Finite Group Databasebeta

← #47984840

Subgroups — #47984840

Order 35456.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 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#31800572C211yesmaximal, characteristic
2#31799974C212yesmaximal
3#31799797C211yesmaximal, characteristic, complemented
4#31799796C212yesmaximal, complemented
5#354×C277C211yesmaximal, characteristic, complemented
6C2×#17984562C2211yescharacteristic
7C2×Q8×D277C411yescharacteristic, complemented
8#22579880C411yescharacteristic, complemented
9#22579878C411yescharacteristic, complemented
10#22579867C2211yescharacteristic
11#22579862—22no—
12#22579857—22no—
13#22579837C411yescharacteristic, complemented
14#22579836—22no—
15#22579795C2212yes—
16#22579794—22no—
17#124×C277C2211yescharacteristic
18#99×C277C2212yes—
19#17984562—42no—
20C2×Dic554—21no—
21C2×Dic554C2×C411yescharacteristic
22C2×Dic554C2×C411yescharacteristic
23Q8×D277—42no—
24C2×C22⋊D277D412yes—
25C22×Dic277D412yes—
26C2×C4×D277—21no—
27C2×C4×D277C2×C411yescharacteristic
28C4⋊Dic277—22no—
29#17984542—21no—
30#17984542—22no—
31#17984542—42no—
32C22⋊Dic277—22no—
33C22⋊Dic277—22no—
34#17984540—21no—
35C4×Dic277—42no—
36C4×Dic277C2×C411yescharacteristic
37C2×Q8×C277C2×C411yescharacteristic
38C2×D4×C277C2311yescharacteristic
39C22⋊C4×C277—22no—
40C22⋊C4×C277—22no—
41C42×C277C2×C411yescharacteristic
42C22×D277C22⋊C411yescharacteristic
43Dic554—41no—
44Dic554—42no—
45Dic554—41no—
46C22⋊D277—42no—
47C2×Dic277—22no—
48C2×Dic277—22no—
49C2×Dic277—21no—
50C2×Dic277—21no—
51C2×Dic277—42no—
52C2×Dic277C22⋊C412yes—
53C2×Dic277C22⋊C411yescharacteristic
54C4×D277—22no—
55C4×D277—42no—
56C23×C277C2×D412yes—
57Q8×C277—41no—
58D4×C277—41no—
59C2×C4×C277—21no—
60C2×C4×C277—21no—
61C2×C4×C277—42no—
62C2×C4×C277C22×C411yescharacteristic
63C22×C277—22no—
64C22×C277—42no—
65C22×C277C2×C22⋊C411yescharacteristic
66C2×D277—21no—
67C2×D277—41no—
68C4×C277—21no—
69C4×C277—41no—
70C4×C277—41no—
71C4×C277—82no—
72Dic277—22no—
73Dic277—41no—
74Dic277—41no—
75Dic277—82no—
76Dic277#9912yes—
77C2×C277—21no—
78C2×C277—42no—
79C2×C277C2×#9911yescharacteristic
80D277—41no—
81C277#212911yescharacteristic, complemented, Sylow
82#2129—2771nomaximal, Sylow
83#536—2771no—
84#410—2772no—
85#354—2771no—
86#354—2772no—
87#354D27711yescharacteristic, complemented
88C2×#41—2771no—
89C22×Q8—2771no—
90C4⋊Q8—2771no—
91C4⋊Q8—2771no—
92#124—2771no—
93#124C2×D27711yescharacteristic
94#123—5542no—
95C22⋊Q8—5542no—
96#117—2771no—
97#117—5542no—
98#99—2772no—
99#99—5542no—
100#99C2×D27712yes—
101#41—11082no—
102C2×Q8—2771no—
103C2×Q8—2771no—
104C2×Q8—5541no—
105C2×Q8—11082no—
106C2×Q8C4×D27711yescharacteristic
107C2×D4—2772no—
108C2×D4C22×D27711yescharacteristic
109C22×C4—2772no—
110C22×C4—2771no—
111C22×C4—5541no—
112C4⋊C4—5541no—
113C4⋊C4—5542no—
114C4⋊C4—5542no—
115C4⋊C4—11082no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—5541no—
119C22⋊C4—5542no—
120C22⋊C4—5542no—
121C42—2771no—
122C42—11082no—
123C42C4×D27711yescharacteristic
124C23—2771no—
125C23D4×D27712yes—
126Q8—41no—
127Q8—11081no—
128Q8—11081no—
129Q8—11082no—
130D4—41no—
131D4—11082no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—2771no—
136C2×C4—2772no—
137C2×C4—5542no—
138C2×C4—5542no—
139C2×C4—5542no—
140C2×C4—5541no—
141C2×C4—5541no—
142C2×C4—11082no—
143C2×C4—11082no—
144C2×C4C2×C4×D27711yescharacteristic
145C22—22no—
146C22—42no—
147C22—5541no—
148C22—11081no—
149C22C22⋊C4×D27711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—2772no—
155C4—5542no—
156C4—11081no—
157C4—11081no—
158C4—22162no—
159C2—21no—
160C2—42no—
161C2—11081no—
162C2#99×D27711yescharacteristic
subgroupG/H
centerC2#99×D277
commutatorC2×C4×C277C22×C4
FrattiniC2×D4C22×D277
Fitting#354×C277C2
socleC2×C277C2×#99
prime radicalC2×#17984562C22
cosocleC2×D4×C277C23
hypercenter#354D277
hypocenterC277#2129
LCSG ▷ C2×C4×C277 ▷ C22×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #31799797C2semidirect
4. #31799796C2semidirect
5. #354×C277C2semidirect
7. C2×Q8×D277C4semidirect
8. #22579880C4semidirect
9. #22579878C4semidirect
13. #22579837C4semidirect
81. C277#2129semidirect
87. #354D277semidirect

Maximal subgroups

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

subgroupindex
#318005722
#317999742
#317997972
#317997962
#354×C2772
#2129277