Finite Group Databasebeta

← C7×#12257 #13189473

Subgroups — C7×#12257 #13189473

Order 1792.

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

normal subgroups only

This group has 342 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×#2571C211yesmaximal, characteristic
2C7×#2060C211yesmaximal, characteristic
3C7×#1617C212yesmaximal
4C7×#1353C212yesmaximal
5C7×#1325C211yesmaximal, characteristic
6C7×#544C2211yescharacteristic
7C7×#484—22no—
8C7×#428—21no—
9C7×#400—21no—
10C7×#400C2212yes—
11C7×#389—22no—
12C7×#384C2211yescharacteristic
13C7×#382—22no—
14C4×C7×C4⋊C4—21no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12257C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×M16—21no—
21C7×C4⋊Q8—41no—
22C7×#125—41no—
23C7×C22⋊Q8—22no—
24C7×#117—21no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C4⋊C4C2311yescharacteristic
32C2×C42×C7—22no—
33C2×C42×C7D411yescharacteristic
34C7×#107—42no—
35C7×#107—41no—
36C7×#106—41no—
37C7×#103—42no—
38C7×C22⋊C8—22no—
39C7×#95—21no—
40C7×#95—22no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95D412yes—
44#2571C2×C711yescharacteristic
45#2060C2×C711yescharacteristic
46#1617C2×C712yes—
47#1353C2×C712yes—
48#1325C2×C711yescharacteristic
49C2×C7×Q8—22no—
50C22×C4×C7—21no—
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7C2×D411yescharacteristic
55C22×C4×C7C2×D412yes—
56C7×M16—22no—
57C7×M16—41no—
58C2×C7×C8—41no—
59C2×C7×C8—42no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—42no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—44no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—81no—
71C7×C22⋊C4—41no—
72C42×C7—21no—
73C42×C7—44no—
74C42×C7—41no—
75C42×C7Q1612yes—
76#544C22×C711yescharacteristic
77#484—22no—
78#428—21no—
79#400—21no—
80#400C22×C712yes—
81#389—22no—
82#384C22×C711yescharacteristic
83#382—22no—
84C4×C4⋊C4—21no—
85#343C22×C711yescharacteristic
86#339—21no—
87#328—22no—
88#324C22×C712yes—
89C23×C7#12011yescharacteristic
90C7×Q8—82no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—42no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—41no—
105C2×C4×C7—41no—
106C2×C4×C7—41no—
107C2×C4×C7—42no—
108C2×C4×C7C2×Q1611yescharacteristic
109C2×C4×C7#13711yescharacteristic
110C7×C8—42no—
111C7×C8—82no—
112C2×M16—21no—
113C4⋊Q8—41no—
114#125—41no—
115C22⋊Q8—22no—
116#117—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C7×D411yescharacteristic
122C2×C4⋊C4C7×D412yes—
123C2×C4⋊C4C23×C711yescharacteristic
124C2×C42—22no—
125C2×C42C7×D411yescharacteristic
126#107—41no—
127#107—42no—
128#106—41no—
129#103—42no—
130C22⋊C8—22no—
131#95—21no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95C7×D412yes—
136C22×C7—21no—
137C22×C7—21no—
138C22×C7#45111yescharacteristic
139C22×C7#45611yescharacteristic
140C22×C7B(4,𝔽2)11yescharacteristic
141C4×C7—24no—
142C4×C7—21no—
143C4×C7—21no—
144C4×C7—42no—
145C4×C7—82no—
146C4×C7—81no—
147C4×C7—81no—
148C4×C7—81no—
149C4×C7—82no—
150C2×Q8—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4C2×C7×D411yescharacteristic
156C22×C4C2×C7×D412yes—
157M16—22no—
158M16—41no—
159C2×C8—41no—
160C2×C8—42no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—81no—
172C22⋊C4—41no—
173C42—21no—
174C42—44no—
175C42—41no—
176C42C7×Q1612yes—
177C2×C7—21no—
178C2×C7—21no—
179C2×C7#160511yescharacteristic
180C2×C7#219811yescharacteristic
181C2×C7#220411yescharacteristic
182C23C7×#12011yescharacteristic
183Q8—82no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4C2×C7×Q1611yescharacteristic
202C2×C4C7×#13711yescharacteristic
203C8—42no—
204C8—82no—
205C7#1225711yescharacteristic, direct product factor, Sylow
206C22—21no—
207C22—21no—
208C22C7×#45111yescharacteristic
209C22C7×#45611yescharacteristic
210C22C7×B(4,𝔽2)11yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4—82no—
219C4—81no—
220C2—21no—
221C2—21no—
222C2C7×#160511yescharacteristic
223C2C7×#219811yescharacteristic
224C2C7×#220411yescharacteristic
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. #12257C7direct (N × H)
205. C7#12257direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25712
C7×#20602
C7×#16172
C7×#13532
C7×#13252
#122577