Subgroups — C₇×#33689 #13210905

Order 1792.

520 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2956	C₂	1	1	yes	maximal, characteristic
3	C₇×#2239	C₂	1	2	yes	maximal, complemented
4	C₇×#2230	C₂	1	2	yes	maximal, complemented
5	C₇×#2214	C₂	1	2	yes	maximal, complemented
6	C₇×#2191	C₂	1	2	yes	maximal
7	C₇×#2180	C₂	1	4	yes	maximal, complemented
8	C₇×#2165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#575	C₂²	1	1	yes	characteristic
10	C₇×#568	—	2	1	no	—
11	C₇×#533	—	2	1	no	—
12	C₇×#510	C₂²	1	4	yes	—
13	C₇×#498	C₂²	1	1	yes	characteristic, complemented
14	C₇×#494	C₂²	1	1	yes	characteristic, complemented
15	C₇×#487	—	2	1	no	—
16	C₇×#470	—	2	1	no	—
17	C₇×#470	C₂²	1	2	yes	—
18	C₇×#444	—	2	1	no	—
19	C₇×#444	C₂²	1	2	yes	—
20	C₇×#443	—	2	1	no	—
21	C₇×#443	C₂²	1	2	yes	—
22	C₇×#441	—	2	1	no	—
23	C₇×#441	C₂²	1	2	yes	complemented
24	C₇×#430	C₂²	1	1	yes	characteristic
25	C₂×C₇×#104	—	2	1	no	—
26	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
27	C₇×#365	C₂²	1	4	yes	complemented
28	C₇×#364	C₂²	1	2	yes	complemented
29	C₇×#362	C₂²	1	4	yes	complemented
30	C₇×#360	C₂²	1	4	yes	—
31	C₇×#349	C₂²	1	2	yes	—
32	C₇×#347	C₂²	1	2	yes	complemented
33	#33689	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	1	no	—
35	C₇×#137	—	2	2	no	—
36	C₇×AGL(1,ℤ/8)	—	2	2	no	—
37	C₇×#135	—	2	1	no	—
38	C₇×#135	C₂³	1	2	yes	—
39	C₂×C₇×Q₁₆	—	2	1	no	—
40	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
41	C₇×#131	—	2	2	no	—
42	C₇×#131	—	2	2	no	—
43	C₇×#131	—	4	2	no	—
44	C₂×C₇×M₁₆	—	2	1	no	—
45	C₂×C₇×M₁₆	—	4	2	no	—
46	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
47	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
48	C₇×C₄⋊Q₈	—	2	1	no	—
49	C₇×C₄⋊Q₈	—	2	1	no	—
50	C₇×#125	—	4	1	no	—
51	C₇×C₂²⋊Q₈	—	2	2	no	—
52	C₇×C₂²⋊Q₈	—	4	1	no	—
53	C₄×C₇×Q₈	—	2	2	no	—
54	C₄×C₇×Q₈	—	4	1	no	—
55	C₇×#117	—	4	1	no	—
56	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
57	C₇×Q₃₂	—	2	4	no	—
58	C₇×SD₃₂	—	2	4	no	—
59	C₇×M₃₂	C₂³	1	4	yes	—
60	C₇×#108	—	2	1	no	—
61	C₇×#108	C₂³	1	2	yes	—
62	C₇×#106	—	2	1	no	—
63	C₇×#106	C₂³	1	2	yes	—
64	C₇×#104	—	2	2	no	—
65	C₇×#104	—	2	2	no	—
66	C₇×#103	—	2	2	no	—
67	C₇×#103	—	2	2	no	—
68	C₇×#97	D₄	1	1	yes	characteristic
69	C₇×#97	C₂³	1	1	yes	characteristic
70	C₄×C₇×C₈	D₄	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	#3185	C₂×C₇	1	1	yes	characteristic, complemented
73	#2956	C₂×C₇	1	1	yes	characteristic
74	#2239	C₂×C₇	1	2	yes	complemented
75	#2230	C₂×C₇	1	2	yes	complemented
76	#2214	C₂×C₇	1	2	yes	complemented
77	#2191	C₂×C₇	1	2	yes	—
