Finite Group Databasebeta

← C7×#13388 #13190604

Subgroups — C7×#13388 #13190604

Order 1792.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3059C211yesmaximal, characteristic
2C7×#2159C211yesmaximal, characteristic, complemented
3C2×C7×#362C211yesmaximal, characteristic, complemented
4C2×C7×#358C211yesmaximal, characteristic, complemented
5C2×C7×#350C211yesmaximal, characteristic
6C7×#1816C211yesmaximal, characteristic
7C7×#1382C211yesmaximal, characteristic, complemented
8C22×C7×Q16C2211yescharacteristic
9C2×C7×#131C2211yescharacteristic
10C7×#536—21no—
11C2×C7×M32C2211yescharacteristic
12C22×C7×C16C2211yescharacteristic
13C7×#470—22no—
14C7×#467—21no—
15C7×#467—21no—
16C2×C7×#108C2211yescharacteristic
17C7×#427C2211yescharacteristic
18C2×C7×#107C2211yescharacteristic
19C7×#419—21no—
20C7×#417—21no—
21C7×#367—21no—
22C7×#366—21no—
23C7×#362—22no—
24C7×#358—22no—
25C7×#350—22no—
26C7×#343—21no—
27#13388C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41D411yescharacteristic
29C22×C7×Q8—21no—
30C2×C7×Q16—21no—
31C2×C7×Q16—21no—
32C2×C7×Q16—21no—
33C2×C7×Q16—21no—
34C2×C7×Q16—21no—
35C2×C7×Q16D412yes—
36C7×#131—21no—
37C7×#131—21no—
38C7×#131—21no—
39C7×#131—21no—
40C2×C7×M16—21no—
41C2×C7×M16—21no—
42C2×C7×M16D411yescharacteristic
43C22×C7×C8—21no—
44C22×C7×C8C2311yescharacteristic
45C7×C4⋊Q8—41no—
46C7×#124—41no—
47C7×#123—42no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—22no—
50C7×#117—21no—
51C2×C7×C4⋊C4—21no—
52C7×M32—21no—
53C7×M32D412yes—
54C2×C7×C16—21no—
55C2×C7×C16—21no—
56C2×C7×C16—21no—
57C2×C7×C16—21no—
58C7×#108—21no—
59C7×#108—21no—
60C7×#107—21no—
61C7×#107—21no—
62C7×#107D411yescharacteristic
63C7×#107D411yescharacteristic
64C7×#106—21no—
65C7×#103—22no—
66C7×#103—22no—
67C7×#103—41no—
68C7×#102—41no—
69#3059C2×C711yescharacteristic
70#2159C2×C711yescharacteristic, complemented
71C2×#362C2×C711yescharacteristic, complemented
72C2×#358C2×C711yescharacteristic, complemented
73C2×#350C2×C711yescharacteristic
74#1816C2×C711yescharacteristic
75#1382C2×C711yescharacteristic, complemented
76C7×#41—21no—
77C7×#41—21no—
78C7×#41—21no—
79C7×#41—21no—
80C2×C7×Q8—22no—
81C2×C7×Q8—41no—
82C2×C7×Q8—41no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8D811yescharacteristic
87C2×C7×D4—21no—
88C2×C7×D4D811yescharacteristic
89C22×C4×C7—21no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7C2×D411yescharacteristic
93C7×Q16—22no—
94C7×Q16—22no—
95C7×Q16—41no—
96C7×Q16—41no—
97C7×M16—21no—
98C7×M16—21no—
99C7×M16—41no—
100C7×M16—41no—
101C7×M16—41no—
102C7×M16—41no—
103C2×C7×C8—21no—
104C2×C7×C8—41no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C2×C7×C8C2×D411yescharacteristic
108C2×C7×C8C2×D411yescharacteristic
109C2×C7×C8C2×D411yescharacteristic
110C2×C7×C8#4111yescharacteristic
111C2×C7×C8#4111yescharacteristic
112C2×C7×C8#4111yescharacteristic
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—42no—
116C7×C4⋊C4—42no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—41no—
122C42×C7—41no—
123C7×C16—22no—
124C7×C16—41no—
125C22×Q16C22×C711yescharacteristic
126C2×#131C22×C711yescharacteristic
127#536—21no—
128C2×M32C22×C711yescharacteristic
129C22×C16C22×C711yescharacteristic
130#470—22no—
131#467—21no—
132#467—21no—
133C2×#108C22×C711yescharacteristic
134#427C22×C711yescharacteristic
135C2×#107C22×C711yescharacteristic
136#419—21no—
137#417—21no—
138#367—21no—
139#366—21no—
140#362—22no—
141#358—22no—
142#350—22no—
143#343—21no—
144C23×C7—21no—
145C23×C7AGL(1,ℤ/8)11yescharacteristic
146C7×Q8—21no—
147C7×Q8—21no—
148C7×Q8—42no—
149C7×Q8—42no—
150C7×Q8—81no—
151C7×Q8—81no—
152C7×D4—21no—
153C7×D4—21no—
154C7×D4—41no—
155C7×D4—41no—
156C2×C4×C7—21no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—42no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7—81no—
165C2×C4×C7—161no—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12111yescharacteristic
