—
34#41×C173—22no—
35#41×C173—22no—
36#41×C173D414yes—
37C2×Q8×C173—28no—
38C2×Q8×C173—24no—
39C2×Q8×C173—24no—
40C2×Q8×C173—22no—
41C2×Q8×C173—21no—
42C2×Q8×C173—22no—
43C2×Q8×C173C2×C411yescharacteristic
44C2×Q8×C173C2×C412yes—
45C2×Q8×C173D414yes—
46C2×Q8×C173D411yescharacteristic
47C2×D4×C173—22no—
48C2×D4×C173—21no—
49C2×D4×C173C2×C411yescharacteristic
50C2×D4×C173D412yes—
51C2×D4×C173D411yescharacteristic
52C22×C4×C173—22no—
53C22×C4×C173—21no—
54C22×C4×C173—22no—
55C22×C4×C173—21no—
56C22×C4×C173C2×C412yes—
57C22×C4×C173C2311yescharacteristic
58C2×Dic173—42no—
59#13063436—42no—
60C23×C173—21no—
61C23×C173C22⋊C412yes—
62C23×C173C2×D411yescharacteristic
63Q8×C173—28no—
64Q8×C173—21no—
65Q8×C173—22no—
66Q8×C173—24no—
67Q8×C173—22no—
68Q8×C173C22⋊C412yes—
69Q8×C173C22⋊C414yes—
70D4×C173—28no—
71D4×C173—24no—
72D4×C173—24no—
73D4×C173—21no—
74D4×C173—22no—
75D4×C173C22⋊C412yes—
76C2×C4×C173—28no—
77C2×C4×C173—28no—
78C2×C4×C173—21no—
79C2×C4×C173—22no—
80C2×C4×C173—24no—
81C2×C4×C173—22no—
82C2×C4×C173—22no—
83C2×C4×C173—24no—
84C2×C4×C173C22⋊C412yes—
85C2×C4×C173C22×C411yescharacteristic
86C2×C4×C173C2×D412yes—
87C2×C4×C173C2×D412yes—
88C2×C4×C173C2×D411yescharacteristic
89C22×C173—22no—
90C22×C173—22no—
91C22×C173—41no—
92C22×C173—42no—
93C22×C173—42no—
94C22×C173C2×C22⋊C411yescharacteristic
95C22×C173#12012yes—
96C4×C173—24no—
97C4×C173—22no—
98C4×C173—22no—
99C4×C173C2×C22⋊C411yescharacteristic
100C4×C173C2×C22⋊C411yescharacteristic
101C4×C173#12012yes—
102Dic173—82no—
103C2×C173—22no—
104C2×C173—21no—
105C2×C173—42no—
106C2×C173—42no—
107C2×C173#37911yescharacteristic
108C173#179611yescharacteristic, complemented, Sylow
109#1796—1731nomaximal, Sylow
110C2×2−1+4D17311yescharacteristic
111#421—1732no—
112#419—1732no—
113#413—1731no—
114#410—1731no—
1152−1+4—24no—
1162−1+4—24no—
117C2×#41—22no—
118C2×#41—21no—
119C2×#41Dic17312yescomplemented
120C2×#41C2×D17311yescharacteristic
121C2×#41C2×D17311yescharacteristic
122C22×Q8—21no—
123C22×Q8Dic17312yescomplemented
124C22×Q8C2×D17311yescharacteristic
125C2×M16—1732no—
126#117—1732no—
127#104—3464no—
128#103—3464no—
129#101—3461no—
130#100—3461no—
131#99—3462no—
132#41—216no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—22no—
137#41—22no—
138#41—22no—
139#41C22⋊D17314yes—
140C2×Q8—28no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8C2×Dic17311yescharacteristic
147C2×Q8C2×Dic17312yes—
148C2×Q8C22⋊D17314yes—
149C2×Q8C22⋊D17311yescharacteristic
150C2×D4—21no—
151C2×D4—22no—
152C2×D4C2×Dic17311yescharacteristic
153C2×D4C22⋊D17311yescharacteristic
154C2×D4C22⋊D17312yes—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4C2×Dic17312yes—
160C22×C4C22×D17311yescharacteristic
161M16—3462no—
162M16—3462no—
163C2×C8—3462no—
164C4⋊C4—3462no—
165C22⋊C4—3462no—
166C42—3462no—
167C23—21no—
168C23C22⋊Dic17312yes—
169C23C2×C22⋊D17311yescharacteristic
170Q8—28no—
171Q8—21no—
172Q8—22no—
173Q8—24no—
174Q8—22no—
175Q8C22⋊Dic17312yes—
176Q8C22⋊Dic17314yes—
177D4—28no—
178D4—24no—
179D4—24no—
180D4—21no—
181D4—22no—
182D4C22⋊Dic17312yes—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—6922no—
192C2×C4C22⋊Dic17312yes—
193C2×C4C22×Dic17311yescharacteristic
194C2×C4C2×C22⋊D17312yes—
195C2×C4C2×C22⋊D17312yes—
196C2×C4C2×C22⋊D17311yescharacteristic
197C8—6922no—
198C22—22no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22C2×C22⋊Dic17311yescharacteristic
204C22#1914964812yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—13842no—
209C4C2×C22⋊Dic17311yescharacteristic
210C4C2×C22⋊Dic17311yescharacteristic
211C4#1914964812yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—42no—
216C2#2601134711yescharacteristic
subgroupG/H
centerC2#26011347
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radicalC2×2−1+4×C173C2
socleC2×C173#379
cosocleC22×C4×C173C23
hypercenterC2×2−1+4D173
hypocenterC173#1796
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #26011508C2semidirect
2. #26011500C2semidirect
3. #26011475C2semidirect
4. #26011465C2semidirect
17. C2×#41×C173C4semidirect
21. C22×Q8×C173C4semidirect
108. C173#1796semidirect
119. C2×#41Dic173semidirect
123. C22×Q8Dic173semidirect

Maximal subgroups

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

subgroupindex
#260115082
#260115002
#260114752
#260114652
C2×2−1+4×C1732
#1796173