Finite Group Databasebeta

← C7×#12224 #13189440

Subgroups — C7×#12224 #13189440

Order 1792.

380 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2811C211yesmaximal, characteristic
2C7×#2028C211yesmaximal, characteristic
3C7×#1605C212yesmaximal
4C7×#1354C212yesmaximal
5C7×#1319C211yesmaximal, characteristic
6C7×#544C2211yescharacteristic
7C7×#451—22no—
8C7×#419—21no—
9C7×#395—22no—
10C7×#395C2212yes—
11C7×#391—21no—
12C7×#391—21no—
13C7×#391—22no—
14C7×#391C2211yescharacteristic
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12224C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8D411yescharacteristic
22C22×C7×Q8D412yes—
23C2×C7×Q16—42no—
24C2×C7×M16—21no—
25C7×C4⋊Q8—41no—
26C7×#125—41no—
27C7×C22⋊Q8—22no—
28C7×#117—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4C2311yescharacteristic
33C2×C42×C7—21no—
34C2×C42×C7D411yescharacteristic
35C7×#103—41no—
36C7×#103—41no—
37C7×#103—42no—
38C7×C22⋊C8—22no—
39C7×#95—22no—
40C7×#95—22no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95D412yes—
44#2811C2×C711yescharacteristic
45#2028C2×C711yescharacteristic
46#1605C2×C712yes—
47#1354C2×C712yes—
48#1319C2×C711yescharacteristic
49C2×C7×Q8—24no—
50C2×C7×Q8—21no—
51C2×C7×Q8—22no—
52C2×C7×Q8—41no—
53C2×C7×Q8—42no—
54C2×C7×Q8—41no—
55C2×C7×Q8—82no—
56C2×C7×Q8—81no—
57C2×C7×Q8—81no—
58C2×C7×Q8Q1612yescomplemented
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D412yes—
65C7×Q16—84no—
66C7×M16—41no—
67C7×M16—41no—
68C2×C7×C8—42no—
69C2×C7×C8—41no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4—22no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—82no—
76C7×C4⋊C4—81no—
77C7×C4⋊C4—81no—
78C7×C22⋊C4—41no—
79C42×C7—41no—
80C42×C7—41no—
81C42×C7—81no—
82#544C22×C711yescharacteristic
83#451—22no—
84#419—21no—
85#395—22no—
86#395C22×C712yes—
87#391—21no—
88#391—21no—
89#391—22no—
90#391C22×C711yescharacteristic
91#343C22×C711yescharacteristic
92#339—21no—
93#328—22no—
94#324C22×C712yes—
95C23×C7#12011yescharacteristic
96C7×Q8—42no—
97C7×Q8—44no—
98C7×Q8—81no—
99C7×Q8—84no—
100C7×Q8—82no—
101C7×Q8—82no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—24no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—41no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7C2×Q1611yescharacteristic
121C2×C4×C7#13711yescharacteristic
122C7×C8—81no—
123C7×C8—82no—
124C22×Q8—21no—
125C22×Q8C7×D411yescharacteristic
126C22×Q8C7×D412yes—
127C2×Q16—42no—
128C2×M16—21no—
129C4⋊Q8—41no—
130#125—41no—
131C22⋊Q8—22no—
132#117—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4C23×C711yescharacteristic
137C2×C42—21no—
138C2×C42C7×D411yescharacteristic
139#103—41no—
140#103—41no—
141#103—42no—
142C22⋊C8—22no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95C7×D412yes—
148C22×C7—21no—
149C22×C7—21no—
150C22×C7#45111yescharacteristic
151C22×C7#45511yescharacteristic
152C22×C7B(4,𝔽2)11yescharacteristic
153C4×C7—21no—
154C4×C7—21no—
155C4×C7—42no—
156C4×C7—44no—
157C4×C7—42no—
158C4×C7—42no—
159C4×C7—82no—
160C4×C7—81no—
161C4×C7—81no—
162C2×Q8—24no—
163C2×Q8—21no—
164C2×Q8—22no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—41no—
168C2×Q8—82no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8C7×Q1612yescomplemented
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4C2×C7×D411yescharacteristic
177C22×C4C2×C7×D412yes—
178Q16—84no—
179M16—41no—
180M16—41no—
181C2×C8—42no—
182C2×C8—41no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C42—41no—
193C42—41no—
194C42—81no—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7#160211yescharacteristic
198C2×C7#219911yescharacteristic
199C2×C7#220511yescharacteristic
200C23C7×#12011yescharacteristic
201Q8—42no—
202Q8—44no—
203Q8—81no—
204Q8—84no—
205Q8—82no—
206Q8—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4C2×C7×Q1611yescharacteristic
226C2×C4C7×#13711yescharacteristic
227C8—81no—
228C8—82no—
229C7#1222411yescharacteristic, direct product factor, Sylow
230C22—21no—
231C22—21no—
232C22C7×#45111yescharacteristic
233C22C7×#45511yescharacteristic
234C22C7×B(4,𝔽2)11yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—81no—
243C4—81no—
244C2—21no—
245C2—21no—
246C2C7×#160211yescharacteristic
247C2C7×#219911yescharacteristic
248C2C7×#220511yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
19. #12224C7direct (N × H)
58. C2×C7×Q8Q16semidirect
171. C2×Q8C7×Q16semidirect
229. C7#12224direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28112
C7×#20282
C7×#16052
C7×#13542
C7×#13192
#122247