Finite Group Databasebeta

← #47985192

Subgroups — #47985192

Order 35456.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#31800718C211yesmaximal, characteristic, complemented
2#31800175C212yesmaximal, complemented
3#31799801C211yesmaximal, characteristic, complemented
4#31799798C212yesmaximal, complemented
5#456×C277C211yesmaximal, characteristic, complemented
6#22579905—22no—
7#22579869—21no—
8#22579869C2212yes—
9#22579855—22no—
10C4×D1108—21no—
11#22579838—22no—
12#22579807—22no—
13#22579799C2212yes—
142−1+4×C277C2211yescharacteristic
15#137×C277—21no—
16C4⋊Q8×C277C2211yescharacteristic
17#104×C277—21no—
18#101×C277C2211yescharacteristic
19C2×D1108—21no—
20C2×D1108—21no—
21C2×D1108D412yes—
22C2×C4×D277—22no—
23#17984550—42no—
24#17984548—42no—
25#17984547—22no—
26C4⋊Dic277—21no—
27#17984540—22no—
28#17984540—22no—
29#17984540—22no—
30C4×Dic277—21no—
31C4×Dic277D412yes—
32#41×C277—21no—
33#41×C277—22no—
34#41×C277—41no—
35C2×Q8×C277—41no—
36C2×Q8×C277D411yescharacteristic
37C2×Q8×C277C2311yescharacteristic
38Q16×C277—41no—
39SD16×C277—41no—
40M16×C277—21no—
41C4⋊C4×C277—22no—
42C42×C277D411yescharacteristic
43C22×D277—22no—
44D1108—42no—
45D1108—42no—
46D1108—81no—
47C2×Dic277—22no—
48C4×D277—44no—
49#14555660—42no—
50Q8×C277—21no—
51Q8×C277—22no—
52Q8×C277—41no—
53Q8×C277—41no—
54D4×C277—21no—
55D4×C277—22no—
56D4×C277—41no—
57C2×C4×C277—21no—
58C2×C4×C277—21no—
59C2×C4×C277—41no—
60C2×C4×C277—42no—
61C2×C4×C277C2×D412yes—
62C2×C4×C277C2×D411yescharacteristic
63C8×C277—41no—
64C22×C277—41no—
65C22×C277#12011yescharacteristic
66C2×D277—42no—
67C2×D277—42no—
68C2×D277—42no—
69C4×C277—22no—
70C4×C277—21no—
71C4×C277—22no—
72C4×C277—41no—
73C4×C277—41no—
74Dic277—42no—
75C2×C277—21no—
76C2×C277—81no—
77C2×C277B(4,𝔽2)11yescharacteristic
78D277—82no—
79C277#220411yescharacteristic, complemented, Sylow
80#2204—2771nomaximal, Sylow
81#555—2771no—
82#456D27711yescharacteristic, complemented
83#454—2772no—
84#356—2771no—
85#355—2772no—
862−1+4C2×D27711yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—5542no—
89C4⋊Q8C2×D27711yescharacteristic
90#124—2772no—
91#124—5541no—
92#121—5542no—
93C4×D4—5541no—
94#117—5542no—
95#104—21no—
96#104—5542no—
97#101—2772no—
98#101C2×D27711yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D27711yescharacteristic
104C2×Q8C22×D27711yescharacteristic
105C2×D4—2772no—
106C2×D4—5541no—
107C2×D4—5541no—
108C22×C4—5542no—
109Q16—41no—
110SD16—41no—
111SD16—11082no—
112D8—11082no—
113M16—21no—
114M16—5542no—
115C4⋊C4—22no—
116C4⋊C4—5541no—
117C22⋊C4—5542no—
118C22⋊C4—5542no—
119C22⋊C4—5542no—
120C42—2772no—
121C42—5541no—
122C42C22⋊D27711yescharacteristic
123C23—5542no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—11082no—
132D4—11082no—
133D4—22161no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—5542no—
139C2×C4—11084no—
140C2×C4D4×D27712yes—
141C2×C4C2×C22⋊D27711yescharacteristic
142C8—41no—
143C8—11082no—
144C22—41no—
145C22—11082no—
146C22—11082no—
147C22—11082no—
148C22#2257984811yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—11082no—
155C2—21no—
156C2—81no—
157C2—22162no—
158C2#3180018811yescharacteristic
subgroupG/H
centerC2#31800188
commutator, cosocleC2×Q8×C277C23
FrattiniC2×Q8C22×D277
Fitting#456×C277C2
socleC2×C277B(4,𝔽2)
hypercenter#456D277
hypocenterC277#2204
LCSG ▷ C2×Q8×C277 ▷ C22×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #31800718C2semidirect
2. #31800175C2semidirect
3. #31799801C2semidirect
4. #31799798C2semidirect
5. #456×C277C2semidirect
79. C277#2204semidirect
82. #456D277semidirect

Maximal subgroups

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

subgroupindex
#318007182
#318001752
#317998012
#317997982
#456×C2772
#2204277