Finite Group Databasebeta

← #47984216

Subgroups — #47984216

Order 35456.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#22579913C212yesmaximal, complemented
2#31800583C211yesmaximal, characteristic
3C2×#22579806C212yesmaximal, complemented
4#31799907C211yesmaximal, characteristic, complemented
5#343×C277C211yesmaximal, characteristic, complemented
6C2×#17984563C2212yes—
7C22×Dic554C2211yescharacteristic
8#22579913—24no—
9#22579913—24no—
10C2×Dic1108—22no—
11C2×#17984552—22no—
12#22579860—22no—
13#22579855—22no—
14#22579854—22no—
15C2×C4×Dic277C2211yescharacteristic
16#22579806—24no—
17#22579787—22no—
18C2×M16×C277C2212yes—
19C2×C4⋊C4×C277C2211yescharacteristic
20#17984563—22no—
21#17984563—22no—
22#17984563—22no—
23#17984563—22no—
24C2×Dic554—22no—
25C2×Dic554—22no—
26C2×Dic554—21no—
27C2×Dic554D412yes—
28C2×Dic554D412yescomplemented
29C2×D1108—22no—
30C2×D1108D412yes—
31C2×C22⋊D277—22no—
32C22×Dic277—21no—
33C22×Dic277—21no—
34C2×C4×D277—22no—
35Dic1108—28no—
36#17984552—28no—
37C4⋊Dic277—22no—
38#17984540—24no—
39C4×Dic277—22no—
40C22×C4×C277D412yes—
41C22×C4×C277C2311yescharacteristic
42M16×C277—22no—
43M16×C277D414yescomplemented
44C2×C8×C277—22no—
45C4⋊C4×C277—22no—
46C22×D277—22no—
47Dic554—24no—
48Dic554—22no—
49Dic554—22no—
50Dic554—42no—
51D1108—22no—
52D1108—22no—
53D1108—82no—
54C22⋊D277—42no—
55C22⋊D277—42no—
56C2×Dic277—22no—
57C2×Dic277—22no—
58C2×Dic277—22no—
59C2×Dic277—41no—
60C2×Dic277—41no—
61C2×Dic277—41no—
62C2×Dic277—41no—
63C4×D277—42no—
64C4×D277—42no—
65C23×C277C2×D411yescharacteristic
66C2×C4×C277—22no—
67C2×C4×C277—42no—
68C2×C4×C277C2×D411yescharacteristic
69C2×C4×C277C2×D412yes—
70C2×C4×C277C2×D412yes—
71C2×C4×C277#4111yescharacteristic
72C8×C277—24no—
73C22×C277—21no—
74C22×C277—21no—
75C22×C277#12011yescharacteristic
76C22×C277#12111yescharacteristic
77C22×C277#12711yescharacteristic
78C2×D277—42no—
79C2×D277—42no—
80C2×D277—42no—
81C4×C277—42no—
82C4×C277#12012yes—
83C4×C277#12112yes—
84Dic277—42no—
85Dic277—42no—
86Dic277—42no—
87C2×C277—21no—
88C2×C277—21no—
89C2×C277#39211yescharacteristic
90C2×C277#45511yescharacteristic
91C2×C277#45611yescharacteristic
92D277—82no—
93C277#200711yescharacteristic, complemented, Sylow
94#2007—2771nomaximal, Sylow
95C2×#137—2772no—
96#537—2771no—
97C2×#104—2772no—
98#391—2771no—
99#343D27711yescharacteristic, complemented
100C2×#41—2772no—
101C22×Q8—2771no—
102#137—5544no—
103#137—5544no—
104C2×Q16—5542no—
105C2×SD16—5542no—
106C2×M16C2×D27712yes—
107C22⋊Q8—5542no—
108#121—5542no—
109#121—5542no—
110C2×C4⋊C4C2×D27711yescharacteristic
111C2×C42—2771no—
112#104—5544no—
113#95—5542no—
114#41—5542no—
115#41—5542no—
116#41—5542no—
117#41—5542no—
118C2×Q8—2772no—
119C2×Q8—2772no—
120C2×Q8—5541no—
121C2×Q8—5542no—
122C2×Q8—5542no—
123C2×D4—2772no—
124C2×D4—5542no—
125C2×D4—5542no—
126C22×C4—5541no—
127C22×C4—5541no—
128C22×C4—5542no—
129C22×C4D110812yes—
130C22×C4C22×D27711yescharacteristic
131Q16—5548no—
132SD16—5548no—
133M16—22no—
134M16D110814yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—5542no—
138C22⋊C4—5544no—
139C42—5542no—
140C23—5542no—
141C23C2×D110811yescharacteristic
142Q8—5544no—
143Q8—5542no—
144Q8—5542no—
145Q8—11082no—
146D4—5542no—
147D4—5542no—
148D4—11082no—
149D4—11082no—
150D4—22162no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—5542no—
154C2×C4—5542no—
155C2×C4—5542no—
156C2×C4—11081no—
157C2×C4—11081no—
158C2×C4—11081no—
159C2×C4—11081no—
160C2×C4—11082no—
161C2×C4—11082no—
162C2×C4D4×D27711yescharacteristic
163C2×C4D4×D27712yes—
164C2×C4C2×D110812yes—
165C2×C4#1798456411yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—11082no—
170C22—11082no—
171C22—11082no—
172C22#2257985011yescharacteristic
173C22#2257985511yescharacteristic
174C22#2257987711yescharacteristic
175C4—42no—
176C4—11082no—
177C4—11082no—
178C4—11082no—
179C4#2257985012yes—
180C4#2257985512yes—
181C2—21no—
182C2—21no—
183C2—22162no—
184C2#3179991211yescharacteristic
185C2#3180018211yescharacteristic
186C2#3180018711yescharacteristic
subgroupG/H
centerC22#22579850
commutator, cosocleC22×C4×C277C23
FrattiniC22×C4C22×D277
Fitting#343×C277C2
socleC22×C277#120
prime radical#31800583C2
hypercenter#343D277
hypocenterC277#2007
LCSG ▷ C22×C4×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#22579913C2semidirect
3. C2×#22579806C2semidirect
4. #31799907C2semidirect
5. #343×C277C2semidirect
28. C2×Dic554D4semidirect
43. M16×C277D4semidirect
93. C277#2007semidirect
99. #343D277semidirect
134. M16D1108semidirect

Maximal subgroups

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

subgroupindex
C2×#225799132
#318005832
C2×#225798062
#317999072
#343×C2772
#2007277