Finite Group Databasebeta

← #31800675

Subgroups — #31800675

Order 17728.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#22579875C211yesmaximal, characteristic, complemented
3#22579866C211yesmaximal, characteristic, complemented
4#22579864C211yesmaximal, characteristic, complemented
5#22579862C211yesmaximal, characteristic, complemented
6#22579860C211yesmaximal, characteristic, complemented
7#22579857C211yesmaximal, characteristic, complemented
8#22579848C211yesmaximal, characteristic
9D4×Dic277C211yesmaximal, characteristic, complemented
10#22579840C211yesmaximal, characteristic, complemented
11#22579839C211yesmaximal, characteristic, complemented
12#22579838C211yesmaximal, characteristic, complemented
13C2×#17984540C211yesmaximal, characteristic, complemented
14C22⋊C4×D277C211yesmaximal, characteristic, complemented
15#123×C277C211yesmaximal, characteristic, complemented
16C23×D277C2211yescharacteristic
17C2×Dic554C2211yescharacteristic, complemented
18C2×C22⋊D277C2211yescharacteristic
19C2×C22⋊D277C2211yescharacteristic
20C22×Dic277C2211yescharacteristic
21C22×Dic277C2211yescharacteristic, complemented
22C2×C4×D277C2211yescharacteristic
23C2×C4×D277C2211yescharacteristic
24C4⋊Dic277C2211yescharacteristic, complemented
25C4⋊Dic277C2211yescharacteristic, complemented
26C4⋊Dic277C2211yescharacteristic, complemented
27C4⋊Dic277C2211yescharacteristic, complemented
28#17984542C2211yescharacteristic, complemented
29#17984542C2211yescharacteristic, complemented
30C22⋊Dic277C2211yescharacteristic
31C22⋊Dic277C2211yescharacteristic
32C22⋊Dic277C2211yescharacteristic, complemented
33#17984540C2211yescharacteristic
34#17984540C2211yescharacteristic
35#17984540C2211yescharacteristic
36#17984540C2211yescharacteristic
37#17984540C2211yescharacteristic
38#17984540C2211yescharacteristic
39#17984540C2211yescharacteristic
40#17984540C2211yescharacteristic
41C4×Dic277C2211yescharacteristic, complemented
42C4×Dic277C2211yescharacteristic, complemented
43C4×Dic277C2211yescharacteristic, complemented
44C2×D4×C277C2211yescharacteristic
45C22×C4×C277C2211yescharacteristic
46C4⋊C4×C277C2211yescharacteristic, complemented
47C4⋊C4×C277C2211yescharacteristic, complemented
48C22⋊C4×C277C2211yescharacteristic
49C22⋊C4×C277C2211yescharacteristic, complemented
50C22⋊C4×C277C2211yescharacteristic, complemented
51C22×D277—22no—
52C22×D277—22no—
53C22×D277—41no—
54C22×D277C2311yescharacteristic
55C22×D277C2311yescharacteristic
56Dic554—41no—
57C22⋊D277—22no—
58C22⋊D277—41no—
59C2×Dic277—41no—
60C2×Dic277—41no—
61C2×Dic277C2311yescharacteristic
62C2×Dic277C2311yescharacteristic
63C2×Dic277C2311yescharacteristic
64C2×Dic277C2311yescharacteristic
65C2×Dic277C2311yescharacteristic
66C2×Dic277C2311yescharacteristic
67C4×D277—22no—
68C4×D277—41no—
69C23×C277C2311yescharacteristic
70C23×C277C2311yescharacteristic
71D4×C277—22no—
72C2×C4×C277—22no—
73C2×C4×C277C2311yescharacteristic
74C2×C4×C277C2311yescharacteristic
75C2×C4×C277C2311yescharacteristic
76C2×C4×C277C2311yescharacteristic
77C2×C4×C277C2311yescharacteristic
78C22×C277—21no—
79C22×C277—21no—
80C22×C277—21no—
81C22×C277—21no—
82C22×C277—21no—
83C22×C277#4112yes—
84C22×C277C2411yescharacteristic
85C2×D277—21no—
86C2×D277—21no—
87C2×D277—21no—
88C2×D277—21no—
89C2×D277—21no—
90C2×D277—44no—
91C2×D277#4112yes—
92C4×C277—21no—
93C4×C277—21no—
94C4×C277—21no—
95C4×C277—21no—
96C4×C277—21no—
97Dic277—21no—
98Dic277—21no—
99Dic277—21no—
100Dic277—21no—
101Dic277—21no—
102Dic277—21no—
103C2×C277—22no—
104C2×C277—41no—
105C2×C277C2×#4111yescharacteristic
106C2×C277C2×#4111yescharacteristic
107C2×C277H(5,𝔽2)11yescharacteristic
108D277—22no—
