Finite Group Databasebeta

← C7×#10099 #13187315

Subgroups — C7×#10099 #13187315

Order 1792.

474 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3236C211yesmaximal, characteristic
2C7×#3139C211yesmaximal, characteristic
3C7×#2660C211yesmaximal, characteristic
4C7×#1558C211yesmaximal, characteristic
5C7×#1528C211yesmaximal, characteristic
6C7×#1506C211yesmaximal, characteristic
7C7×#1300C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#125C2211yescharacteristic
11C7×#501—21no—
12C7×#491—21no—
13C7×#491—22no—
14C7×#487—21no—
15C7×#487—21no—
16C7×#484—21no—
17C7×#483—21no—
18C7×#479—22no—
19C7×#432C2211yescharacteristic
20C7×#432C2211yescharacteristic
21C7×#419—22no—
22C4×C7×M16C2211yescharacteristic
23C7×#398—22no—
24C7×#395—22no—
25C7×#384—21no—
26C7×#382—21no—
27C7×#376C2211yescharacteristic
28C7×#333—22no—
29C7×#330—22no—
30C7×#328—22no—
31#10099C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D412yes—
33C2×C7×M16—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8D411yescharacteristic
39C7×C4⋊Q8D411yescharacteristic
40C7×#125—21no—
41C7×#125—21no—
42C7×#125—21no—
43C7×#125—21no—
44C7×#125—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C7×#117—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C42×C7—21no—
58C2×C42×C7C2311yescharacteristic
59C7×#107—22no—
60C7×#106—22no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8D412yes—
63C7×#103—22no—
64C7×#103—22no—
65C7×#103—24no—
66C7×#103—24no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8—21no—
69C7×#97—21no—
70C7×#97—21no—
71C7×#97D412yes—
72C4×C7×C8—21no—
73C7×#95—24no—
74C7×#95—21no—
75C7×#95D412yes—
76#3236C2×C711yescharacteristic
77#3139C2×C711yescharacteristic
78#2660C2×C711yescharacteristic
79#1558C2×C711yescharacteristic
80#1528C2×C711yescharacteristic
81#1506C2×C711yescharacteristic
82#1300C2×C711yescharacteristic
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—21no—
86C2×C7×Q8—22no—
87C2×C7×Q8—44no—
88C22×C4×C7—22no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—21no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C7×M16—42no—
96C2×C7×C8—22no—
97C2×C7×C8—22no—
98C2×C7×C8—41no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—84no—
118C7×C22⋊C4—41no—
119C42×C7—24no—
120C42×C7—41no—
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124C42×C7#4111yescharacteristic
125#533C22×C711yescharacteristic
126C2×C4⋊Q8C22×C711yescharacteristic
127C2×#125C22×C711yescharacteristic
128#501—21no—
129#491—21no—
130#491—22no—
131#487—21no—
132#487—21no—
133#484—21no—
134#483—21no—
135#479—22no—
136#432C22×C711yescharacteristic
137#432C22×C711yescharacteristic
138#419—22no—
139C4×M16C22×C711yescharacteristic
140#398—22no—
141#395—22no—
142#384—21no—
143#382—21no—
144#376C22×C711yescharacteristic
145#333—22no—
146#330—22no—
147#328—22no—
148C23×C7#12011yescharacteristic
149C7×Q8—44no—
150C7×Q8—42no—
151C7×Q8—81no—
152C7×Q8—82no—
153C2×C4×C7—22no—
154C2×C4×C7—22no—
155C2×C4×C7—24no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—44no—
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12711yescharacteristic
173C7×C8—42no—
174C7×C8—42no—
175C7×C8—81no—
176C22×Q8C7×D412yes—
177C2×M16—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C7×D411yescharacteristic
183C4⋊Q8C7×D411yescharacteristic
184#125—21no—
185#125—21no—
186#125—21no—
187#125—21no—
188#125—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193#117—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C7×D411yescharacteristic
200C2×C4⋊C4C7×D411yescharacteristic
201C2×C42—21no—
202C2×C42C23×C711yescharacteristic
203#107—22no—
204#106—22no—
205C4⋊C8—21no—
206C4⋊C8C7×D412yes—
207#103—24no—
208#103—24no—
209#103—22no—
210#103—22no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213#97—21no—
214#97—21no—
215#97C7×D412yes—
216C4×C8—21no—
217#95—24no—
218#95—21no—
219#95C7×D412yes—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7#39211yescharacteristic
223C22×C7#45512yes—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—44no—
230C4×C7—81no—
231C4×C7—82no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7—81no—
235C4×C7#45612yes—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—44no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4C2×C7×D411yescharacteristic
246C22×C4C2×C7×D411yescharacteristic
247C22×C4C2×C7×D411yescharacteristic
248M16—42no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—84no—
271C22⋊C4—41no—
272C42—24no—
273C42—41no—
274C42C2×C7×D411yescharacteristic
275C42C2×C7×D411yescharacteristic
276C42C2×C7×D411yescharacteristic
277C42C7×#4111yescharacteristic
278C2×C7—22no—
279C2×C7#200411yescharacteristic
280C2×C7#200511yescharacteristic
281C2×C7#200911yescharacteristic
282C23C7×#12011yescharacteristic
283Q8—44no—
284Q8—42no—
285Q8—81no—
286Q8—82no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4C7×#12011yescharacteristic
302C2×C4C7×#12011yescharacteristic
303C2×C4C7×#12111yescharacteristic
304C2×C4C7×#12111yescharacteristic
305C2×C4C7×#12111yescharacteristic
306C2×C4C7×#12711yescharacteristic
307C8—42no—
308C8—42no—
309C8—81no—
310C7#1009911yescharacteristic, direct product factor, Sylow
311C22—21no—
312C22—21no—
313C22C7×#39211yescharacteristic
314C22C7×#45512yes—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—82no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4C7×#45612yes—
327C2—22no—
328C2C7×#200411yescharacteristic
329C2C7×#200511yescharacteristic
330C2C7×#200911yescharacteristic
subgroupG/H
center, socleC22×C7#392
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
31. #10099C7direct (N × H)
310. C7#10099direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32362
C7×#31392
C7×#26602
C7×#15582
C7×#15282
C7×#15062
C7×#13002
#100997