Finite Group Databasebeta

← #31779421

Subgroups — #31779421

Order 17536.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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#22577937C211yesmaximal, characteristic
2#22577460C211yesmaximal, characteristic, complemented
3#22577144C212yesmaximal, complemented
4#22577142C212yesmaximal, complemented
5#432×C137C211yesmaximal, characteristic, complemented
6#17984082—21no—
7#17984080C2211yescharacteristic
8#17984076C2211yescharacteristic
9#17984071—21no—
10#17984067C411yescharacteristic
11#17984062C411yescharacteristic
12C4×Dic274C411yescharacteristic
13C4×D548C411yescharacteristic
14C4×C22⋊D137—21no—
15#17984045—21no—
16C4⋊#11764040—21no—
17C22⋊#11764040—21no—
18#17983999C2212yes—
19C2×C42×C137C2211yescharacteristic
20C4⋊C8×C137—21no—
21C22⋊C8×C137—21no—
22#97×C137C2212yes—
23C2×Dic274C2×C411yescharacteristic
24C2×D548C2×C411yescharacteristic
25C2×C22⋊D137—21no—
26C2×C4×D137—21no—
27C2×#11764040—22no—
28C4⋊Dic137C2×C411yescharacteristic
29C4⋊Dic137C2×C411yescharacteristic
30#14555489—21no—
31#14555489—21no—
32#14555489—21no—
33#14555489—21no—
34C22⋊Dic137—21no—
35#14555487—21no—
36#14555487—21no—
37#14555487—21no—
38#14555487—21no—
39C4×Dic137—21no—
40C22×C4×C137C2×C411yescharacteristic
41C22×C4×C137D411yescharacteristic
42C22×C4×C137D411yescharacteristic
43C2×C8×C137—22no—
44C42×C137C2×C411yescharacteristic
45C42×C137D411yescharacteristic
46C42×C137D411yescharacteristic
47C42×C137C2311yescharacteristic
48C22×D137—21no—
49Dic274—41no—
50D548—41no—
51C22⋊D137—42no—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C4×D137—42no—
56#11764040—42no—
57C23×C137C22⋊C411yescharacteristic
58C2×C4×C137—41no—
59C2×C4×C137—41no—
60C2×C4×C137—41no—
61C2×C4×C137C22⋊C411yescharacteristic
62C2×C4×C137C22⋊C411yescharacteristic
63C2×C4×C137C22⋊C411yescharacteristic
64C2×C4×C137C22×C411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C2×C4×C137C2×D411yescharacteristic
67C8×C137—42no—
68C22×C137—21no—
69C22×C137—21no—
70C22×C137—21no—
71C22×C137C2×C22⋊C411yescharacteristic
72C2×D137—41no—
73C2×D137—41no—
74C2×D137—41no—
75C4×C137—21no—
76C4×C137—21no—
77C4×C137—21no—
78C4×C137—21no—
79C4×C137—21no—
80C4×C137—21no—
81Dic137—41no—
82Dic137—41no—
83Dic137—41no—
84C2×C137—41no—
85C2×C137#41311yescharacteristic
86C2×C137#42111yescharacteristic
87C2×C137#42111yescharacteristic
88D137—81no—
89C137#153511yescharacteristic, complemented, Sylow
90#1535—1371nomaximal, Sylow
91#529—1371no—
92#432—1371no—
93#432D13711yescharacteristic, complemented
94#330—1372no—
95#329—1372no—
96#126—2741no—
97#125—1371no—
98#124—1371no—
99#123—2741no—
100C22⋊Q8—1371no—
101#121—1371no—
102C4×Q8—1371no—
103C4×D4—2741no—
104C4×D4—1371no—
105#117—2741no—
106C2×C42C2×D13711yescharacteristic
107C4⋊C8—21no—
108C4⋊C8—2741no—
109C22⋊C8—21no—
110C22⋊C8—2741no—
111#97—1372no—
112#97C2×D13712yes—
113C2×Q8—1371no—
114C2×D4—2741no—
115C2×D4—1371no—
116C22×C4—2741no—
117C22×C4C4×D13711yescharacteristic
118C22×C4C22⋊D13711yescharacteristic
119C22×C4D54811yescharacteristic
120C2×C8—22no—
121C2×C8—2742no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—1371no—
127C4⋊C4—1371no—
128C22⋊C4—2741no—
129C22⋊C4—2741no—
130C22⋊C4—2741no—
131C22⋊C4—2741no—
132C22⋊C4—2741no—
133C42—2741no—
134C42C4×D13711yescharacteristic
135C42C22⋊D13711yescharacteristic
136C42D54811yescharacteristic
137C42C22×D13711yescharacteristic
138C23—2741no—
139C23#1455548711yescharacteristic
140Q8—5481no—
141D4—5481no—
142D4—5482no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—2741no—
147C2×C4—2741no—
148C2×C4—2741no—
149C2×C4—5482no—
150C2×C4#1455548711yescharacteristic
151C2×C4#1455548711yescharacteristic
152C2×C4#1455548711yescharacteristic
153C2×C4C2×C4×D13711yescharacteristic
154C2×C4C2×C22⋊D13711yescharacteristic
155C2×C4C2×D54811yescharacteristic
156C8—42no—
157C8—5482no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—5481no—
162C22—5481no—
163C22—5481no—
164C22C2×#1455548711yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—5481no—
172C4—5481no—
173C4—5481no—
174C2—41no—
175C2—10961no—
176C2#2257738511yescharacteristic
177C2#2257741911yescharacteristic
178C2#2257742111yescharacteristic
subgroupG/H
centerC22C2×#14555487
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#432×C137C2
socleC22×C137C2×C22⋊C4
prime radical#17984062C4
cosocleC42×C137C23
hypercenter#432D137
hypocenterC137#1535
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #22577460C2semidirect
3. #22577144C2semidirect
4. #22577142C2semidirect
5. #432×C137C2semidirect
89. C137#1535semidirect
93. #432D137semidirect

Maximal subgroups

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

subgroupindex
#225779372
#225774602
#225771442
#225771422
#432×C1372
#1535137