Finite Group Databasebeta

← C97×#2044 #27138002

Subgroups — C97×#2044 #27138002

Order 12416.

382 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 84 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#534×C97C211yesmaximal, characteristic
2C2×#99×C97C212yesmaximal, complemented
3#386×C97C212yesmaximal, complemented
4#385×C97C211yesmaximal, characteristic, complemented
5#342×C97C211yesmaximal, characteristic, complemented
6C22×D4×C97C2212yes—
7C23×C4×C97C2211yescharacteristic
8#123×C97—22no—
9#121×C97—21no—
10#121×C97—21no—
11#120×C97—22no—
12C2×C4⋊C4×C97—21no—
13C2×C22⋊C4×C97—22no—
14C2×C22⋊C4×C97C2212yes—
15C2×C22⋊C4×C97C2211yescharacteristic
16C2×C22⋊C4×C97C2211yescharacteristic
17#99×C97—24no—
18#95×C97—21no—
19#95×C97—22no—
20C24×C97D412yes—
21C24×C97C2311yescharacteristic
22C2×D4×C97—22no—
23C2×D4×C97—22no—
24C2×D4×C97—22no—
25C2×D4×C97—22no—
26C2×D4×C97—22no—
27C2×D4×C97—22no—
28C2×D4×C97—21no—
29C2×D4×C97—21no—
30C22×C4×C97—22no—
31C22×C4×C97—21no—
32C22×C4×C97—21no—
33C22×C4×C97—22no—
34C22×C4×C97—42no—
35C22×C4×C97D412yes—
36C22×C4×C97D411yescharacteristic
37C22×C4×C97D411yescharacteristic
38C4⋊C4×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97—42no—
41C22⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C22⋊C4×C97—22no—
44C22⋊C4×C97—21no—
45C22⋊C4×C97—22no—
46C22⋊C4×C97—22no—
47C22⋊C4×C97—44no—
48C22⋊C4×C97D412yescomplemented
49C23×C97—22no—
50C23×C97—22no—
51C23×C97—22no—
52C23×C97—41no—
53C23×C97—41no—
54C23×C97—42no—
55C23×C97—42no—
56C23×C97C2×D412yes—
57C23×C97C2×D411yescharacteristic
58C23×C97C2×D411yescharacteristic
59C23×C97#4111yescharacteristic
60C23×C97#4112yes—
61D4×C97—42no—
62D4×C97—42no—
63D4×C97—42no—
64D4×C97—42no—
65D4×C97—81no—
66D4×C97—81no—
67C2×C4×C97—24no—
68C2×C4×C97—22no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—42no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97—41no—
82C2×C4×C97—42no—
83C2×C4×C97—42no—
84C2×C4×C97—42no—
85C22×C97—21no—
86C22×C97—22no—
87C22×C97—21no—
88C22×C97—22no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—42no—
92C22×C97—44no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97—42no—
96C22×C97—82no—
97C22×C97#12011yescharacteristic
98C22×C97#12111yescharacteristic
99C22×C97#12112yes—
100C22×C97#12312yes—
101C22×C97#12411yescharacteristic
102C4×C97—42no—
103C4×C97—42no—
104C4×C97—42no—
105C4×C97—41no—
106C4×C97—41no—
107C4×C97—82no—
108C2×C97—21no—
109C2×C97—22no—
110C2×C97—22no—
111C2×C97—21no—
112C2×C97—82no—
113C2×C97#39411yescharacteristic
114C2×C97B(4,𝔽2)11yescharacteristic
115C2×C97#45811yescharacteristic
116#2044C9711yesmaximal, characteristic, direct product factor, Sylow
117C97#204411yescharacteristic, direct product factor, Sylow
118#534C2×C9711yescharacteristic
119C2×#99C2×C9712yescomplemented
120#386C2×C9712yescomplemented
121#385C2×C9711yescharacteristic, complemented
122#342C2×C9711yescharacteristic, complemented
123C22×D4C22×C9712yes—
124C23×C4C22×C9711yescharacteristic
125#123—22no—
126#121—21no—
127#121—21no—
128#120—22no—
129C2×C4⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4C22×C9712yes—
132C2×C22⋊C4C22×C9711yescharacteristic
133C2×C22⋊C4C22×C9711yescharacteristic
134#99—24no—
135#95—21no—
136#95—22no—
137C24D4×C9712yes—
138C24C23×C9711yescharacteristic
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4D4×C9712yes—
153C22×C4D4×C9711yescharacteristic
154C22×C4D4×C9711yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—21no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—44no—
165C22⋊C4D4×C9712yescomplemented
166C23—22no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23C2×D4×C9712yes—
174C23C2×D4×C9711yescharacteristic
175C23C2×D4×C9711yescharacteristic
176C23#41×C9711yescharacteristic
177C23#41×C9712yes—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—81no—
183D4—81no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22#120×C9711yescharacteristic
215C22#121×C9711yescharacteristic
216C22#121×C9712yes—
217C22#123×C9712yes—
218C22#124×C9711yescharacteristic
219C4—42no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—82no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#394×C9711yescharacteristic
231C2B(4,𝔽2)×C9711yescharacteristic
232C2#458×C9711yescharacteristic
subgroupG/H
center, socleC22×C97#120
commutator, Frattini, cosocleC24C23×C97
prime radical#534×C97C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C97 ◁ C24×C97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#99×C97C2semidirect
3. #386×C97C2semidirect
4. #385×C97C2semidirect
5. #342×C97C2semidirect
48. C22⋊C4×C97D4semidirect
116. #2044C97direct (N × H)
117. C97#2044direct (N × H)
119. C2×#99C2×C97semidirect
120. #386C2×C97semidirect
121. #385C2×C97semidirect
122. #342C2×C97semidirect
165. C22⋊C4D4×C97semidirect

Maximal subgroups

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

subgroupindex
#534×C972
C2×#99×C972
#386×C972
#385×C972
#342×C972
#204497