109D277—41no—
110C277#55111yescharacteristic, complemented, Sylow
111#551—2771nomaximal, Sylow
112#126—2771no—
113#126—2771no—
114#124—2771no—
115#123—2771no—
116#123—2771no—
117#123D27711yescharacteristic, complemented
118C22⋊Q8—2771no—
119C22⋊Q8—2771no—
120#120—2771no—
121C4×D4—2771no—
122C4×D4—2771no—
123#117—2771no—
124#117—2771no—
125C2×C22⋊C4—2771no—
126C2×C22⋊C4—2771no—
127C24—2771no—
128C2×Q8—2771no—
129C2×D4—2771no—
130C2×D4—2771no—
131C2×D4C2×D27711yescharacteristic
132C22×C4—2771no—
133C22×C4—2771no—
134C22×C4—2771no—
135C22×C4—2771no—
136C22×C4C2×D27711yescharacteristic
137C4⋊C4—2771no—
138C4⋊C4—2771no—
139C4⋊C4—2771no—
140C4⋊C4—2771no—
141C4⋊C4—2771no—
142C4⋊C4—2771no—
143C4⋊C4C2×D27711yescharacteristic, complemented
144C4⋊C4C2×D27711yescharacteristic, complemented
145C22⋊C4—2771no—
146C22⋊C4—2771no—
147C22⋊C4—2771no—
148C22⋊C4—2771no—
149C22⋊C4—2771no—
150C22⋊C4—2771no—
151C22⋊C4—2771no—
152C22⋊C4—2771no—
153C22⋊C4—2771no—
154C22⋊C4—2771no—
155C22⋊C4—2771no—
156C22⋊C4C2×D27711yescharacteristic
157C22⋊C4C2×D27711yescharacteristic, complemented
158C22⋊C4C2×D27711yescharacteristic, complemented
159C42—2771no—
160C42—2771no—
161C42—2771no—
162C23—2771no—
163C23—2771no—
164C23—5542no—
165C23—5542no—
166C23—11081no—
167C23C22×D27711yescharacteristic
168C23C22×D27711yescharacteristic
169Q8—11081no—
170D4—22no—
171D4—5542no—
172D4—11081no—
173C2×C4—22no—
174C2×C4—2771no—
175C2×C4—2771no—
176C2×C4—2771no—
177C2×C4—2771no—
178C2×C4—2771no—
179C2×C4—2771no—
180C2×C4—5542no—
181C2×C4—11081no—
182C2×C4—11081no—
183C2×C4—11081no—
184C2×C4C22×D27711yescharacteristic
185C2×C4C22×D27711yescharacteristic
186C2×C4C22×D27711yescharacteristic
187C2×C4C22×D27711yescharacteristic
188C2×C4C22×D27711yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—2772no—
195C22—5541no—
196C22—5541no—
197C22—5541no—
198C22—5541no—
199C22—5541no—
200C22—11084no—
201C22#1798456212yes—
202C22C23×D27711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—5541no—
209C4—5541no—
210C4—5541no—
211C4—5541no—
212C4—5541no—
213C4—5541no—
214C2—22no—
215C2—41no—
216C2—5542no—
217C2—11081no—
218C2#41×D27711yescharacteristic
219C2C2×#1798456211yescharacteristic
220C2#2257992611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D277
commutator, socle, cosocleC22×C277C24
Fitting#123×C277C2
prime radical#22579848C2
hypercenter#123D277
hypocenterC277#551
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #22579875C2semidirect
2. #22579875C2semidirect
3. #22579866C2semidirect
4. #22579864C2semidirect
5. #22579862C2semidirect
6. #22579860C2semidirect
7. #22579857C2semidirect
9. D4×Dic277C2semidirect
10. #22579840C2semidirect
11. #22579839C2semidirect
12. #22579838C2semidirect
13. C2×#17984540C2semidirect
14. C22⋊C4×D277C2semidirect
15. #123×C277C2semidirect
17. C2×Dic554C22semidirect
21. C22×Dic277C22semidirect
24. C4⋊Dic277C22semidirect
25. C4⋊Dic277C22semidirect
26. C4⋊Dic277C22semidirect
27. C4⋊Dic277C22semidirect
28. #17984542C22semidirect
29. #17984542C22semidirect
32. C22⋊Dic277C22semidirect
41. C4×Dic277C22semidirect
42. C4×Dic277C22semidirect
43. C4×Dic277C22semidirect
46. C4⋊C4×C277C22semidirect
47. C4⋊C4×C277C22semidirect
49. C22⋊C4×C277C22semidirect
50. C22⋊C4×C277C22semidirect
110. C277#551semidirect
117. #123D277semidirect
143. C4⋊C4C2×D277semidirect
144. C4⋊C4C2×D277semidirect
157. C22⋊C4C2×D277semidirect
158. C22⋊C4C2×D277semidirect

Maximal subgroups

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

subgroupindex
#225798752
#225798662
#225798642
#225798622
#225798602
#225798572
#225798482
D4×Dic2772
#225798402
#225798392
#225798382
C2×#179845402
C22⋊C4×D2772
#123×C2772
#551277