Finite Group Databasebeta

← #47983117

Subgroups — #47983117

Order 35456.

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

normal subgroups only

This group has 548 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#31800838C211yesmaximal, characteristic
2#31799986C211yesmaximal, characteristic, complemented
3#31799984C212yesmaximal, complemented
4#31799968C212yesmaximal, complemented
5#408×C277C211yesmaximal, characteristic, complemented
6#22579928—21no—
7#22579927—22no—
8#22579926—22no—
9#22579925—21no—
10#22579925C414yes—
11C2×#17984563—22no—
12C2×#17984563C2212yes—
13#41×D277—24no—
14C2×#17984547C2212yescomplemented
15#22579799—21no—
16#22579798—22no—
17#22579797—21no—
18#22579796—22no—
19#22579792—24no—
20C2×#41×C277C2211yescharacteristic
21C2×M16×C277C2211yescharacteristic, complemented
22C22×C8×C277C2211yescharacteristic, complemented
23C22⋊C8×C277—22no—
24#17984564—24no—
25#17984563—24no—
26#17984563—24no—
27#17984563—24no—
28#17984563—22no—
29#17984563—22no—
30#17984563—22no—
31#17984563—22no—
32#17984562—24no—
33#17984562—28no—
34C2×Dic554—22no—
35C2×Dic554D412yes—
36Q8×D277—24no—
37C2×D1108—22no—
38C2×D1108D412yes—
39C2×C22⋊D277—24no—
40C2×C22⋊D277C2×C414yes—
41D4×D277—24no—
42D4×D277—28no—
43C2×C4×D277—24no—
44C2×C4×D277D414yes—
45#17984547—22no—
46#17984547—22no—
47C2×#14555660—22no—
48#41×C277—22no—
49#41×C277—22no—
50C2×Q8×C277D411yescharacteristic
51C2×D4×C277C2×C412yes—
52C2×D4×C277D411yescharacteristic
53C22×C4×C277D412yes—
54C22×C4×C277C2311yescharacteristic
55M16×C277—22no—
56C2×C8×C277—21no—
57C2×C8×C277—21no—
58C2×C8×C277—22no—
59C22×D277—24no—
60C22×D277C22⋊C414yes—
61Dic554—24no—
62Dic554—22no—
63Dic554—22no—
64D1108—24no—
65D1108—22no—
66D1108—22no—
67C22⋊D277—28no—
68C22⋊D277—24no—
69C22⋊D277—24no—
70C2×Dic277—24no—
71C2×Dic277C22⋊C414yes—
72C4×D277—28no—
73C4×D277—24no—
74C4×D277—24no—
75#14555660—42no—
76C23×C277C22⋊C412yes—
77C23×C277C22×C411yescharacteristic
78Q8×C277—22no—
79D4×C277—24no—
80D4×C277—22no—
81C2×C4×C277—22no—
82C2×C4×C277—22no—
83C2×C4×C277C22⋊C412yes—
84C2×C4×C277C2×D412yes—
85C2×C4×C277C2×D412yes—
86C2×C4×C277C2×D411yescharacteristic
87C2×C4×C277C2×D411yescharacteristic
88C8×C277—41no—
89C8×C277—41no—
90C22×C277—22no—
91C22×C277—41no—
92C22×C277—42no—
93C22×C277C2×C22⋊C412yes—
94C22×C277C2×C22⋊C411yescharacteristic
95C2×D277—24no—
96C2×D277—44no—
97C2×D277—84no—
98C4×C277—22no—
99C4×C277#12011yescharacteristic
100C4×C277#12011yescharacteristic
101C4×C277#12012yes—
102Dic277—24no—
103C2×C277—22no—
104C2×C277—21no—
105C2×C277—42no—
106C2×C277#37911yescharacteristic
107D277—44no—
108C277#179111yescharacteristic, complemented, Sylow
109#1791—2771nomaximal, Sylow
110#585—2771no—
111#413—2771no—
112#413—2772no—
113#408—2772no—
114#408D27711yescharacteristic, complemented
1152−1+4—5542no—
1162−1+4—5541no—
117H(5,𝔽2)—2774no—
118H(5,𝔽2)—5542no—
119H(5,𝔽2)—5541no—
120C2×#41—2772no—
121C2×#41—5544no—
122C2×#41—5542no—
123C2×#41C2×D27711yescharacteristic
124C2×M16—2772no—
125C2×M16C2×D27711yescharacteristic, complemented
126C22×C8C2×D27711yescharacteristic, complemented
127#101—5542no—
128#101—5541no—
129#100—5542no—
130#100—5541no—
131C22⋊C8—22no—
132C22⋊C8—5544no—
133#41—22no—
134#41—22no—
135#41—5544no—
136#41—5544no—
137#41—5548no—
138#41—5544no—
139#41—5544no—
140#41—5544no—
141#41—5542no—
142#41—5542no—
143#41—5542no—
144#41—5542no—
145C2×Q8—2772no—
146C2×Q8—5542no—
147C2×Q8—5544no—
148C2×Q8D110811yescharacteristic
149C2×D4—2774no—
150C2×D4—2772no—
151C2×D4—5542no—
152C2×D4—5548no—
153C2×D4—5544no—
154C2×D4—5544no—
155C2×D4C4×D27712yes—
156C2×D4D110811yescharacteristic
157C22×C4—2774no—
158C22×C4—5544no—
159C22×C4C22⋊D27712yes—
160C22×C4C22×D27711yescharacteristic
161M16—22no—
162M16—5542no—
163M16—5542no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—5542no—
168C23—2774no—
169C23—5544no—
170C23#1798454012yes—
171C23C2×C4×D27711yescharacteristic
172Q8—22no—
173Q8—5544no—
174Q8—5542no—
175Q8—5542no—
176D4—24no—
177D4—22no—
178D4—5544no—
179D4—5548no—
180D4—5544no—
181D4—5544no—
182D4—5542no—
183D4—5542no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—2774no—
187C2×C4—5548no—
188C2×C4—5544no—
189C2×C4—5544no—
190C2×C4—5544no—
191C2×C4#1798454012yes—
192C2×C4D4×D27712yes—
193C2×C4D4×D27712yes—
194C2×C4C2×C22⋊D27711yescharacteristic
195C2×C4C2×D110811yescharacteristic
196C8—41no—
197C8—41no—
198C8—11082no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—5544no—
203C22—11084no—
204C22—22164no—
205C22C22⋊C4×D27712yes—
206C22C2×#1798454011yescharacteristic
207C4—22no—
208C4—5544no—
209C4#2257984811yescharacteristic
210C4#2257984811yescharacteristic
211C4#2257985012yes—
212C2—21no—
213C2—22no—
214C2—42no—
215C2—11084no—
216C2#3179985711yescharacteristic
subgroupG/H
centerC4#22579848
commutatorC23×C277C22×C4
FrattiniC22×C4C22×D277
Fitting#408×C277C2
socleC2×C277#379
prime radical#31800838C2
cosocleC22×C4×C277C23
hypercenter#408D277
hypocenterC277#1791
LCSG ▷ C23×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #31799986C2semidirect
3. #31799984C2semidirect
4. #31799968C2semidirect
5. #408×C277C2semidirect
14. C2×#17984547C22semidirect
21. C2×M16×C277C22semidirect
22. C22×C8×C277C22semidirect
108. C277#1791semidirect
114. #408D277semidirect
125. C2×M16C2×D277semidirect
126. C22×C8C2×D277semidirect

Maximal subgroups

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

subgroupindex
#318008382
#317999862
#317999842
#317999682
#408×C2772
#1791277