Finite Group Databasebeta

← C7×#12225 #13189441

Subgroups — C7×#12225 #13189441

Order 1792.

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

normal subgroups only

This group has 336 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×#2809C211yesmaximal, characteristic
2C7×#2028C211yesmaximal, characteristic
3C7×#1617C212yesmaximal
4C7×#1353C212yesmaximal
5C7×#1319C211yesmaximal, characteristic
6C7×#544C2211yescharacteristic
7C7×#484—22no—
8C7×#419—21no—
9C7×#400C2212yes—
10C7×#398—22no—
11C7×#391—21no—
12C7×#391C2211yescharacteristic
13C7×#389—21no—
14C7×#387—22no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12225C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8D411yescharacteristic
21C2×C7×M16—21no—
22C7×C4⋊Q8—41no—
23C7×#125—41no—
24C7×C22⋊Q8—22no—
25C7×#117—21no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C4⋊C4C2311yescharacteristic
32C2×C42×C7—21no—
33C2×C42×C7D411yescharacteristic
34C7×#107—42no—
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#2809C2×C711yescharacteristic
45#2028C2×C711yescharacteristic
46#1617C2×C712yes—
47#1353C2×C712yes—
48#1319C2×C711yescharacteristic
49C2×C7×Q8—21no—
50C2×C7×Q8—22no—
51C2×C7×Q8—41no—
52C2×C7×Q8—41no—
53C2×C7×Q8Q1612yes—
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D412yes—
60C7×M16—41no—
61C7×M16—41no—
62C2×C7×C8—42no—
63C2×C7×C8—41no—
64C7×C4⋊C4—21no—
65C7×C4⋊C4—21no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—41no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—44no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—81no—
73C7×C4⋊C4—81no—
74C7×C22⋊C4—41no—
75C42×C7—41no—
76C42×C7—41no—
77C42×C7—81no—
78#544C22×C711yescharacteristic
79#484—22no—
80#419—21no—
81#400C22×C712yes—
82#398—22no—
83#391—21no—
84#391C22×C711yescharacteristic
85#389—21no—
86#387—22no—
87#343C22×C711yescharacteristic
88#339—21no—
89#328—22no—
90#324C22×C712yes—
91C23×C7#12011yescharacteristic
92C7×Q8—42no—
93C7×Q8—81no—
94C7×Q8—82no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7C2×Q1611yescharacteristic
114C2×C4×C7#13711yescharacteristic
115C7×C8—81no—
116C7×C8—82no—
117C22×Q8C7×D411yescharacteristic
118C2×M16—21no—
119C4⋊Q8—41no—
120#125—41no—
121C22⋊Q8—22no—
122#117—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C7×D412yes—
128C2×C4⋊C4C23×C711yescharacteristic
129C2×C42—21no—
130C2×C42C7×D411yescharacteristic
131#107—42no—
132#103—41no—
133#103—41no—
134#103—42no—
135C22⋊C8—22no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95C7×D412yes—
141C22×C7—21no—
142C22×C7—21no—
143C22×C7#45111yescharacteristic
144C22×C7#45511yescharacteristic
145C22×C7B(4,𝔽2)11yescharacteristic
146C4×C7—21no—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7—42no—
150C4×C7—42no—
151C4×C7—82no—
152C4×C7—81no—
153C4×C7—81no—
154C4×C7—82no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C7×Q1612yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4C2×C7×D411yescharacteristic
165C22×C4C2×C7×D412yes—
166M16—41no—
167M16—41no—
168C2×C8—42no—
169C2×C8—41no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C42—41no—
182C42—41no—
183C42—81no—
184C2×C7—21no—
185C2×C7—21no—
186C2×C7#160211yescharacteristic
187C2×C7#220111yescharacteristic
188C2×C7#220411yescharacteristic
189C23C7×#12011yescharacteristic
190Q8—42no—
191Q8—81no—
192Q8—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4C2×C7×Q1611yescharacteristic
212C2×C4C7×#13711yescharacteristic
213C8—81no—
214C8—82no—
215C7#1222511yescharacteristic, direct product factor, Sylow
216C22—21no—
217C22—21no—
218C22C7×#45111yescharacteristic
219C22C7×#45511yescharacteristic
220C22C7×B(4,𝔽2)11yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4—81no—
230C2—21no—
231C2—21no—
232C2C7×#160211yescharacteristic
233C2C7×#220111yescharacteristic
234C2C7×#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. #12225C7direct (N × H)
215. C7#12225direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28092
C7×#20282
C7×#16172
C7×#13532
C7×#13192
#122257