Finite Group Databasebeta

← C7×#10381 #13187597

Subgroups — C7×#10381 #13187597

Order 1792.

352 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2606C211yesmaximal, characteristic
2C7×#1986C211yesmaximal, characteristic
3C7×#1515C211yesmaximal, characteristic
4C7×#1506C211yesmaximal, characteristic
5C7×#1300C211yesmaximal, characteristic
6C7×#1288C211yesmaximal, characteristic
7C7×#1285C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C7×#432C2211yescharacteristic
10C7×#432C2211yescharacteristic
11C4×C7×M16C2211yescharacteristic
12C7×#398—21no—
13C7×#398—21no—
14C7×#389—21no—
15C7×#389—21no—
16C7×#387—21no—
17C7×#387—21no—
18C7×#383C2211yescharacteristic
19C4×C7×C4⋊C4C2211yescharacteristic
20C7×#376C2211yescharacteristic
21C7×#339—21no—
22C7×#339—21no—
23C7×#328—21no—
24C7×#328—21no—
25C7×#328—21no—
26C7×#328—21no—
27#10381C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×M16—21no—
29C7×C4⋊Q8Q811yescharacteristic
30C7×C4⋊Q8Q811yescharacteristic
31C7×#125—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C7×#117—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C42×C7—21no—
45C2×C42×C7D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×C42×C7C2311yescharacteristic
48C7×C4⋊C8—21no—
49C7×C4⋊C8—21no—
50C7×C22⋊C8—21no—
51C7×C22⋊C8—21no—
52C7×#97—21no—
53C7×#97—21no—
54C7×#97—21no—
55C4×C7×C8—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95D411yescharacteristic
63C7×#95D411yescharacteristic
64C7×#95D411yescharacteristic
65C7×#95D411yescharacteristic
66#2606C2×C711yescharacteristic
67#1986C2×C711yescharacteristic
68#1515C2×C711yescharacteristic
69#1506C2×C711yescharacteristic
70#1300C2×C711yescharacteristic
71#1288C2×C711yescharacteristic
72#1285C2×C711yescharacteristic
73C2×C7×Q8—21no—
74C2×C7×Q8—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C7×M16—22no—
85C7×M16—41no—
86C2×C7×C8—41no—
87C2×C7×C8—41no—
88C2×C7×C8—41no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—81no—
101C7×C4⋊C4—81no—
102C7×C22⋊C4—41no—
103C42×C7—41no—
104C42×C7—41no—
105C42×C7—42no—
106C42×C7—41no—
107C42×C7C2×Q811yescharacteristic
108C42×C7#4111yescharacteristic
109C42×C7#4111yescharacteristic
110C42×C7#4111yescharacteristic
111#533C22×C711yescharacteristic
112#432C22×C711yescharacteristic
113#432C22×C711yescharacteristic
114C4×M16C22×C711yescharacteristic
115#398—21no—
116#398—21no—
117#389—21no—
118#389—21no—
119#387—21no—
120#387—21no—
121#383C22×C711yescharacteristic
122C4×C4⋊C4C22×C711yescharacteristic
123#376C22×C711yescharacteristic
124#339—21no—
125#339—21no—
126#328—21no—
127#328—21no—
128#328—21no—
129#328—21no—
130C23×C7#12011yescharacteristic
131C7×Q8—81no—
132C7×Q8—81no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7C22⋊Q811yescharacteristic
158C2×C4×C7C22⋊Q811yescharacteristic
159C2×C4×C7C22⋊Q811yescharacteristic
160C2×C4×C7#12411yescharacteristic
161C2×C4×C7#12411yescharacteristic
162C2×C4×C7#12411yescharacteristic
163C7×C8—42no—
164C7×C8—81no—
165C7×C8—81no—
166C2×M16—21no—
167C4⋊Q8C7×Q811yescharacteristic
168C4⋊Q8C7×Q811yescharacteristic
169#125—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C42—21no—
183C2×C42C7×D411yescharacteristic
184C2×C42C7×D411yescharacteristic
185C2×C42C23×C711yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190#97—21no—
191#97—21no—
192#97—21no—
193C4×C8—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95C7×D411yescharacteristic
201#95C7×D411yescharacteristic
202#95C7×D411yescharacteristic
203#95C7×D411yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7#39311yescharacteristic
207C22×C7#45511yescharacteristic
208C22×C7#45611yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—42no—
216C4×C7—42no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7—81no—
220C4×C7—81no—
221C4×C7—81no—
222C4×C7—81no—
223C2×Q8—21no—
224C2×Q8—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C2×C7×D411yescharacteristic
232C22×C4C2×C7×D411yescharacteristic
233C22×C4C2×C7×D411yescharacteristic
234M16—22no—
235M16—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C22⋊C4—41no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—41no—
257C42C2×C7×Q811yescharacteristic
258C42C7×#4111yescharacteristic
259C42C7×#4111yescharacteristic
260C42C7×#4111yescharacteristic
261C2×C7—21no—
262C2×C7—21no—
263C2×C7#202811yescharacteristic
264C2×C7#203111yescharacteristic
265C2×C7#203111yescharacteristic
266C23C7×#12011yescharacteristic
267Q8—81no—
268Q8—81no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4C7×C22⋊Q811yescharacteristic
294C2×C4C7×C22⋊Q811yescharacteristic
295C2×C4C7×C22⋊Q811yescharacteristic
296C2×C4C7×#12411yescharacteristic
297C2×C4C7×#12411yescharacteristic
298C2×C4C7×#12411yescharacteristic
299C8—42no—
300C8—81no—
301C8—81no—
302C7#1038111yescharacteristic, direct product factor, Sylow
303C22—21no—
304C22—21no—
305C22C7×#39311yescharacteristic
306C22C7×#45511yescharacteristic
307C22C7×#45611yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C2—21no—
323C2—21no—
324C2C7×#202811yescharacteristic
325C2C7×#203111yescharacteristic
326C2C7×#203111yescharacteristic
subgroupG/H
center, socleC22×C7#393
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
27. #10381C7direct (N × H)
302. C7#10381direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26062
C7×#19862
C7×#15152
C7×#15062
C7×#13002
C7×#12882
C7×#12852
#103817