Finite Group Databasebeta

← #36369725

Subgroups — #36369725

Order 22144.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D173⋊#108C211yesmaximal, characteristic
2#26012484C212yesmaximal
3D173⋊#101C212yesmaximal
4#26012454C211yesmaximal, characteristic
5C2×Q16×D173C211yesmaximal, characteristic
6D173⋊M16C2212yes—
7D173⋊M16C2211yescharacteristic
8#19149758C2211yescharacteristic
9#19149750—21no—
10#19149749—21no—
11#19149742—24no—
12#19149740—24no—
13#19149733—21no—
14C8×#10621924—21no—
15C2×Q8×D173C2212yes—
16C2×Dic692C411yescharacteristic, complemented
17C2×#14583816C412yescomplemented
18Q16×D173—21no—
19Q16×D173—21no—
20Q16×D173—21no—
21Q16×D173—21no—
22C2×C8×D173C2211yescharacteristic
23C2×Q16×C173C411yescharacteristic, complemented
24#14583836—22no—
25#14583836—21no—
26#14583835—22no—
27#14583835D412yes—
28D173⋊C8—21no—
29D173⋊C8—22no—
30C4⋊#10621924D412yes—
31C22⋊#10621924—21no—
32C4×#10621924—21no—
33C2×Dic346—22no—
34C2×Dic346C2×C412yes—
35Q8×D173—22no—
36Q8×D173—22no—
37Q8×D173—22no—
38Q8×D173—22no—
39C2×C4×D173—22no—
40C2×C4×D173C2311yescharacteristic
41Dic692—21no—
42Dic692—21no—
43#14583816—22no—
44#14583816—22no—
45C2×#13063436C2×C411yescharacteristic
46C8×D173D411yescharacteristic
47C8×D173D411yescharacteristic
48C8×D173D411yescharacteristic
49C8×D173D411yescharacteristic
50C2×Q8×C173C2×C412yes—
51Q16×C173—21no—
52Q16×C173—21no—
53C2×C8×C173C2×C411yescharacteristic
54C2×#10621924—22no—
55#13063443—22no—
56#13063443—42no—
57C22×D173#4111yescharacteristic
58Dic346—22no—
59Dic346—22no—
60Dic346—42no—
61Dic346—42no—
62C2×Dic173—22no—
63C2×Dic173C2×D411yescharacteristic
64C4×D173—42no—
65C4×D173—42no—
66C4×D173C2×D411yescharacteristic
67C4×D173C2×D411yescharacteristic
68C4×D173C2×D411yescharacteristic
69C4×D173#4111yescharacteristic
70#13063436C22⋊C411yescharacteristic
71#13063436C22⋊C411yescharacteristic
72Q8×C173—22no—
73Q8×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173C22×C411yescharacteristic
76C8×C173C22⋊C411yescharacteristic
77C8×C173C22⋊C411yescharacteristic
78C22×C173C4×D411yescharacteristic
79C2×D173—22no—
80C2×D173#12111yescharacteristic
81C2×D173#12411yescharacteristic
82#10621924—42no—
83C4×C173—42no—
84C4×C173C2×C22⋊C411yescharacteristic
85C4×C173C4×D411yescharacteristic
86Dic173—42no—
87Dic173#12111yescharacteristic
88Dic173#12711yescharacteristic
89C2×C173—21no—
90C2×C173#39011yescharacteristic
91D173—21no—
92D173#47312yes—
93C173#197911yescharacteristic, complemented, Sylow
94#1979—1731nomaximal, Sylow
95C22×Q16—1731no—
96C2×#108—1731no—
97#419—1732no—
98C2×#101—1732no—
99#405—1731no—
100C22×Q8—1732no—
101C2×Q16—3461no—
102C2×Q16—3461no—
103C2×Q16—3461no—
104C2×Q16—3461no—
105C2×Q16—1732no—
106C2×Q16—1731no—
107C2×Q16#1062192411yescharacteristic, complemented
108C2×M16—1732no—
109C2×M16—1731no—
110C22×C8—1731no—
111#117—1731no—
112#108—3461no—
113#108—3461no—
114#103—3464no—
115#101—3464no—
116#97—3461no—
117C4×C8—3461no—
118C2×Q8—3462no—
119C2×Q8—3462no—
120C2×Q8—3462no—
121C2×Q8—3462no—
122C2×Q8—3462no—
123C2×Q8—1732no—
124C2×Q8C2×#1062192412yes—
125C22×C4—3462no—
126C22×C4—1731no—
127Q16—21no—
128Q16—21no—
129Q16—3462no—
130Q16—3462no—
131Q16—3461no—
132Q16—3461no—
133M16—3461no—
134M16—3462no—
135M16—3462no—
136M16—1732no—
137C2×C8—3461no—
138C2×C8—3462no—
139C2×C8—1731no—
140C2×C8—1731no—
141C2×C8—1731no—
142C2×C8—1731no—
143C2×C8—1731no—
144C2×C8C2×#1062192411yescharacteristic
145C4⋊C4—1732no—
146C22⋊C4—3461no—
147C42—3461no—
148C23—1731no—
149Q8—22no—
150Q8—22no—
151Q8—3462no—
152Q8—3462no—
153Q8—6922no—
154Q8—6922no—
155C2×C4—22no—
156C2×C4—3462no—
157C2×C4—3462no—
158C2×C4—1731no—
159C2×C4—1731no—
160C2×C4—1731no—
161C2×C4—1731no—
162C2×C4—1731no—
163C2×C4—6922no—
164C2×C4—6922no—
165C2×C4C22×#1062192411yescharacteristic
166C8—3462no—
167C8—1731no—
168C8—1731no—
169C8—6922no—
170C8C22⋊#1062192411yescharacteristic
171C8C22⋊#1062192411yescharacteristic
172C22—3462no—
173C22—1731no—
174C22—1731no—
175C22D4×#1062192411yescharacteristic
176C4—42no—
177C4—1731no—
178C4—1731no—
179C4—6922no—
180C4—6922no—
181C4C2×C22⋊#1062192411yescharacteristic
182C4D4×#1062192411yescharacteristic
183C2—21no—
184C2—3461no—
185C2—1732no—
186C2#2601244311yescharacteristic
subgroupG/H
centerC2#26012443
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×Q16×C173C4
socleC2×C173#390
prime radicalC22×D173#41
cosocleC2×C4×D173C23
hypercenterC2×Q16#10621924
hypocenterC173#1979
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
16. C2×Dic692C4semidirect
17. C2×#14583816C4semidirect
23. C2×Q16×C173C4semidirect
93. C173#1979semidirect
107. C2×Q16#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊#1082
#260124842
D173⋊#1012
#260124542
C2×Q16×D1732
#1979173