Finite Group Databasebeta

← #36352629

Subgroups — #36352629

Order 22144.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012161C211yesmaximal, characteristic, complemented
2#26011867C211yesmaximal, characteristic, complemented
3#26011692C211yesmaximal, characteristic, complemented
4#26011687C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#331×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9C2×#14583815—21no—
10C2×D1384—21no—
11C2×#14583811—22no—
12#19149676C2211yescharacteristic
13#19149675—21no—
14#19149651—22no—
15#19149650—21no—
16C4×Dic346C2211yescharacteristic
17C4×D692C2211yescharacteristic
18D4×Dic173—21no—
19#19149601—21no—
20#19149601—21no—
21C4×#13063436C2211yescharacteristic
22#127×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25#14583825—42no—
26#14583825—42no—
27C2×Dic346D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692D411yescharacteristic
32C2×C22⋊D173—22no—
33C2×C22⋊D173—22no—
34C22×Dic173—22no—
35C2×C4×D173—21no—
36#14583815—42no—
37D1384—42no—
38#14583811—44no—
39C2×#13063436—21no—
40C4⋊Dic173D411yescharacteristic
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43C22⋊Dic173—22no—
44#14583803—21no—
45C4×Dic173—21no—
46C2×D4×C173—21no—
47C2×D4×C173—21no—
48C2×D4×C173D412yes—
49C2×C8×C173—21no—
50C2×C8×C173—21no—
51C42×C173C2311yescharacteristic
52C22×D173—21no—
53C22×D173—41no—
54Dic346—21no—
55Dic346D812yescomplemented
56D692—21no—
57D692—21no—
58D692—42no—
59D692—41no—
60D692—41no—
61D692—41no—
62D692—41no—
63C22⋊D173—44no—
64C22⋊D173—82no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—44no—
69C4×D173—41no—
70C4×D173—41no—
71#13063436—22no—
72C23×C173—22no—
73D4×C173—42no—
74D4×C173—41no—
75D4×C173—41no—
76D4×C173—41no—
77D4×C173—41no—
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C8×C173—41no—
82C8×C173—41no—
83C22×C173—42no—
84C22×C173—42no—
85C22×C173—42no—
86C22×C173#12011yescharacteristic
87C2×D173—41no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173—81no—
91C2×D173—81no—
92C2×D173—81no—
93C4×C173—21no—
94C4×C173C2×D811yescharacteristic
95C4×C173#13511yescharacteristic
96C4×C173AGL(1,ℤ/8)11yescharacteristic
97C4×C173AGL(1,ℤ/8)11yescharacteristic
98Dic173—22no—
99Dic173—41no—
100Dic173—41no—
101C2×C173—82no—
102C2×C173#44711yescharacteristic
103C2×C173#44911yescharacteristic
104C2×C173#45311yescharacteristic
105D173—81no—
106D173—161no—
107C173#162111yescharacteristic, complemented, Sylow
108#1621—1731nomaximal, Sylow
109#550—1731no—
110#493—1731no—
111#460—1731no—
112#459—1731no—
113#331D17311yescharacteristic, complemented
114Q8⋊C8—1731no—
115D4⋊C8—1731no—
116C2×#41—3461no—
117C2×SD16—3461no—
118C2×D8—3461no—
119C2×D8—3462no—
120#127—3461no—
121#127—1731no—
122#127C2×D17311yescharacteristic
123#121—3462no—
124#121—3461no—
125C4×Q8—1731no—
126C4×D4—3461no—
127C4×D4—1731no—
128C4⋊C8C2×D17311yescharacteristic
129C4⋊C8C2×D17311yescharacteristic
130#102—3461no—
131#102—3461no—
132C4×C8—1731no—
133#41—6922no—
134#41—6922no—
135C2×Q8—1731no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—3462no—
139C2×D4—3462no—
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—3461no—
143C2×D4—1731no—
144C2×D4D69212yes—
145C22×C4—3462no—
146C22×C4—3461no—
147SD16—6922no—
148D8—6922no—
149D8—6924no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—3461no—
153C4⋊C4—3461no—
154C4⋊C4—1731no—
155C4⋊C4—1731no—
156C22⋊C4—3462no—
157C22⋊C4—3461no—
158C42—3461no—
159C42C22×D17311yescharacteristic
160C23—22no—
161C23—3461no—
162C23—6921no—
163Q8—3461no—
164Q8—1732no—
165D4—42no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—3461no—
171D4—3461no—
172D4—13842no—
173D4—6922no—
174D4—6924no—
175D4—6921no—
176D4—6921no—
177D4—6921no—
178D4—6921no—
179C2×C4—3461no—
180C2×C4—3461no—
181C2×C4—3461no—
182C2×C4—6924no—
183C2×C4—6921no—
184C2×C4—6921no—
185C2×C4D4×D17311yescharacteristic
186C2×C4D4×D17311yescharacteristic
187C2×C4C2×D69211yescharacteristic
188C8—41no—
189C8—41no—
190C8—3462no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—13841no—
195C22—13841no—
196C22—13841no—
197C22—6921no—
198C22—6921no—
199C22—6921no—
200C22#1914964911yescharacteristic
201C4—21no—
202C4—3462no—
203C4—6921no—
204C4—6921no—
205C4D8×D17311yescharacteristic
206C4#1914970011yescharacteristic
207C4#1914970311yescharacteristic
208C4#1914970611yescharacteristic
209C2—82no—
210C2—13841no—
211C2—27681no—
212C2#2601162611yescharacteristic
213C2#2601163711yescharacteristic
214C2#2601166111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#331×C173C2
socleC22×C173#120
hypercenter#331D173
hypocenterC173#1621
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #26012161C2semidirect
2. #26011867C2semidirect
3. #26011692C2semidirect
4. #26011687C2semidirect
5. #26011221C2semidirect
6. #26011218C2semidirect
7. #331×C173C2semidirect
55. Dic346D8semidirect
107. C173#1621semidirect
113. #331D173semidirect

Maximal subgroups

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

subgroupindex
#260121612
#260118672
#260116922
#260116872
#260112212
#260112182
#331×C1732
#1621173