Finite Group Databasebeta

← #31800588

Subgroups — #31800588

Order 17728.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#22579875C211yesmaximal, characteristic, complemented
2#22579873C211yesmaximal, characteristic, complemented
3#22579866C211yesmaximal, characteristic, complemented
4#22579866C211yesmaximal, characteristic, complemented
5#22579865C211yesmaximal, characteristic, complemented
6#22579860C211yesmaximal, characteristic, complemented
7#22579854C211yesmaximal, characteristic, complemented
8#22579853C211yesmaximal, characteristic, complemented
9#22579851C211yesmaximal, characteristic, complemented
10#22579850C211yesmaximal, characteristic, complemented
11#22579848C211yesmaximal, characteristic, complemented
12#22579842C211yesmaximal, characteristic, complemented
13C4×C22⋊D277C211yesmaximal, characteristic, complemented
14C22⋊C4×D277C211yesmaximal, characteristic, complemented
15#123×C277C211yesmaximal, characteristic, complemented
16C23×D277C2211yescharacteristic
17C2×Dic554C2211yescharacteristic, complemented
18C2×D1108C2211yescharacteristic
19C2×D1108C2211yescharacteristic
20C2×C22⋊D277C2211yescharacteristic
21C2×C22⋊D277C2211yescharacteristic
22C2×C22⋊D277C2211yescharacteristic
23C2×C22⋊D277C2211yescharacteristic, complemented
24C2×C4×D277C2211yescharacteristic
25C2×C4×D277C2211yescharacteristic
26C2×C4×D277C2211yescharacteristic, complemented
27C4⋊Dic277C2211yescharacteristic, complemented
28C4⋊Dic277C2211yescharacteristic, complemented
29#17984542C2211yescharacteristic, complemented
30#17984542C2211yescharacteristic, complemented
31C22⋊Dic277C2211yescharacteristic
32C22⋊Dic277C2211yescharacteristic
33C22⋊Dic277C2211yescharacteristic, complemented
34#17984540C2211yescharacteristic
35#17984540C2211yescharacteristic
36#17984540C2211yescharacteristic
37#17984540C2211yescharacteristic
38#17984540C2211yescharacteristic
39#17984540C2211yescharacteristic
40#17984540C2211yescharacteristic, complemented
41#17984540C2211yescharacteristic, complemented
42C4×Dic277C2211yescharacteristic, complemented
43C4×Dic277C2211yescharacteristic, complemented
44C2×D4×C277C2211yescharacteristic
45C22×C4×C277C2211yescharacteristic, complemented
46C4⋊C4×C277C2211yescharacteristic, complemented
47C4⋊C4×C277C2211yescharacteristic, complemented
48C22⋊C4×C277C2211yescharacteristic
49C22⋊C4×C277C2211yescharacteristic
50C22⋊C4×C277C2211yescharacteristic, complemented
51C22×D277—22no—
52C22×D277—41no—
53C22×D277—41no—
54C22×D277C2311yescharacteristic
55C22×D277C2311yescharacteristic
56C22×D277C2311yescharacteristic
57Dic554—41no—
58D1108—22no—
59D1108—41no—
60C22⋊D277—22no—
61C22⋊D277—41no—
62C22⋊D277—41no—
63C22⋊D277—41no—
64C2×Dic277C2311yescharacteristic
65C2×Dic277C2311yescharacteristic
66C2×Dic277C2311yescharacteristic
67C2×Dic277C2311yescharacteristic
68C2×Dic277C2311yescharacteristic
69C4×D277—41no—
70C4×D277—41no—
71C4×D277—41no—
72C23×C277C2311yescharacteristic
73C23×C277C2311yescharacteristic
74D4×C277—41no—
75C2×C4×C277—41no—
76C2×C4×C277C2311yescharacteristic
77C2×C4×C277C2311yescharacteristic
78C2×C4×C277C2311yescharacteristic
79C2×C4×C277C2311yescharacteristic
80C2×C4×C277C2311yescharacteristic
81C22×C277—21no—
82C22×C277—21no—
83C22×C277—21no—
84C22×C277—21no—
85C22×C277—21no—
86C22×C277—21no—
87C22×C277C2411yescharacteristic
88C2×D277—21no—
89C2×D277—21no—
90C2×D277—21no—
91C2×D277—21no—
92C2×D277—21no—
93C2×D277—21no—
94C2×D277—21no—
95C2×D277—21no—
96C2×D277—44no—
97C2×D277#4112yes—
98C4×C277—21no—
99C4×C277—21no—
100C4×C277—21no—
101C4×C277—21no—
102C4×C277—21no—
103Dic277—21no—
104Dic277—21no—
105Dic277—21no—
106Dic277—21no—
107Dic277—21no—
108C2×C277—41no—
109C2×C277—41no—
110C2×C277C2×#4111yescharacteristic
111C2×C277H(5,𝔽2)11yescharacteristic
