Finite Group Databasebeta

← C7×#13299 #13190515

Subgroups — C7×#13299 #13190515

Order 1792.

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

normal subgroups only

This group has 500 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×#2563C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C4×C7×#99C211yesmaximal, characteristic
4C7×#1393C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#99C2212yes—
8C2×C7×#99C2212yes—
9C7×#393—21no—
10C7×#393—21no—
11C7×#390C2212yes—
12C7×#382C412yescomplemented
13C7×#380—21no—
14C7×#380—21no—
15C7×#380—21no—
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C4×C7×C22⋊C4—21no—
18C7×#342—21no—
19C7×#342—21no—
20#13299C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4C2311yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—41no—
27C4×C7×D4—41no—
28C4×C7×D4—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—41no—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7—41no—
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7C2×C412yes—
43C7×#99—22no—
44C7×#99—24no—
45C7×#99Q814yescomplemented
46C7×#95—21no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—41no—
50C7×#95—41no—
51#2563C2×C711yescharacteristic
52#1799C2×C711yescharacteristic
53C4×#99C2×C711yescharacteristic
54#1393C2×C714yes—
55C24×C7C2×D411yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—42no—
58C2×C7×Q8—82no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4C2×Q812yes—
62C2×C7×D4#4112yes—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C7×C4⋊C4—41no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—82no—
84C7×C22⋊C4—24no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—82no—
91C7×C22⋊C4—82no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—42no—
95C42×C7—82no—
96C2×C4⋊Q8C4×C711yescharacteristic, complemented
97C2×C4×D4C22×C711yescharacteristic
98C2×#99C22×C712yes—
99C2×#99C22×C712yes—
100#393—21no—
101#393—21no—
102#390C22×C712yes—
103#382C4×C712yescomplemented
104#380—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4C4×C711yescharacteristic, complemented
108C4×C22⋊C4—21no—
109#342—21no—
110#342—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7C2×C22⋊C411yescharacteristic
120C23×C7C22⋊Q812yes—
121C23×C7#12411yescharacteristic
122C23×C7C4⋊Q811yescharacteristic
123C7×Q8—84no—
124C7×D4—42no—
125C7×D4—42no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—44no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7—82no—
147C2×C4×C7—82no—
148C2×C4×C7—82no—
149C2×C4×C7—82no—
150C2×C4×C7—82no—
151C2×C4×C7#9911yescharacteristic
152C2×C4×C7#9911yescharacteristic
153C2×C4×C7C4×Q812yes—
154C22×Q8—21no—
155C22×D4C23×C711yescharacteristic
156C23×C4C7×D411yescharacteristic
157C23×C4C7×D411yescharacteristic
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4×D4—41no—
161C4×D4—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C2×C4×C712yes—
166C2×C4⋊C4C2×C4×C711yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4C7×D411yescharacteristic
172C2×C22⋊C4C7×D411yescharacteristic
173C2×C42—41no—
174C2×C42C2×C4×C711yescharacteristic
175C2×C42C2×C4×C712yes—
176#99—22no—
177#99—24no—
178#99C7×Q814yescomplemented
179#95—21no—
180#95—22no—
181#95—22no—
182#95—41no—
183#95—41no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—22no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—42no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—81no—
197C22×C7—81no—
198C22×C7#39111yescharacteristic
199C22×C7C2×#9911yescharacteristic
200C22×C7#41011yescharacteristic
201C4×C7—21no—
202C4×C7—22no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—82no—
207C4×C7—84no—
208C4×C7—82no—
209C4×C7—162no—
210C4×C7#35412yes—
211C24C2×C7×D411yescharacteristic
212C24C2×C7×D411yescharacteristic
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4C2×C7×Q812yes—
218C2×D4C7×#4112yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C7×C22⋊C411yescharacteristic
230C22×C4C7×C22⋊C411yescharacteristic
231C22×C4C7×C22⋊C411yescharacteristic
232C22×C4C7×C22⋊C411yescharacteristic
233C22×C4C22×C4×C711yescharacteristic
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—82no—
240C22⋊C4—24no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C42—42no—
251C42—82no—
252C2×C7—21no—
253C2×C7—21no—
254C2×C7—42no—
255C2×C7—41no—
256C2×C7—41no—
257C2×C7#199611yescharacteristic
258C2×C7C2×#35411yescharacteristic
259C2×C7#212811yescharacteristic
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23C2×C7×C22⋊C411yescharacteristic
269C23C7×C22⋊Q812yes—
270C23C7×#12411yescharacteristic
271C23C7×C4⋊Q811yescharacteristic
272Q8—84no—
273D4—42no—
274D4—42no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4C7×#9911yescharacteristic
301C2×C4C7×#9911yescharacteristic
302C2×C4C4×C7×Q812yes—
303C7#1329911yescharacteristic, direct product factor, Sylow
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22C7×#39111yescharacteristic
319C22C2×C7×#9911yescharacteristic
320C22C7×#41011yescharacteristic
321C4—22no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—84no—
328C4—82no—
329C4—162no—
330C4C7×#35412yes—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—41no—
335C2—41no—
336C2C7×#199611yescharacteristic
337C2C2×C7×#35411yescharacteristic
338C2C7×#212811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C7×C4⋊Q8C4semidirect
12. C7×#382C4semidirect
16. C4×C7×C4⋊C4C4semidirect
20. #13299C7direct (N × H)
45. C7×#99Q8semidirect
96. C2×C4⋊Q8C4×C7semidirect
103. #382C4×C7semidirect
107. C4×C4⋊C4C4×C7semidirect
178. #99C7×Q8semidirect
303. C7#13299direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25632
C7×#17992
C4×C7×#992
C7×#13932
#132997