Finite Group Databasebeta

← #36364646

Subgroups — #36364646

Order 22144.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012103C211yesmaximal, characteristic, complemented
2#26011886C211yesmaximal, characteristic, complemented
3#26011870C211yesmaximal, characteristic, complemented
4#26011859C211yesmaximal, characteristic, complemented
5#26011845C211yesmaximal, characteristic, complemented
6#26011710C212yesmaximal, complemented
7#26011698C212yesmaximal, complemented
8#26011653C212yesmaximal
9#26011647C212yesmaximal, complemented
10#26011552C211yesmaximal, characteristic, complemented
11#536×C173C211yesmaximal, characteristic
12C2×#14583816C2212yes—
13C2×#14583816C2212yes—
14C2×#14583816C2212yes—
15C2×#14583814C2212yes—
16#19149679C2211yescharacteristic
17#19149672—22no—
18#19149671C2211yescharacteristic
19#19149669—22no—
20#19149658C2212yes—
21C4×Dic346C2212yes—
22C4⋊#13063436C2212yes—
23#19149603C2212yes—
24#19149603C2212yes—
25#19149603C2212yes—
26#19149600C2212yes—
27C22⋊#13063436C2212yes—
28#19149590C2211yescharacteristic
29C4×#13063436C2211yescharacteristic
30C2×#41×C173C2211yescharacteristic
31C22×Q8×C173C2211yescharacteristic
32C4⋊Q8×C173C2211yescharacteristic
33C4⋊Q8×C173C2211yescharacteristic
34#124×C173C2211yescharacteristic
35#124×C173C2211yescharacteristic
36#123×C173—22no—
37C22⋊Q8×C173—22no—
38#117×C173C2211yescharacteristic
39C2×Dic346C2312yes—
40#14583816—42no—
41#14583816—42no—
42#14583816—42no—
43#14583814—42no—
44C2×#13063436C2312yes—
45C2×#13063436C2312yes—
46C4⋊Dic173—22no—
47C4⋊Dic173C2312yes—
48#14583805—22no—
49#14583805—22no—
50#14583805—22no—
51C22⋊Dic173—22no—
52C4×Dic173—22no—
53#41×C173—42no—
54C2×Q8×C173—21no—
55C2×Q8×C173—42no—
56C2×Q8×C173C2311yescharacteristic
57C2×Q8×C173C2311yescharacteristic
58C2×Q8×C173C2311yescharacteristic
59C2×D4×C173—21no—
60C2×D4×C173C2311yescharacteristic
61C22×C4×C173—21no—
62C22×C4×C173—21no—
63C22×C4×C173C2311yescharacteristic
64C4⋊C4×C173—22no—
65C4⋊C4×C173—22no—
66C4⋊C4×C173D412yes—
67C22⋊C4×C173—22no—
68C22⋊C4×C173—22no—
69C22⋊C4×C173D412yes—
70C42×C173C2311yescharacteristic
71C42×C173C2311yescharacteristic
72Dic346—22no—
73Dic346—22no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76#13063436—22no—
77#13063436—22no—
78C23×C173—21no—
79C23×C173C2×D411yescharacteristic
80Q8×C173—22no—
81Q8×C173—22no—
82Q8×C173—22no—
83Q8×C173—42no—
84D4×C173—22no—
85D4×C173—42no—
86C2×C4×C173—22no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—21no—
90C2×C4×C173—42no—
91C2×C4×C173—42no—
92C2×C4×C173C2×D412yes—
93C2×C4×C173C2×D412yes—
94C2×C4×C173C2×D411yescharacteristic
95C2×C4×C173C2411yescharacteristic
96C22×C173—21no—
97C22×C173—22no—
98C22×C173—41no—
99C22×C173—42no—
100C22×C173C22×D411yescharacteristic
101C4×C173—21no—
102C4×C173—22no—
103C4×C173—22no—
104C4×C173—41no—
105C4×C173—41no—
106C4×C173—41no—
107C4×C173H(5,𝔽2)12yes—
108Dic173—42no—
109Dic173—42no—
110C2×C173—41no—
111C2×C173—81no—
112C2×C173#53511yescharacteristic
113C2×C173#57812yes—
114C173#322311yescharacteristic, complemented, Sylow
115#3223—1731nomaximal, Sylow
116#541—1731no—
117#536D17311yescharacteristic
118#497—1731no—
119#494—1731no—
120#491—1731no—
121#488—1731no—
122#464—1732no—
123#462—1732no—
124#452—1732no—
125#451—1732no—
126#433—1731no—
127C2×#41C2×D17311yescharacteristic
128C22×Q8C2×D17311yescharacteristic
129C2×Q16—1732no—
130C2×Q16—1732no—
131C2×Q16—1732no—
132C2×SD16—1732no—
133C4⋊Q8—1731no—
134C4⋊Q8C2×D17311yescharacteristic
135C4⋊Q8C2×D17311yescharacteristic
136#126—3462no—
137#125—3462no—
138#125—1731no—
139#124C2×D17311yescharacteristic
140#124C2×D17311yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—1732no—
144C4×Q8—1732no—
145#117C2×D17311yescharacteristic
146C4⋊C8—1732no—
147#103—1732no—
148#103—1732no—
149#103—1732no—
150#102—1732no—
151C22⋊C8—1732no—
152#97—1731no—
153C4×C8—1731no—
154#41—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—1732no—
158C2×Q8C22×D17311yescharacteristic
159C2×Q8C22×D17311yescharacteristic
160C2×Q8C22×D17311yescharacteristic
161C2×D4—21no—
162C2×D4C22×D17311yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D17311yescharacteristic
166Q16—6922no—
167Q16—6922no—
168Q16—6922no—
169SD16—6922no—
170C2×C8—1732no—
171C2×C8—1732no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—3462no—
175C4⋊C4—3462no—
176C4⋊C4—3462no—
177C4⋊C4—3462no—
178C4⋊C4—1732no—
179C4⋊C4C22⋊D17312yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—3462no—
183C22⋊C4C22⋊D17312yes—
184C42—3462no—
185C42C22×D17311yescharacteristic
186C42C22×D17311yescharacteristic
187C23—21no—
188C23C2×C22⋊D17311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—3462no—
194Q8—3462no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4C2×C22⋊D17312yes—
206C2×C4C2×C22⋊D17312yes—
207C2×C4C2×C22⋊D17311yescharacteristic
208C2×C4C23×D17311yescharacteristic
209C8—3462no—
210C8—3462no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D17311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—6922no—
223C4—6922no—
224C4#1914972412yes—
225C2—41no—
226C2—81no—
227C2#2601205811yescharacteristic
228C2#2601230612yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#536×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×C173C22
hypercenter#536D173
hypocenterC173#3223
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #26012103C2semidirect
2. #26011886C2semidirect
3. #26011870C2semidirect
4. #26011859C2semidirect
5. #26011845C2semidirect
6. #26011710C2semidirect
7. #26011698C2semidirect
9. #26011647C2semidirect
10. #26011552C2semidirect
114. C173#3223semidirect

Maximal subgroups

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

subgroupindex
#260121032
#260118862
#260118702
#260118592
#260118452
#260117102
#260116982
#260116532
#260116472
#260115522
#536×C1732
#3223173