112C2×C277H(5,𝔽2)11yescharacteristic
113D277—22no—
114D277—41no—
115D277—41no—
116C277#53811yescharacteristic, complemented, Sylow
117#538—2771nomaximal, Sylow
118#126—2771no—
119#126—2771no—
120#124—2771no—
121#124—2771no—
122#123—2771no—
123#123D27711yescharacteristic, complemented
124C22⋊Q8—2771no—
125#121—2771no—
126#121—2771no—
127#121—2771no—
128#120—2771no—
129#120—2771no—
130C4×D4—2771no—
131C4×D4—2771no—
132C2×C22⋊C4—2771no—
133C24—2771no—
134C2×Q8—2771no—
135C2×D4—2771no—
136C2×D4—2771no—
137C2×D4—2771no—
138C2×D4—2771no—
139C2×D4—2771no—
140C2×D4—2771no—
141C2×D4C2×D27711yescharacteristic
142C22×C4—2771no—
143C22×C4—2771no—
144C22×C4—2771no—
145C22×C4C2×D27711yescharacteristic, complemented
146C4⋊C4—2771no—
147C4⋊C4—2771no—
148C4⋊C4—2771no—
149C4⋊C4—2771no—
150C4⋊C4C2×D27711yescharacteristic, complemented
151C4⋊C4C2×D27711yescharacteristic, complemented
152C22⋊C4—2771no—
153C22⋊C4—2771no—
154C22⋊C4—2771no—
155C22⋊C4—2771no—
156C22⋊C4—2771no—
157C22⋊C4—2771no—
158C22⋊C4—2771no—
159C22⋊C4—2771no—
160C22⋊C4—2771no—
161C22⋊C4—2771no—
162C22⋊C4—2771no—
163C22⋊C4C2×D27711yescharacteristic
164C22⋊C4C2×D27711yescharacteristic
165C22⋊C4C2×D27711yescharacteristic, complemented
166C42—2771no—
167C42—2771no—
168C23—2771no—
169C23—2771no—
170C23—2771no—
171C23—5542no—
172C23—11081no—
173C23—11081no—
174C23C22×D27711yescharacteristic
175C23C22×D27711yescharacteristic
176Q8—11081no—
177D4—41no—
178D4—5542no—
179D4—5542no—
180D4—11081no—
181D4—11081no—
182D4—11081no—
183D4—11081no—
184C2×C4—41no—
185C2×C4—2771no—
186C2×C4—2771no—
187C2×C4—2771no—
188C2×C4—2771no—
189C2×C4—2771no—
190C2×C4—11081no—
191C2×C4—11081no—
192C2×C4—11081no—
193C2×C4C22×D27711yescharacteristic
194C2×C4C22×D27711yescharacteristic
195C2×C4C22×D27711yescharacteristic
196C2×C4C22×D27711yescharacteristic
197C2×C4C22×D27711yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—2772no—
205C22—5541no—
206C22—5541no—
207C22—5541no—
208C22—5541no—
209C22—5541no—
210C22—5541no—
211C22—5541no—
212C22—5541no—
213C22—11084no—
214C22C23×D27711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—5541no—
221C4—5541no—
222C4—5541no—
223C4—5541no—
224C4—5541no—
225C2—41no—
226C2—41no—
227C2—5542no—
228C2—11081no—
229C2—11081no—
230C2#41×D27711yescharacteristic
231C2#2257992511yescharacteristic
232C2#2257992611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D277
commutator, socle, cosocleC22×C277C24
Fitting#123×C277C2
hypercenter#123D277
hypocenterC277#538
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #22579875C2semidirect
2. #22579873C2semidirect
3. #22579866C2semidirect
4. #22579866C2semidirect
5. #22579865C2semidirect
6. #22579860C2semidirect
7. #22579854C2semidirect
8. #22579853C2semidirect
9. #22579851C2semidirect
10. #22579850C2semidirect
11. #22579848C2semidirect
12. #22579842C2semidirect
13. C4×C22⋊D277C2semidirect
14. C22⋊C4×D277C2semidirect
15. #123×C277C2semidirect
17. C2×Dic554C22semidirect
23. C2×C22⋊D277C22semidirect
26. C2×C4×D277C22semidirect
27. C4⋊Dic277C22semidirect
28. C4⋊Dic277C22semidirect
29. #17984542C22semidirect
30. #17984542C22semidirect
33. C22⋊Dic277C22semidirect
40. #17984540C22semidirect
41. #17984540C22semidirect
42. C4×Dic277C22semidirect
43. C4×Dic277C22semidirect
45. C22×C4×C277C22semidirect
46. C4⋊C4×C277C22semidirect
47. C4⋊C4×C277C22semidirect
50. C22⋊C4×C277C22semidirect
116. C277#538semidirect
123. #123D277semidirect
145. C22×C4C2×D277semidirect
150. C4⋊C4C2×D277semidirect
151. C4⋊C4C2×D277semidirect
165. C22⋊C4C2×D277semidirect

Maximal subgroups

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

subgroupindex
#225798752
#225798732
#225798662
#225798652
#225798602
#225798542
#225798532
#225798512
#225798502
#225798482
#225798422
C4×C22⋊D2772
C22⋊C4×D2772
#123×C2772
#538277