Finite Group Databasebeta

← C7×#10087 #13187303

Subgroups — C7×#10087 #13187303

Order 1792.

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

normal subgroups only

This group has 500 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
1C2×C7×#501C212yesmaximal
2C7×#2822C211yesmaximal, characteristic
3C7×#1960C211yesmaximal, characteristic
4C2×C7×#330C212yesmaximal
5C7×#1295C211yesmaximal, characteristic
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×#125C2212yes—
8C7×#501—24no—
9C7×#501—24no—
10C2×C7×#107—22no—
11C2×C7×#106—22no—
12C2×C7×#103—22no—
13C2×C7×#97C2211yescharacteristic
14C2×C7×#97C2212yes—
15C7×#400—22no—
16C7×#398—22no—
17C7×#398—22no—
18C7×#382C2211yescharacteristic
19C7×#330—24no—
20#10087C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×C8—22no—
23C22×C7×C8—21no—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—21no—
26C7×C4⋊Q8D412yes—
27C7×#125—22no—
28C7×#125—22no—
29C7×#125—22no—
30C7×#125—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4D412yes—
36C2×C7×C4⋊C4D412yes—
37C2×C7×C4⋊C4D412yes—
38C2×C42×C7C2311yescharacteristic
39C7×#107—28no—
40C7×#106—28no—
41C7×#103—44no—
42C7×#97—22no—
43C7×#97—22no—
44C7×#97D414yes—
45C7×#95—24no—
46C7×#95—22no—
47C7×#95—22no—
48C2×#501C2×C712yes—
49#2822C2×C711yescharacteristic
50#1960C2×C711yescharacteristic
51C2×#330C2×C712yes—
52#1295C2×C711yescharacteristic
53C2×C7×Q8—22no—
54C2×C7×Q8—42no—
55C2×C7×Q8—42no—
56C2×C7×Q8—41no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—41no—
61C22×C4×C7C2×D412yes—
62C22×C4×C7C2×D411yescharacteristic
63C2×C7×C8—28no—
64C2×C7×C8—24no—
65C2×C7×C8—22no—
66C2×C7×C8—41no—
67C2×C7×C8—41no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—24no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—22no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—82no—
80C42×C7C2×D411yescharacteristic
81C42×C7C2×D412yes—
82C42×C7#4111yescharacteristic
83C2×C4⋊Q8C22×C711yescharacteristic
84C2×#125C22×C712yes—
85#501—24no—
86#501—24no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#103—22no—
90C2×#97C22×C711yescharacteristic
91C2×#97C22×C712yes—
92#400—22no—
93#398—22no—
94#398—22no—
95#382C22×C711yescharacteristic
96#330—24no—
97C23×C7#12011yescharacteristic
98C7×Q8—44no—
99C7×Q8—82no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—24no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—44no—
110C2×C4×C7—44no—
111C2×C4×C7—81no—
112C2×C4×C7—81no—
113C2×C4×C7#12012yes—
114C2×C4×C7#12112yes—
115C2×C4×C7#12111yescharacteristic
116C2×C4×C7#12711yescharacteristic
117C7×C8—28no—
118C7×C8—42no—
119C22×Q8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C4⋊Q8—22no—
123C4⋊Q8—21no—
124C4⋊Q8C7×D412yes—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C7×D412yes—
134C2×C4⋊C4C7×D412yes—
135C2×C4⋊C4C7×D412yes—
136C2×C42C23×C711yescharacteristic
137#107—28no—
138#106—28no—
139#103—44no—
140#97—22no—
141#97—22no—
142#97C7×D414yes—
143#95—24no—
144#95—22no—
145#95—22no—
146C22×C7#39211yescharacteristic
147C22×C7#45311yescharacteristic
148C22×C7#45411yescharacteristic
149C22×C7#45614yes—
150C4×C7—22no—
151C4×C7—22no—
152C4×C7—21no—
153C4×C7—21no—
154C4×C7—82no—
155C4×C7—82no—
156C4×C7—82no—
157C4×C7—82no—
158C2×Q8—22no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—41no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4C2×C7×D412yes—
167C22×C4C2×C7×D411yescharacteristic
168C2×C8—28no—
169C2×C8—24no—
170C2×C8—22no—
171C2×C8—41no—
172C2×C8—41no—
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C42C2×C7×D411yescharacteristic
186C42C2×C7×D412yes—
187C42C7×#4111yescharacteristic
188C2×C7#166012yes—
189C2×C7#166312yes—
190C2×C7#200512yes—
191C2×C7#200611yescharacteristic
192C23C7×#12011yescharacteristic
193Q8—44no—
194Q8—82no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4C7×#12012yes—
209C2×C4C7×#12112yes—
210C2×C4C7×#12111yescharacteristic
211C2×C4C7×#12711yescharacteristic
212C8—28no—
213C8—42no—
214C7#1008711yescharacteristic, direct product factor, Sylow
215C22C7×#39211yescharacteristic
216C22C7×#45311yescharacteristic
217C22C7×#45411yescharacteristic
218C22C7×#45614yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—82no—
226C4—82no—
227C2C7×#166012yes—
228C2C7×#166312yes—
229C2C7×#200512yes—
230C2C7×#200611yescharacteristic
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
20. #10087C7direct (N × H)
214. C7#10087direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5012
C7×#28222
C7×#19602
C2×C7×#3302
C7×#12952
#100877