Finite Group Databasebeta

← #27141711

Subgroups — #27141711

Order 12416.

536 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 121 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#20260752C212yesmaximal, complemented
2#20260744C212yesmaximal, complemented
3#20260719C211yesmaximal, characteristic, complemented
4#20260709C211yesmaximal, characteristic, complemented
5C2×2−1+4×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7#15690769C2212yes—
8#15690737—24no—
9#15690734—24no—
10#15690729—21no—
11#15690727—21no—
12#15690725—22no—
132−1+4×C97—24no—
142−1+4×C97—24no—
15C2×#41×C97—21no—
16C2×#41×C97—22no—
17C2×#41×C97C412yescomplemented
18C2×#41×C97C2211yescharacteristic
19C2×#41×C97C2211yescharacteristic
20C22×Q8×C97—21no—
21C22×Q8×C97C412yescomplemented
22C22×Q8×C97C2211yescharacteristic
23#13095495—22no—
24#13095495—22no—
25C2×#11715895—22no—
26C4⋊Dic97—22no—
27C22⋊Dic97—22no—
28C4×Dic97—22no—
29#41×C97—216no—
30#41×C97—28no—
31#41×C97—24no—
32#41×C97—24no—
33#41×C97—22no—
34#41×C97—22no—
35#41×C97—22no—
36#41×C97D414yes—
37C2×Q8×C97—28no—
38C2×Q8×C97—24no—
39C2×Q8×C97—24no—
40C2×Q8×C97—22no—
41C2×Q8×C97—21no—
42C2×Q8×C97—22no—
43C2×Q8×C97C2×C411yescharacteristic
44C2×Q8×C97C2×C412yes—
45C2×Q8×C97D414yes—
46C2×Q8×C97D411yescharacteristic
47C2×D4×C97—22no—
48C2×D4×C97—21no—
49C2×D4×C97C2×C411yescharacteristic
50C2×D4×C97D412yes—
51C2×D4×C97D411yescharacteristic
52C22×C4×C97—22no—
53C22×C4×C97—21no—
54C22×C4×C97—22no—
55C22×C4×C97—21no—
56C22×C4×C97C2×C412yes—
57C22×C4×C97C2311yescharacteristic
58C23×C97—21no—
59C23×C97C22⋊C412yes—
60C23×C97C2×D411yescharacteristic
61Q8×C97—28no—
62Q8×C97—21no—
63Q8×C97—22no—
64Q8×C97—24no—
65Q8×C97—22no—
66Q8×C97C22⋊C412yes—
67Q8×C97C22⋊C414yes—
68D4×C97—28no—
69D4×C97—24no—
70D4×C97—24no—
71D4×C97—21no—
72D4×C97—22no—
73D4×C97C22⋊C412yes—
74C2×C4×C97—28no—
75C2×C4×C97—28no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—24no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—24no—
82C2×C4×C97C22⋊C412yes—
83C2×C4×C97C22×C411yescharacteristic
84C2×C4×C97C2×D412yes—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D411yescharacteristic
87C2×Dic97—42no—
88#11715895—42no—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—41no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97C2×C22⋊C411yescharacteristic
95C22×C97#12012yes—
96C4×C97—24no—
97C4×C97—22no—
98C4×C97—22no—
99C4×C97C2×C22⋊C411yescharacteristic
100C4×C97C2×C22⋊C411yescharacteristic
101C4×C97#12012yes—
102Dic97—82no—
103C2×C97—22no—
104C2×C97—21no—
105C2×C97—42no—
106C2×C97—42no—
107C2×C97#37911yescharacteristic
108#1796—971nomaximal, Sylow
109C97#179611yescharacteristic, complemented, Sylow
110C2×2−1+4D9711yescharacteristic
111#421—972no—
112#419—972no—
113#413—971no—
114#410—971no—
1152−1+4—24no—
1162−1+4—24no—
117C2×#41—21no—
118C2×#41—22no—
119C2×#41Dic9712yescomplemented
120C2×#41C2×D9711yescharacteristic
121C2×#41C2×D9711yescharacteristic
122C22×Q8—21no—
123C22×Q8Dic9712yescomplemented
124C22×Q8C2×D9711yescharacteristic
125C2×M16—972no—
126#117—972no—
127#104—1944no—
128#103—1944no—
129#101—1941no—
130#100—1941no—
131#99—1942no—
132#41—216no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—22no—
137#41—22no—
138#41—22no—
139#41C22⋊D9714yes—
140C2×Q8—28no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8C2×Dic9711yescharacteristic
147C2×Q8C2×Dic9712yes—
148C2×Q8C22⋊D9714yes—
149C2×Q8C22⋊D9711yescharacteristic
150C2×D4—22no—
151C2×D4—21no—
152C2×D4C2×Dic9711yescharacteristic
153C2×D4C22⋊D9712yes—
154C2×D4C22⋊D9711yescharacteristic
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4C2×Dic9712yes—
160C22×C4C22×D9711yescharacteristic
161M16—1942no—
162M16—1942no—
163C2×C8—1942no—
164C4⋊C4—1942no—
165C22⋊C4—1942no—
166C42—1942no—
167C23—21no—
168C23C22⋊Dic9712yes—
169C23C2×C22⋊D9711yescharacteristic
170Q8—28no—
171Q8—21no—
172Q8—22no—
173Q8—24no—
174Q8—22no—
175Q8C22⋊Dic9712yes—
176Q8C22⋊Dic9714yes—
177D4—28no—
178D4—24no—
179D4—24no—
180D4—21no—
181D4—22no—
182D4C22⋊Dic9712yes—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—3882no—
192C2×C4C22⋊Dic9712yes—
193C2×C4C22×Dic9711yescharacteristic
194C2×C4C2×C22⋊D9712yes—
195C2×C4C2×C22⋊D9712yes—
196C2×C4C2×C22⋊D9711yescharacteristic
197C8—3882no—
198C22—22no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22C2×C22⋊Dic9711yescharacteristic
204C22#1569077912yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—7762no—
209C4C2×C22⋊Dic9711yescharacteristic
210C4C2×C22⋊Dic9711yescharacteristic
211C4#1569077912yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—42no—
216C2#2026059111yescharacteristic
subgroupG/H
centerC2#20260591
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×2−1+4×C97C2
socleC2×C97#379
cosocleC22×C4×C97C23
hypercenterC2×2−1+4D97
hypocenterC97#1796
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20260752C2semidirect
2. #20260744C2semidirect
3. #20260719C2semidirect
4. #20260709C2semidirect
17. C2×#41×C97C4semidirect
21. C22×Q8×C97C4semidirect
109. C97#1796semidirect
119. C2×#41Dic97semidirect
123. C22×Q8Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607522
#202607442
#202607192
#202607092
C2×2−1+4×C972
#179697