Finite Group Databasebeta

← #36355585

Subgroups — #36355585

Order 22144.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012240C211yesmaximal, characteristic, complemented
2#26011676C211yesmaximal, characteristic, complemented
3#26011673C211yesmaximal, characteristic, complemented
4#26011666C211yesmaximal, characteristic
5#26011289C211yesmaximal, characteristic, complemented
6#26011287C211yesmaximal, characteristic, complemented
7#356×C173C211yesmaximal, characteristic, complemented
8#19149727C2211yescharacteristic
9#19149712—21no—
10#19149710—21no—
11#19149707—21no—
12#19149677C2211yescharacteristic
13#19149674—21no—
14#19149673—21no—
15#19149669—21no—
16#19149668C2211yescharacteristic
17#19149664—21no—
18#19149657—21no—
19#19149655—21no—
20#19149607—21no—
21#19149605—21no—
22#19149605—21no—
23#19149598C2211yescharacteristic
24C4⋊Q8×C173C2211yescharacteristic
25#101×C173C2211yescharacteristic
26#101×C173C2211yescharacteristic
27#14583827—41no—
28#14583826—21no—
29#14583826—21no—
30#14583826—21no—
31C2×Dic346D411yescharacteristic
32Q8×D173—41no—
33C2×D692D411yescharacteristic
34C2×C4×D173—21no—
35Dic692—41no—
36#14583816—41no—
37#14583815—41no—
38#14583815—41no—
39#14583813—41no—
40D1384—41no—
41#14583810—21no—
42C4⋊Dic173—21no—
43#14583805—21no—
44#14583805—21no—
45#14583805—21no—
46#14583805—21no—
47#14583805—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51#14583803—21no—
52#14583803—21no—
53C4×Dic173D411yescharacteristic
54C4×Dic173D411yescharacteristic
55C2×Q8×C173D411yescharacteristic
56C2×Q8×C173C2311yescharacteristic
57M16×C173—21no—
58M16×C173—21no—
59C4⋊C4×C173—21no—
60C4⋊C4×C173—21no—
61C42×C173D411yescharacteristic
62C22×D173—21no—
63Dic346—21no—
64Dic346—21no—
65Dic346—21no—
66Dic346—41no—
67D692—21no—
68D692—21no—
69D692—21no—
70D692—41no—
71C22⋊D173—41no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C4×D173—41no—
76C4×D173—41no—
77C4×D173—41no—
78C4×D173—81no—
79#13063436—41no—
80Q8×C173—41no—
81Q8×C173—41no—
82C2×C4×C173—21no—
83C2×C4×C173—21no—
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2×D411yescharacteristic
87C8×C173—41no—
88C8×C173—41no—
89C22×C173#12011yescharacteristic
90C2×D173—42no—
91C2×D173—41no—
92C2×D173—41no—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—41no—
97C4×C173—41no—
98Dic173—41no—
99Dic173—41no—
100Dic173—41no—
101Dic173—41no—
102C2×C173—21no—
103C2×C173B(4,𝔽2)11yescharacteristic
104D173—81no—
105D173—81no—
106C173#220311yescharacteristic, complemented, Sylow
107#2203—1731nomaximal, Sylow
108#563—1731no—
109#456—1731no—
110#456—1731no—
111#454—1731no—
112#356—1731no—
113#356D17311yescharacteristic, complemented
114#355—1731no—
1152−1+4—1731no—
116#137—3461no—
117#137—3461no—
118AGL(1,ℤ/8)—3461no—
119C4⋊Q8—1731no—
120C4⋊Q8C2×D17311yescharacteristic
121#126—3461no—
122#126—3461no—
123#125—3461no—
124#124—1731no—
125#123—3461no—
126C22⋊Q8—3461no—
127C22⋊Q8—3461no—
128#104—3461no—
129#104—3461no—
130#104—3461no—
131#101—1731no—
132#101C2×D17311yescharacteristic
133#101C2×D17311yescharacteristic
134#41—3461no—
135#41—3461no—
136#41—3461no—
137#41—6921no—
138C2×Q8—1731no—
139C2×Q8—6921no—
140C2×Q8D69211yescharacteristic
141C2×Q8C22×D17311yescharacteristic
142C2×D4—1731no—
143C22×C4—3461no—
144Q16—6921no—
145Q16—6921no—
146SD16—6921no—
147SD16—6921no—
148SD16—6921no—
149D8—6921no—
150M16—21no—
151M16—21no—
152M16—3461no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C22⋊C4—3461no—
164C22⋊C4—3461no—
165C22⋊C4—3461no—
166C42—1731no—
167C42—1731no—
168C42D69211yescharacteristic
169C23—3461no—
170Q8—41no—
171Q8—41no—
172Q8—3461no—
173Q8—3461no—
174Q8—3461no—
175Q8—6921no—
176D4—3461no—
177D4—3461no—
178D4—3461no—
179D4—6921no—
180D4—6921no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4—3461no—
186C2×C4—13841no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4D4×D17311yescharacteristic
191C2×C4D4×D17311yescharacteristic
192C2×C4C2×D69211yescharacteristic
193C8—41no—
194C8—41no—
195C8—6921no—
196C22—6922no—
197C22—6921no—
198C22—6921no—
199C22#1914964911yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—6921no—
206C4—6921no—
207C4—6921no—
208C4—6921no—
209C2—21no—
210C2—13841no—
211C2—13841no—
212C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×Q8×C173C23
FrattiniC2×Q8C22×D173
Fitting#356×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011666C2
hypercenter#356D173
hypocenterC173#2203
LCSG ▷ C2×Q8×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #26012240C2semidirect
2. #26011676C2semidirect
3. #26011673C2semidirect
5. #26011289C2semidirect
6. #26011287C2semidirect
7. #356×C173C2semidirect
106. C173#2203semidirect
113. #356D173semidirect

Maximal subgroups

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

subgroupindex
#260122402
#260116762
#260116732
#260116662
#260112892
#260112872
#356×C1732
#2203173