Finite Group Databasebeta

← #1796×C173 #36349581

Subgroups — #1796×C173 #36349581

Order 22144.

536 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C2×2−1+4×C173C211yesmaximal, characteristic
2#421×C173C212yesmaximal, complemented
3#419×C173C212yesmaximal, complemented
4#413×C173C211yesmaximal, characteristic, complemented
5#410×C173C211yesmaximal, characteristic, complemented
62−1+4×C173—24no—
72−1+4×C173—24no—
8C2×#41×C173—21no—
9C2×#41×C173—22no—
10C2×#41×C173C412yescomplemented
11C2×#41×C173C2211yescharacteristic
12C2×#41×C173C2211yescharacteristic
13C22×Q8×C173—21no—
14C22×Q8×C173C412yescomplemented
15C22×Q8×C173C2211yescharacteristic
16C2×M16×C173C2212yes—
17#117×C173C2212yes—
18#104×C173—24no—
19#103×C173—24no—
20#101×C173—21no—
21#100×C173—21no—
22#99×C173—22no—
23#41×C173—216no—
24#41×C173—28no—
25#41×C173—24no—
26#41×C173—24no—
27#41×C173—22no—
28#41×C173—22no—
29#41×C173—22no—
30#41×C173D414yes—
31C2×Q8×C173—28no—
32C2×Q8×C173—24no—
33C2×Q8×C173—24no—
34C2×Q8×C173—22no—
35C2×Q8×C173—21no—
36C2×Q8×C173—22no—
37C2×Q8×C173C2×C411yescharacteristic
38C2×Q8×C173C2×C412yes—
39C2×Q8×C173D414yes—
40C2×Q8×C173D411yescharacteristic
41C2×D4×C173—22no—
42C2×D4×C173—21no—
43C2×D4×C173C2×C411yescharacteristic
44C2×D4×C173D412yes—
45C2×D4×C173D411yescharacteristic
46C22×C4×C173—22no—
47C22×C4×C173—21no—
48C22×C4×C173—22no—
49C22×C4×C173—21no—
50C22×C4×C173C2×C412yes—
51C22×C4×C173C2311yescharacteristic
52M16×C173—22no—
53M16×C173—22no—
54C2×C8×C173—22no—
55C4⋊C4×C173—22no—
56C22⋊C4×C173—22no—
57C42×C173—22no—
58C23×C173—21no—
59C23×C173C22⋊C412yes—
60C23×C173C2×D411yescharacteristic
61Q8×C173—28no—
62Q8×C173—21no—
63Q8×C173—22no—
64Q8×C173—24no—
65Q8×C173—22no—
66Q8×C173C22⋊C412yes—
67Q8×C173C22⋊C414yes—
68D4×C173—28no—
69D4×C173—24no—
70D4×C173—24no—
71D4×C173—21no—
72D4×C173—22no—
73D4×C173C22⋊C412yes—
74C2×C4×C173—28no—
75C2×C4×C173—28no—
76C2×C4×C173—21no—
77C2×C4×C173—22no—
78C2×C4×C173—24no—
79C2×C4×C173—22no—
80C2×C4×C173—22no—
81C2×C4×C173—24no—
82C2×C4×C173—42no—
83C2×C4×C173C22⋊C412yes—
84C2×C4×C173C22×C411yescharacteristic
85C2×C4×C173C2×D412yes—
86C2×C4×C173C2×D412yes—
87C2×C4×C173C2×D411yescharacteristic
88C8×C173—42no—
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×C173—82no—
100C4×C173C2×C22⋊C411yescharacteristic
101C4×C173C2×C22⋊C411yescharacteristic
102C4×C173#12012yes—
103C2×C173—22no—
104C2×C173—21no—
105C2×C173—42no—
106C2×C173—42no—
107C2×C173#37911yescharacteristic
108C173#179611yescharacteristic, direct product factor, Sylow
109#1796C17311yesmaximal, characteristic, direct product factor, Sylow
110C2×2−1+4C2×C17311yescharacteristic
111#421C2×C17312yescomplemented
112#419C2×C17312yescomplemented
113#413C2×C17311yescharacteristic, complemented
114#410C2×C17311yescharacteristic, complemented
1152−1+4—24no—
1162−1+4—24no—
117C2×#41—21no—
118C2×#41—22no—
119C2×#41C4×C17312yescomplemented
120C2×#41C22×C17311yescharacteristic
121C2×#41C22×C17311yescharacteristic
122C22×Q8—21no—
123C22×Q8C4×C17312yescomplemented
124C22×Q8C22×C17311yescharacteristic
125C2×M16C22×C17312yes—
126#117C22×C17312yes—
127#104—24no—
128#103—24no—
129#101—21no—
130#100—21no—
131#99—22no—
132#41—216no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—22no—
137#41—22no—
138#41—22no—
139#41D4×C17314yes—
140C2×Q8—28no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8C2×C4×C17311yescharacteristic
147C2×Q8C2×C4×C17312yes—
148C2×Q8D4×C17314yes—
149C2×Q8D4×C17311yescharacteristic
150C2×D4—22no—
151C2×D4—21no—
152C2×D4C2×C4×C17311yescharacteristic
153C2×D4D4×C17312yes—
154C2×D4D4×C17311yescharacteristic
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4C2×C4×C17312yes—
160C22×C4C23×C17311yescharacteristic
161M16—22no—
162M16—22no—
163C2×C8—22no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—21no—
168C23C22⋊C4×C17312yes—
169C23C2×D4×C17311yescharacteristic
170Q8—28no—
171Q8—21no—
172Q8—22no—
173Q8—24no—
174Q8—22no—
175Q8C22⋊C4×C17312yes—
176Q8C22⋊C4×C17314yes—
177D4—28no—
178D4—24no—
179D4—24no—
180D4—21no—
181D4—22no—
182D4C22⋊C4×C17312yes—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—42no—
192C2×C4C22⋊C4×C17312yes—
193C2×C4C22×C4×C17311yescharacteristic
194C2×C4C2×D4×C17312yes—
195C2×C4C2×D4×C17312yes—
196C2×C4C2×D4×C17311yescharacteristic
197C8—42no—
198C22—22no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22C2×C22⋊C4×C17311yescharacteristic
204C22#120×C17312yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—82no—
209C4C2×C22⋊C4×C17311yescharacteristic
210C4C2×C22⋊C4×C17311yescharacteristic
211C4#120×C17312yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—42no—
216C2#379×C17311yescharacteristic
subgroupG/H
center, socleC2×C173#379
commutatorC2×C4C22×C4×C173
Frattini, cosocleC22×C4C23×C173
prime radicalC2×2−1+4×C173C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C173 ◁ C22×C4×C173 ◁ G

Complements

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

NH ≅ G/Ntype
2. #421×C173C2semidirect
3. #419×C173C2semidirect
4. #413×C173C2semidirect
5. #410×C173C2semidirect
10. C2×#41×C173C4semidirect
14. C22×Q8×C173C4semidirect
108. C173#1796direct (N × H)
109. #1796C173direct (N × H)
111. #421C2×C173semidirect
112. #419C2×C173semidirect
113. #413C2×C173semidirect
114. #410C2×C173semidirect
119. C2×#41C4×C173semidirect
123. C22×Q8C4×C173semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4×C1732
#421×C1732
#419×C1732
#413×C1732
#410×C1732
#1796173