Finite Group Databasebeta

← #36352625

Subgroups — #36352625

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#26011688C211yesmaximal, characteristic, complemented
3#26011683C211yesmaximal, characteristic, complemented
4#26011235C211yesmaximal, characteristic, complemented
5#26011223C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#493×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9C2×#14583814—21no—
10C2×#14583811—21no—
11#19149675—21no—
12#19149651—22no—
13#19149650—21no—
14C4×Dic346C2211yescharacteristic
15C4×D692C2211yescharacteristic
16D4×Dic173—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149600—21no—
20#19149600—21no—
21C2×D8×C173—22no—
22#127×C173C2211yescharacteristic
23#127×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic
25#14583825—42no—
26#14583825—42no—
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—22no—
30C2×C22⋊D173—22no—
31C22×Dic173—22no—
32C2×C4×D173—21no—
33#14583814—42no—
34#14583811—42no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—21no—
40C22⋊Dic173—22no—
41#14583803—21no—
42C4×Dic173—21no—
43C2×D4×C173—21no—
44C2×D4×C173—21no—
45C2×D4×C173—21no—
46C2×D4×C173—21no—
47C2×D4×C173—21no—
48C2×D4×C173D412yes—
49D8×C173—44no—
50C2×C8×C173—21no—
51C42×C173C2311yescharacteristic
52C22×D173—21no—
53Dic346—21no—
54Dic346D812yescomplemented
55D692—21no—
56D692—21no—
57C22⋊D173—44no—
58C22⋊D173—82no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—44no—
63C4×D173—41no—
64C4×D173—41no—
65#13063436—41no—
66#13063436—41no—
67C23×C173—22no—
68C23×C173—41no—
69D4×C173—42no—
70D4×C173—42no—
71D4×C173—41no—
72D4×C173—41no—
73D4×C173—41no—
74D4×C173—41no—
75D4×C173—41no—
76D4×C173—41no—
77D4×C173—41no—
78D4×C173—41no—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C8×C173—22no—
83C22×C173—42no—
84C22×C173—42no—
85C22×C173—42no—
86C22×C173—81no—
87C22×C173—81no—
88C22×C173—81no—
89C22×C173#12011yescharacteristic
90C2×D173—41no—
91C2×D173—41no—
92C2×D173—41no—
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—161no—
103C2×C173#44711yescharacteristic
104C2×C173#44911yescharacteristic
105C2×C173#45311yescharacteristic
106D173—81no—
107C173#162111yescharacteristic, complemented, Sylow
108#1621—1731nomaximal, Sylow
109#550—1731no—
110#493D17311yescharacteristic, complemented
111#460—1731no—
112#459—1731no—
113#331—1731no—
114Q8⋊C8—1731no—
115D4⋊C8—1731no—
116C2×#41—3461no—
117C2×SD16—3461no—
118C2×D8—22no—
119C2×D8—3461no—
120#127—3461no—
121#127C2×D17311yescharacteristic
122#127C2×D17311yescharacteristic
123#121—3462no—
124#121—3461no—
125C4×Q8—1731no—
126C4×D4—3461no—
127C4×D4—1731no—
128C4⋊C8—1731no—
129C4⋊C8—1731no—
130#102—3461no—
131#102—3461no—
132C4×C8C2×D17311yescharacteristic
133#41—6922no—
134#41—6922no—
135C2×Q8—1731no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—3462no—
142C2×D4—3462no—
143C2×D4—1731no—
144C2×D4C22⋊D17312yes—
145C22×C4—3462no—
146C22×C4—3461no—
147SD16—6922no—
148D8—44no—
149D8—6922no—
150C2×C8—21no—
151C2×C8—3461no—
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—41no—
162C23—3461no—
163Q8—3461no—
164Q8—1732no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—3461no—
176D4—3461no—
177D4—13842no—
178D4—6924no—
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×C22⋊D17311yescharacteristic
188C8—22no—
189C8—6921no—
190C8—6921no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—6921no—
198C22—6921no—
199C22—6921no—
200C22#1914964711yescharacteristic
201C4—21no—
202C4—3462no—
203C4—6921no—
204C4—6921no—
205C4D8×D17311yescharacteristic
206C4#1914969811yescharacteristic
207C4#1914970311yescharacteristic
208C4#1914970311yescharacteristic
209C2—82no—
210C2—161no—
211C2—13841no—
212C2#2601162311yescharacteristic
213C2#2601163311yescharacteristic
214C2#2601165811yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#493×C173C2
socleC22×C173#120
hypercenter#493D173
hypocenterC173#1621
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #26012161C2semidirect
2. #26011688C2semidirect
3. #26011683C2semidirect
4. #26011235C2semidirect
5. #26011223C2semidirect
6. #26011220C2semidirect
7. #493×C173C2semidirect
54. Dic346D8semidirect
107. C173#1621semidirect
110. #493D173semidirect

Maximal subgroups

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

subgroupindex
#260121612
#260116882
#260116832
#260112352
#260112232
#260112202
#493×C1732
#1621173