Finite Group Databasebeta

← #36352628

Subgroups — #36352628

Order 22144.

278 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 61 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#26011867C211yesmaximal, characteristic, complemented
2#26011691C211yesmaximal, characteristic, complemented
3#26011686C211yesmaximal, characteristic, complemented
4#26011235C211yesmaximal, characteristic, complemented
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic, complemented
7#550×C173C211yesmaximal, characteristic, complemented
8C2×#14583813—21no—
9C2×#14583811—21no—
10C2×#14583811—22no—
11#19149676C2211yescharacteristic
12C4⋊#13063436C2211yescharacteristic
13C4⋊#13063436C2211yescharacteristic
14#19149599—21no—
15#19149599—21no—
16C4×#13063436C2211yescharacteristic
17C2×#41×C173—21no—
18#127×C173—21no—
19#127×C173C2211yescharacteristic
20#121×C173—22no—
21#121×C173—21no—
22C4×Q8×C173C2211yescharacteristic
23C4×D4×C173—21no—
24C4×D4×C173C2211yescharacteristic
25C2×D692—21no—
26C2×D692—21no—
27C2×D692—21no—
28#14583813—42no—
29#14583811—42no—
30#14583811—44no—
31C2×#13063436—21no—
32C2×#13063436—21no—
33C2×#13063436—21no—
34#41×C173—42no—
35#41×C173—42no—
36C2×Q8×C173D411yescharacteristic
37C2×D4×C173—22no—
38C2×D4×C173—22no—
39C2×D4×C173—21no—
40C2×D4×C173—21no—
41C2×D4×C173D411yescharacteristic
42C2×D4×C173D412yes—
43C22×C4×C173—22no—
44C22×C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173D411yescharacteristic
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—22no—
49C22⋊C4×C173—21no—
50C42×C173—21no—
51C42×C173C2311yescharacteristic
52C22×D173—41no—
53D692—42no—
54D692—41no—
55D692—41no—
56D692—41no—
57D692—41no—
58#13063436—22no—
59#13063436—41no—
60#13063436—41no—
61C23×C173—21no—
62C23×C173—22no—
63Q8×C173—21no—
64Q8×C173D812yescomplemented
65D4×C173—21no—
66D4×C173—21no—
67D4×C173—42no—
68D4×C173—44no—
69D4×C173—41no—
70D4×C173—41no—
71D4×C173—41no—
72D4×C173—41no—
73D4×C173—82no—
74C2×C4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—21no—
77C2×C4×C173—44no—
78C2×C4×C173—41no—
79C2×C4×C173—41no—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C22×C173—42no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—42no—
87C22×C173—42no—
88C22×C173—41no—
89C22×C173#12011yescharacteristic
90C2×D173—81no—
91C2×D173—81no—
92C2×D173—81no—
93C4×C173—22no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173—41no—
97C4×C173C2×D811yescharacteristic
98C4×C173#13511yescharacteristic
99C4×C173AGL(1,ℤ/8)11yescharacteristic
100C4×C173AGL(1,ℤ/8)11yescharacteristic
101C2×C173—81no—
102C2×C173—82no—
103C2×C173#44711yescharacteristic
104C2×C173#44911yescharacteristic
105C2×C173#45311yescharacteristic
106D173—161no—
107C173#162111yescharacteristic, complemented, Sylow
108#1621—1731nomaximal, Sylow
109#550D17311yescharacteristic, complemented
110#493—1731no—
111#460—1731no—
112#459—1731no—
113#331—1731no—
114Q8⋊C8—1731no—
115D4⋊C8—1731no—
116C2×#41—21no—
117C2×SD16—3461no—
118C2×D8—3461no—
119C2×D8—3462no—
120#127—21no—
121#127—1731no—
122#127C2×D17311yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C2×D17311yescharacteristic
126C4×D4—21no—
127C4×D4C2×D17311yescharacteristic
128C4⋊C8—1731no—
129C4⋊C8—1731no—
130#102—3461no—
131#102—3461no—
132C4×C8—1731no—
133#41—42no—
134#41—42no—
135C2×Q8C22⋊D17311yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—3461no—
143C2×D4C22⋊D17311yescharacteristic
144C2×D4C22⋊D17312yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—6922no—
148D8—6924no—
149D8—6922no—
150C2×C8—3461no—
151C2×C8—3461no—
152C2×C8—3461no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D17311yescharacteristic
155C4⋊C4C22⋊D17311yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C22×D17311yescharacteristic
160C23—22no—
161C23—21no—
162C23—6921no—
163Q8—21no—
164Q8#1458381112yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—82no—
174D4—6922no—
175D4—6921no—
176D4—6921no—
177D4—6921no—
178D4—6921no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C22⋊D17311yescharacteristic
186C2×C4C2×C22⋊D17311yescharacteristic
187C2×C4C2×C22⋊D17311yescharacteristic
188C8—3462no—
189C8—6921no—
190C8—6921no—
191C22—42no—
192C22—41no—
193C22—42no—
194C22—41no—
195C22—41no—
196C22—42no—
197C22—13841no—
198C22—13841no—
199C22—13841no—
200C22#1914964811yescharacteristic
201C4—21no—
202C4—22no—
203C4—41no—
204C4—41no—
205C4C2×#1458381111yescharacteristic
206C4#1914969911yescharacteristic
207C4#1914970411yescharacteristic
208C4#1914970511yescharacteristic
209C2—82no—
210C2—81no—
211C2—27681no—
212C2#2601162511yescharacteristic
213C2#2601163611yescharacteristic
214C2#2601165911yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#550×C173C2
socleC22×C173#120
hypercenter#550D173
hypocenterC173#1621
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #26011867C2semidirect
2. #26011691C2semidirect
3. #26011686C2semidirect
4. #26011235C2semidirect
5. Q8⋊#13063436C2semidirect
6. D4⋊#13063436C2semidirect
7. #550×C173C2semidirect
64. Q8×C173D8semidirect
107. C173#1621semidirect
109. #550D173semidirect
164. Q8#14583811semidirect

Maximal subgroups

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

subgroupindex
#260118672
#260116912
#260116862
#260112352
Q8⋊#130634362
D4⋊#130634362
#550×C1732
#1621173