Finite Group Databasebeta

← C7×#10011 #13187227

Subgroups — C7×#10011 #13187227

Order 1792.

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

normal subgroups only

This group has 496 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×#3113C211yesmaximal, characteristic
3C7×#2584C211yesmaximal, characteristic
4C7×#1986C211yesmaximal, characteristic
5C7×#1528C211yesmaximal, characteristic
6C7×#1491C211yesmaximal, characteristic
7C7×#1285C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C7×#125C2211yescharacteristic
10C2×C4×C7×Q8C2211yescharacteristic
11C7×#501—21no—
12C7×#491—21no—
13C7×#484—21no—
14C7×#483—21no—
15C7×#483—21no—
16C7×#479—22no—
17C7×#475—21no—
18C7×#462—21no—
19C7×#451—21no—
20C7×#441—22no—
21C7×#432C2211yescharacteristic
22C7×#432C2211yescharacteristic
23C4×C7×M16C2211yescharacteristic
24C7×#398—21no—
25C7×#395—21no—
26C7×#389—21no—
27C7×#389—21no—
28C7×#389—22no—
29C4×C7×C4⋊C4C2211yescharacteristic
30C7×#339—22no—
31C7×#330—22no—
32C7×Q8⋊C8—22no—
33#10011C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8D411yescharacteristic
35C2×C7×Q16—22no—
36C2×C7×M16—21no—
37C7×C4⋊Q8D411yescharacteristic
38C7×C4⋊Q8D411yescharacteristic
39C7×#125—21no—
40C7×#125—21no—
41C7×#125—21no—
42C7×#125—21no—
43C7×#125—21no—
44C7×C22⋊Q8—21no—
45C7×C22⋊Q8—21no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C7×#117—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C42×C7—21no—
63C2×C42×C7D412yes—
64C2×C42×C7C2311yescharacteristic
65C7×#107—22no—
66C7×#106—22no—
67C7×#106—22no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8D412yes—
70C7×#103—22no—
71C7×#103—22no—
72C7×#103—22no—
73C7×#103—22no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8—21no—
76C7×#97—21no—
77C7×#97—21no—
78C7×#97D412yes—
79C4×C7×C8—21no—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—21no—
83#3236C2×C711yescharacteristic
84#3113C2×C711yescharacteristic
85#2584C2×C711yescharacteristic
86#1986C2×C711yescharacteristic
87#1528C2×C711yescharacteristic
88#1491C2×C711yescharacteristic
89#1285C2×C711yescharacteristic
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—42no—
95C22×C4×C7—22no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C7×Q16—44no—
104C7×M16—22no—
105C7×M16—41no—
106C2×C7×C8—22no—
107C2×C7×C8—22no—
108C2×C7×C8—41no—
109C7×C4⋊C4—24no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—44no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—82no—
130C7×C22⋊C4—41no—
131C42×C7—42no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7#4111yescharacteristic
139#533C22×C711yescharacteristic
140C2×#125C22×C711yescharacteristic
141C2×C4×Q8C22×C711yescharacteristic
142#501—21no—
143#491—21no—
144#484—21no—
145#483—21no—
146#483—21no—
147#479—22no—
148#475—21no—
149#462—21no—
150#451—21no—
151#441—22no—
152#432C22×C711yescharacteristic
153#432C22×C711yescharacteristic
154C4×M16C22×C711yescharacteristic
155#398—21no—
156#395—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160C4×C4⋊C4C22×C711yescharacteristic
161#339—22no—
162#330—22no—
163Q8⋊C8—22no—
164C23×C7#12011yescharacteristic
165C7×Q8—22no—
166C7×Q8—22no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—42no—
170C7×Q8—42no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—22no—
174C2×C4×C7—24no—
175C2×C4×C7—22no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12711yescharacteristic
197C7×C8—42no—
198C7×C8—42no—
199C7×C8—42no—
200C22×Q8C7×D411yescharacteristic
201C2×Q16—22no—
202C2×M16—21no—
203C4⋊Q8C7×D411yescharacteristic
204C4⋊Q8C7×D411yescharacteristic
205#125—21no—
206#125—21no—
207#125—21no—
208#125—21no—
209#125—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218#117—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C4⋊C4C7×D411yescharacteristic
227C2×C4⋊C4C7×D411yescharacteristic
228C2×C42—21no—
229C2×C42C7×D412yes—
230C2×C42C23×C711yescharacteristic
231#107—22no—
232#106—22no—
233#106—22no—
234C4⋊C8—21no—
235C4⋊C8C7×D412yes—
236#103—22no—
237#103—22no—
238#103—22no—
239#103—22no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242#97—21no—
243#97—21no—
244#97C7×D412yes—
245C4×C8—21no—
246#95—22no—
247#95—22no—
248#95—21no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7#39211yescharacteristic
252C22×C7#45511yescharacteristic
253C22×C7#45611yescharacteristic
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—44no—
260C4×C7—42no—
261C4×C7—81no—
262C4×C7—81no—
263C4×C7—81no—
264C4×C7—81no—
265C4×C7—81no—
266C4×C7#45612yes—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—42no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4C2×C7×D411yescharacteristic
278C22×C4C2×C7×D411yescharacteristic
279C22×C4C2×C7×D411yescharacteristic
280Q16—44no—
281M16—22no—
282M16—41no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C4⋊C4—24no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—44no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C22⋊C4—41no—
308C42—42no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42C2×C7×D411yescharacteristic
313C42C2×C7×D411yescharacteristic
314C42C2×C7×D411yescharacteristic
315C42C7×#4111yescharacteristic
316C2×C7—21no—
317C2×C7—21no—
318C2×C7#200511yescharacteristic
319C2×C7#200711yescharacteristic
320C2×C7#201011yescharacteristic
321C23C7×#12011yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—41no—
325Q8—41no—
326Q8—42no—
327Q8—42no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4C7×#12011yescharacteristic
349C2×C4C7×#12011yescharacteristic
350C2×C4C7×#12111yescharacteristic
351C2×C4C7×#12111yescharacteristic
352C2×C4C7×#12111yescharacteristic
353C2×C4C7×#12711yescharacteristic
354C8—42no—
355C8—42no—
356C8—42no—
357C7#1001111yescharacteristic, direct product factor, Sylow
358C22—21no—
359C22—21no—
360C22C7×#39211yescharacteristic
361C22C7×#45511yescharacteristic
362C22C7×#45611yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—44no—
369C4—42no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4C7×#45612yes—
376C2—21no—
377C2—21no—
378C2C7×#200511yescharacteristic
379C2C7×#200711yescharacteristic
380C2C7×#201011yescharacteristic
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
33. #10011C7direct (N × H)
357. C7#10011direct (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×#31132
C7×#25842
C7×#19862
C7×#15282
C7×#14912
C7×#12852
#100117