Finite Group Databasebeta

← C7×#2044 #11730513

Subgroups — C7×#2044 #11730513

Order 896.

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
1C7×#534C211yesmaximal, characteristic
2C2×C7×#99C212yesmaximal, complemented
3C7×#386C212yesmaximal, complemented
4C7×#385C211yesmaximal, characteristic, complemented
5C7×#342C211yesmaximal, characteristic, complemented
6C22×C7×D4C2212yes—
7C23×C4×C7C2211yescharacteristic
8C7×#123—22no—
9C7×#121—21no—
10C7×#121—21no—
11C7×#120—22no—
12C2×C7×C4⋊C4—21no—
13C2×C7×C22⋊C4—22no—
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C7×#99—24no—
18C7×#95—21no—
19C7×#95—22no—
20#2044C711yesmaximal, characteristic, direct product factor, Sylow
21C24×C7D412yes—
22C24×C7C2311yescharacteristic
23C2×C7×D4—22no—
24C2×C7×D4—22no—
25C2×C7×D4—22no—
26C2×C7×D4—22no—
27C2×C7×D4—22no—
28C2×C7×D4—22no—
29C2×C7×D4—21no—
30C2×C7×D4—21no—
31C22×C4×C7—22no—
32C22×C4×C7—21no—
33C22×C4×C7—21no—
34C22×C4×C7—22no—
35C22×C4×C7—42no—
36C22×C4×C7D412yes—
37C22×C4×C7D411yescharacteristic
38C22×C4×C7D411yescharacteristic
39C7×C4⋊C4—21no—
40C7×C4⋊C4—21no—
41C7×C4⋊C4—42no—
42C7×C22⋊C4—22no—
43C7×C22⋊C4—22no—
44C7×C22⋊C4—22no—
45C7×C22⋊C4—21no—
46C7×C22⋊C4—22no—
47C7×C22⋊C4—22no—
48C7×C22⋊C4—44no—
49C7×C22⋊C4D412yescomplemented
50#534C2×C711yescharacteristic
51C2×#99C2×C712yescomplemented
52#386C2×C712yescomplemented
53#385C2×C711yescharacteristic, complemented
54#342C2×C711yescharacteristic, complemented
55C23×C7—22no—
56C23×C7—22no—
57C23×C7—22no—
58C23×C7—41no—
59C23×C7—41no—
60C23×C7—42no—
61C23×C7—42no—
62C23×C7C2×D412yes—
63C23×C7C2×D411yescharacteristic
64C23×C7C2×D411yescharacteristic
65C23×C7#4111yescharacteristic
66C23×C7#4112yes—
67C7×D4—42no—
68C7×D4—42no—
69C7×D4—42no—
70C7×D4—42no—
71C7×D4—81no—
72C7×D4—81no—
73C2×C4×C7—24no—
74C2×C4×C7—22no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—41no—
82C2×C4×C7—41no—
83C2×C4×C7—41no—
84C2×C4×C7—42no—
85C2×C4×C7—41no—
86C2×C4×C7—41no—
87C2×C4×C7—41no—
88C2×C4×C7—42no—
89C2×C4×C7—42no—
90C2×C4×C7—42no—
91C22×D4C22×C712yes—
92C23×C4C22×C711yescharacteristic
93#123—22no—
94#121—21no—
95#121—21no—
96#120—22no—
97C2×C4⋊C4—21no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4C22×C712yes—
100C2×C22⋊C4C22×C711yescharacteristic
101C2×C22⋊C4C22×C711yescharacteristic
102#99—24no—
103#95—21no—
104#95—22no—
105C22×C7—21no—
106C22×C7—22no—
107C22×C7—21no—
108C22×C7—22no—
109C22×C7—41no—
110C22×C7—41no—
111C22×C7—42no—
112C22×C7—44no—
113C22×C7—42no—
114C22×C7—42no—
115C22×C7—42no—
116C22×C7—82no—
117C22×C7#12011yescharacteristic
118C22×C7#12111yescharacteristic
119C22×C7#12112yes—
120C22×C7#12312yes—
121C22×C7#12411yescharacteristic
122C4×C7—42no—
123C4×C7—42no—
124C4×C7—42no—
125C4×C7—41no—
126C4×C7—41no—
127C4×C7—82no—
128C24C7×D412yes—
129C24C23×C711yescharacteristic
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4C7×D412yes—
144C22×C4C7×D411yescharacteristic
145C22×C4C7×D411yescharacteristic
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—42no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C22⋊C4—21no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—44no—
156C22⋊C4C7×D412yescomplemented
157C2×C7—21no—
158C2×C7—22no—
159C2×C7—22no—
160C2×C7—21no—
161C2×C7—82no—
162C2×C7#39411yescharacteristic
163C2×C7B(4,𝔽2)11yescharacteristic
164C2×C7#45811yescharacteristic
165C23—22no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23C2×C7×D412yes—
173C23C2×C7×D411yescharacteristic
174C23C2×C7×D411yescharacteristic
175C23C7×#4111yescharacteristic
176C23C7×#4112yes—
177D4—42no—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—81no—
182D4—81no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C7#204411yescharacteristic, direct product factor, Sylow
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22C7×#12011yescharacteristic
215C22C7×#12111yescharacteristic
216C22C7×#12112yes—
217C22C7×#12312yes—
218C22C7×#12411yescharacteristic
219C4—42no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—82no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2C7×#39411yescharacteristic
231C2C7×B(4,𝔽2)11yescharacteristic
232C2C7×#45811yescharacteristic
subgroupG/H
center, socleC22×C7#120
commutator, Frattini, cosocleC24C23×C7
prime radicalC7×#534C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#99C2semidirect
3. C7×#386C2semidirect
4. C7×#385C2semidirect
5. C7×#342C2semidirect
20. #2044C7direct (N × H)
49. C7×C22⋊C4D4semidirect
51. C2×#99C2×C7semidirect
52. #386C2×C7semidirect
53. #385C2×C7semidirect
54. #342C2×C7semidirect
156. C22⋊C4C7×D4semidirect
201. C7#2044direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5342
C2×C7×#992
C7×#3862
C7×#3852
C7×#3422
#20447