Finite Group Databasebeta

← C7×#13302 #13190518

Subgroups — C7×#13302 #13190518

Order 1792.

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2567C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic
3C4×C7×#101C211yesmaximal, characteristic
4C7×#1394C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C4×C7×Q8C2211yescharacteristic
7C7×#423—21no—
8C2×C7×#101C2212yes—
9C2×C7×#101C2212yes—
10C4×C7×M16—21no—
11C7×#395—21no—
12C7×#395—21no—
13C7×#391C2212yes—
14C7×#389—21no—
15C7×#389—21no—
16C7×#384C412yes—
17C4×C7×C4⋊C4C411yescharacteristic
18C7×#343—21no—
19C7×#343—21no—
20#13302C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8C2311yescharacteristic
23C2×C7×M16—22no—
24C2×C7×M16—22no—
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—41no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7C2×C412yes—
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C7×C4⋊C8—41no—
42C7×C4⋊C8—41no—
43C7×#101—22no—
44C7×#101—24no—
45C7×#101Q814yescomplemented
46C7×#97—41no—
47C4×C7×C8—41no—
48C7×#95—21no—
49C7×#95—22no—
50C7×#95—22no—
51#2567C2×C711yescharacteristic
52#1801C2×C711yescharacteristic
53C4×#101C2×C711yescharacteristic
54#1394C2×C714yes—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—42no—
58C2×C7×Q8—82no—
59C2×C7×Q8C2×Q812yes—
60C2×C7×Q8#4112yes—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C7×M16—24no—
71C7×M16—42no—
72C7×M16—42no—
73C7×M16—42no—
74C2×C7×C8—42no—
75C2×C7×C8—42no—
76C7×C4⋊C4—24no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—82no—
84C7×C4⋊C4—82no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—42no—
92C2×C4⋊Q8C4×C711yescharacteristic
93C2×C4×Q8C22×C711yescharacteristic
94#423—21no—
95C2×#101C22×C712yes—
96C2×#101C22×C712yes—
97C4×M16—21no—
98#395—21no—
99#395—21no—
100#391C22×C712yes—
101#389—21no—
102#389—21no—
103#384C4×C712yes—
104C4×C4⋊C4C4×C711yescharacteristic
105#343—21no—
106#343—21no—
107C23×C7C2×C22⋊C411yescharacteristic
108C7×Q8—42no—
109C7×Q8—42no—
110C7×Q8—84no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—24no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7#9911yescharacteristic
131C2×C4×C7#9911yescharacteristic
132C2×C4×C7C4×Q812yes—
133C2×C4×C7C22⋊Q812yes—
134C2×C4×C7#12411yescharacteristic
135C2×C4×C7C4⋊Q811yescharacteristic
136C7×C8—44no—
137C7×C8—82no—
138C22×Q8—21no—
139C22×Q8C23×C711yescharacteristic
140C2×M16—22no—
141C2×M16—22no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×C4×C712yes—
151C2×C4⋊C4C2×C4×C711yescharacteristic
152C2×C4⋊C4C7×D411yescharacteristic
153C2×C4⋊C4C7×D411yescharacteristic
154C2×C42C2×C4×C711yescharacteristic
155C2×C42C2×C4×C712yes—
156C2×C42C7×D411yescharacteristic
157C2×C42C7×D411yescharacteristic
158C4⋊C8—41no—
159C4⋊C8—41no—
160#101—22no—
161#101—24no—
162#101C7×Q814yescomplemented
163#97—41no—
164C4×C8—41no—
165#95—21no—
166#95—22no—
167#95—22no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7#39111yescharacteristic
171C22×C7C2×#9911yescharacteristic
172C22×C7#41011yescharacteristic
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—21no—
176C4×C7—22no—
177C4×C7—21no—
178C4×C7—44no—
179C4×C7—41no—
180C4×C7—41no—
181C4×C7—41no—
182C4×C7—82no—
183C4×C7#35612yes—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—82no—
188C2×Q8C2×C7×Q812yes—
189C2×Q8C7×#4112yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4C7×C22⋊C411yescharacteristic
193C22×C4C7×C22⋊C411yescharacteristic
194C22×C4C7×C22⋊C411yescharacteristic
195C22×C4C7×C22⋊C411yescharacteristic
196C22×C4C22×C4×C711yescharacteristic
197C22×C4C2×C7×D411yescharacteristic
198C22×C4C2×C7×D411yescharacteristic
199M16—24no—
200M16—42no—
201M16—42no—
202M16—42no—
203C2×C8—42no—
204C2×C8—42no—
205C4⋊C4—24no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—42no—
221C2×C7—21no—
222C2×C7—21no—
223C2×C7#199611yescharacteristic
224C2×C7C2×#35611yescharacteristic
225C2×C7#213411yescharacteristic
226C23C2×C7×C22⋊C411yescharacteristic
227Q8—42no—
228Q8—42no—
229Q8—84no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4C7×#9911yescharacteristic
250C2×C4C7×#9911yescharacteristic
251C2×C4C4×C7×Q812yes—
252C2×C4C7×C22⋊Q812yes—
253C2×C4C7×#12411yescharacteristic
254C2×C4C7×C4⋊Q811yescharacteristic
255C8—44no—
256C8—82no—
257C7#1330211yescharacteristic, direct product factor, Sylow
258C22—21no—
259C22—21no—
260C22C7×#39111yescharacteristic
261C22C2×C7×#9911yescharacteristic
262C22C7×#41011yescharacteristic
263C4—22no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4C7×#35612yes—
274C2—21no—
275C2—21no—
276C2C7×#199611yescharacteristic
277C2C2×C7×#35611yescharacteristic
278C2C7×#213411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×Q8C23×C7
prime radicalC23×C7C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
20. #13302C7direct (N × H)
45. C7×#101Q8semidirect
162. #101C7×Q8semidirect
257. C7#13302direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25672
C7×#18012
C4×C7×#1012
C7×#13942
#133027