Finite Group Databasebeta

← C7×#10584 #13187800

Subgroups — C7×#10584 #13187800

Order 1792.

374 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2844C211yesmaximal, characteristic
2C7×#1847C211yesmaximal, characteristic
3C7×#1836C212yesmaximal
4C7×#1295C212yesmaximal
5C7×#1295C211yesmaximal, characteristic
6C2×C7×#107—21no—
7C2×C7×#106—21no—
8C2×C7×#97C2212yes—
9C2×C7×#97C2211yescharacteristic
10C7×#400—22no—
11C7×#400—21no—
12C7×#395—22no—
13C7×#395—21no—
14C7×#384C2211yescharacteristic
15C7×#382C2212yes—
16C7×#382C2211yescharacteristic
17C7×#340—22no—
18C7×#340—22no—
19#10584C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×C8—21no—
22C22×C7×C8—22no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4D412yes—
28C2×C7×C4⋊C4D412yes—
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C42×C7C2311yescharacteristic
32C7×#107—42no—
33C7×#106—42no—
34C7×#97—21no—
35C7×#97—22no—
36C7×#97Q812yescomplemented
37C7×#97Q814yescomplemented
38C7×#95—22no—
39C7×#95—21no—
40C7×#95—21no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—22no—
44#2844C2×C711yescharacteristic
45#1847C2×C711yescharacteristic
46#1836C2×C712yes—
47#1295C2×C712yes—
48#1295C2×C711yescharacteristic
49C2×C7×Q8—81no—
50C2×C7×Q8—82no—
51C22×C4×C7—21no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7—22no—
55C22×C4×C7—41no—
56C22×C4×C7C2×D411yescharacteristic
57C22×C4×C7C2×D412yes—
58C2×C7×C8—22no—
59C2×C7×C8—22no—
60C2×C7×C8—22no—
61C2×C7×C8—24no—
62C2×C7×C8—42no—
63C2×C7×C8—42no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—44no—
71C7×C4⋊C4—81no—
72C7×C4⋊C4—82no—
73C42×C7C2×Q811yescharacteristic
74C42×C7C2×Q812yes—
75C42×C7#4111yescharacteristic
76C2×#107—21no—
77C2×#106—21no—
78C2×#97C22×C712yes—
79C2×#97C22×C711yescharacteristic
80#400—22no—
81#400—21no—
82#395—22no—
83#395—21no—
84#384C22×C711yescharacteristic
85#382C22×C712yes—
86#382C22×C711yescharacteristic
87#340—22no—
88#340—22no—
89C23×C7#12011yescharacteristic
90C7×Q8—84no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—41no—
96C2×C4×C7—41no—
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—42no—
105C2×C4×C7—81no—
106C2×C4×C7—82no—
107C2×C4×C7C22⋊Q811yescharacteristic
108C2×C4×C7C22⋊Q812yes—
109C2×C4×C7C4⋊Q811yescharacteristic
110C2×C4×C7C4⋊Q812yes—
111C7×C8—24no—
112C7×C8—44no—
113C22×Q8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4C7×D412yes—
121C2×C4⋊C4C7×D412yes—
122C2×C4⋊C4C7×D411yescharacteristic
123C2×C4⋊C4C7×D411yescharacteristic
124C2×C42C23×C711yescharacteristic
125#107—42no—
126#106—42no—
127#97—21no—
128#97—22no—
129#97C7×Q812yescomplemented
130#97C7×Q814yescomplemented
131#95—22no—
132#95—21no—
133#95—21no—
134#95—22no—
135#95—22no—
136#95—22no—
137C22×C7#39511yescharacteristic
138C22×C7#45311yescharacteristic
139C22×C7#45412yes—
140C22×C7#45512yes—
141C22×C7#45611yescharacteristic
142C4×C7—21no—
143C4×C7—21no—
144C4×C7—22no—
145C4×C7—22no—
146C4×C7—81no—
147C4×C7—81no—
148C4×C7—82no—
149C4×C7—82no—
150C4×C7—161no—
151C2×Q8—81no—
152C2×Q8—82no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4C2×C7×D411yescharacteristic
159C22×C4C2×C7×D412yes—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—24no—
164C2×C8—42no—
165C2×C8—42no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—44no—
173C4⋊C4—81no—
174C4⋊C4—82no—
175C42C2×C7×Q811yescharacteristic
176C42C2×C7×Q812yes—
177C42C7×#4111yescharacteristic
178C2×C7#165711yescharacteristic
179C2×C7#165911yescharacteristic
180C2×C7#166412yes—
181C2×C7#206011yescharacteristic
182C2×C7#206112yes—
183C23C7×#12011yescharacteristic
184Q8—84no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4C7×C22⋊Q811yescharacteristic
202C2×C4C7×C22⋊Q812yes—
203C2×C4C7×C4⋊Q811yescharacteristic
204C2×C4C7×C4⋊Q812yes—
205C8—24no—
206C8—44no—
207C7#1058411yescharacteristic, direct product factor, Sylow
208C22C7×#39511yescharacteristic
209C22C7×#45311yescharacteristic
210C22C7×#45412yes—
211C22C7×#45512yes—
212C22C7×#45611yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—161no—
222C2C7×#165711yescharacteristic
223C2C7×#165911yescharacteristic
224C2C7×#166412yes—
225C2C7×#206011yescharacteristic
226C2C7×#206112yes—
subgroupG/H
center, socle, prime radicalC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
19. #10584C7direct (N × H)
36. C7×#97Q8semidirect
37. C7×#97Q8semidirect
129. #97C7×Q8semidirect
130. #97C7×Q8semidirect
207. C7#10584direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28442
C7×#18472
C7×#18362
C7×#12952
#105847