168C2×C4×C7#12311yescharacteristic
169C2×C4×C7C2×D811yescharacteristic
170C2×C4×C7#13511yescharacteristic
171C2×C4×C7#13711yescharacteristic
172C7×C8—41no—
173C7×C8—81no—
174C7×C8#12111yescharacteristic
175C7×C8#12111yescharacteristic
176C7×C8#12311yescharacteristic
177C7×C8#12411yescharacteristic
178C2×#41C7×D411yescharacteristic
179C22×Q8—21no—
180C2×Q16—21no—
181C2×Q16—21no—
182C2×Q16—21no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×Q16C7×D412yes—
186#131—21no—
187#131—21no—
188#131—21no—
189#131—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16C7×D411yescharacteristic
193C22×C8—21no—
194C22×C8C23×C711yescharacteristic
195C4⋊Q8—41no—
196#124—41no—
197#123—42no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200#117—21no—
201C2×C4⋊C4—21no—
202M32—21no—
203M32C7×D412yes—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208#108—21no—
209#108—21no—
210#107—21no—
211#107—21no—
212#107C7×D411yescharacteristic
213#107C7×D411yescharacteristic
214#106—21no—
215#103—22no—
216#103—22no—
217#103—41no—
218#102—41no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7#44711yescharacteristic
225C22×C7#46811yescharacteristic
226C22×C7#48211yescharacteristic
227C4×C7—41no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7—82no—
231C4×C7—161no—
232C4×C7#39411yescharacteristic
233C4×C7#45211yescharacteristic
234C4×C7#46611yescharacteristic
235C4×C7#48011yescharacteristic
236#41—21no—
237#41—21no—
238#41—21no—
239#41—21no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C7×D811yescharacteristic
247C2×D4—21no—
248C2×D4C7×D811yescharacteristic
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×C7×D411yescharacteristic
253Q16—22no—
254Q16—22no—
255Q16—41no—
256Q16—41no—
257M16—21no—
258M16—21no—
259M16—41no—
260M16—41no—
261M16—41no—
262M16—41no—
263C2×C8—21no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8C2×C7×D411yescharacteristic
268C2×C8C2×C7×D411yescharacteristic
269C2×C8C2×C7×D411yescharacteristic
270C2×C8C7×#4111yescharacteristic
271C2×C8C7×#4111yescharacteristic
272C2×C8C7×#4111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C42—41no—
283C16—22no—
284C16—41no—
285C2×C7—21no—
286C2×C7—21no—
287C2×C7—81no—
288C2×C7#203311yescharacteristic
289C2×C7#219311yescharacteristic
290C2×C7#219511yescharacteristic
291C23—21no—
292C23C7×AGL(1,ℤ/8)11yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—42no—
297Q8—81no—
298Q8—81no—
299D4—21no—
300D4—21no—
301D4—41no—
302D4—41no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—161no—
313C2×C4C7×#12011yescharacteristic
314C2×C4C7×#12111yescharacteristic
315C2×C4C7×#12311yescharacteristic
316C2×C4C2×C7×D811yescharacteristic
317C2×C4C7×#13511yescharacteristic
318C2×C4C7×#13711yescharacteristic
319C8—41no—
320C8—81no—
321C8C7×#12111yescharacteristic
322C8C7×#12111yescharacteristic
323C8C7×#12311yescharacteristic
324C8C7×#12411yescharacteristic
325C7#1338811yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22C7×#44711yescharacteristic
332C22C7×#46811yescharacteristic
333C22C7×#48211yescharacteristic
334C4—41no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—161no—
339C4C7×#39411yescharacteristic
340C4C7×#45211yescharacteristic
341C4C7×#46611yescharacteristic
342C4C7×#48011yescharacteristic
343C2—21no—
344C2—21no—
345C2—81no—
346C2C7×#203311yescharacteristic
347C2C7×#219311yescharacteristic
348C2C7×#219511yescharacteristic
subgroupG/H
center, socleC22×C7#447
commutator, Frattini, cosocleC22×C8C23×C7
prime radicalC2×C7×#41D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2159C2semidirect
3. C2×C7×#362C2semidirect
4. C2×C7×#358C2semidirect
7. C7×#1382C2semidirect
27. #13388C7direct (N × H)
70. #2159C2×C7semidirect
71. C2×#362C2×C7semidirect
72. C2×#358C2×C7semidirect
75. #1382C2×C7semidirect
325. C7#13388direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30592
C7×#21592
C2×C7×#3622
C2×C7×#3582
C2×C7×#3502
C7×#18162
C7×#13822
#133887