78	#2180	C₂×C₇	1	4	yes	complemented
79	#2165	C₂×C₇	1	1	yes	characteristic, complemented
80	C₇×#41	—	2	2	no	—
81	C₇×#41	—	2	2	no	—
82	C₇×#41	—	4	1	no	—
83	C₇×#41	—	4	1	no	—
84	C₂×C₇×Q₈	—	2	1	no	—
85	C₂×C₇×Q₈	—	2	2	no	—
86	C₂×C₇×Q₈	—	4	1	no	—
87	C₂×C₇×D₄	—	2	1	no	—
88	C₂×C₇×D₄	—	4	1	no	—
89	C₂²×C₄×C₇	—	2	1	no	—
90	C₂²×C₄×C₇	—	4	1	no	—
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₇×Q₁₆	—	2	1	no	—
93	C₇×Q₁₆	—	2	4	no	—
94	C₇×Q₁₆	—	4	2	no	—
95	C₇×Q₁₆	#41	1	2	yes	—
96	C₇×SD₁₆	—	2	2	no	—
97	C₇×SD₁₆	—	2	2	no	—
98	C₇×D₈	—	2	1	no	—
99	C₇×D₈	#41	1	2	yes	—
100	C₇×M₁₆	—	2	2	no	—
101	C₇×M₁₆	—	2	2	no	—
102	C₇×M₁₆	—	2	2	no	—
103	C₇×M₁₆	—	4	2	no	—
104	C₇×M₁₆	—	4	1	no	—
105	C₇×M₁₆	—	4	2	no	—
106	C₂×C₇×C₈	—	4	2	no	—
107	C₂×C₇×C₈	—	4	1	no	—
108	C₂×C₇×C₈	—	4	2	no	—
109	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
113	C₇×C₄⋊C₄	—	2	2	no	—
114	C₇×C₄⋊C₄	—	4	1	no	—
115	C₇×C₄⋊C₄	—	4	1	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₇×C₄⋊C₄	—	4	1	no	—
118	C₇×C₄⋊C₄	—	4	2	no	—
119	C₇×C₂²⋊C₄	—	4	2	no	—
120	C₄²×C₇	—	2	1	no	—
121	C₄²×C₇	—	4	2	no	—
122	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₇×C₁₆	—	2	4	no	—
125	#575	C₂²×C₇	1	1	yes	characteristic
126	#568	—	2	1	no	—
127	#533	—	2	1	no	—
128	#510	C₂²×C₇	1	4	yes	—
129	#498	C₂²×C₇	1	1	yes	characteristic, complemented
130	#494	C₂²×C₇	1	1	yes	characteristic, complemented
131	#487	—	2	1	no	—
132	#470	—	2	1	no	—
133	#470	C₂²×C₇	1	2	yes	—
134	#444	—	2	1	no	—
135	#444	C₂²×C₇	1	2	yes	—
136	#443	—	2	1	no	—
137	#443	C₂²×C₇	1	2	yes	—
138	#441	—	2	1	no	—
139	#441	C₂²×C₇	1	2	yes	complemented
140	#430	C₂²×C₇	1	1	yes	characteristic
141	C₂×#104	—	2	1	no	—
142	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
143	#365	C₂²×C₇	1	4	yes	complemented
144	#364	C₂²×C₇	1	2	yes	complemented
145	#362	C₂²×C₇	1	4	yes	complemented
146	#360	C₂²×C₇	1	4	yes	—
147	#349	C₂²×C₇	1	2	yes	—
148	#347	C₂²×C₇	1	2	yes	complemented
149	C₂³×C₇	—	4	1	no	—
150	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₇×Q₈	—	2	2	no	—
152	C₇×Q₈	—	4	1	no	—
153	C₇×Q₈	—	4	2	no	—
154	C₇×Q₈	—	4	2	no	—
155	C₇×Q₈	—	8	2	no	—
156	C₇×D₄	—	2	2	no	—
157	C₇×D₄	—	4	1	no	—
158	C₇×D₄	—	4	2	no	—
159	C₇×D₄	—	4	2	no	—
160	C₂×C₄×C₇	—	2	1	no	—
161	C₂×C₄×C₇	—	2	1	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	4	2	no	—
165	C₂×C₄×C₇	—	4	2	no	—
166	C₂×C₄×C₇	—	4	1	no	—
167	C₂×C₄×C₇	—	4	2	no	—
168	C₂×C₄×C₇	—	4	2	no	—
169	C₂×C₄×C₇	—	4	2	no	—
170	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	4	2	no	—
175	C₇×C₈	AGL(1,ℤ/8)	1	2	yes	—
176	C₇×C₈	C₂×#41	1	2	yes	—
177	C₂×#41	—	2	1	no	—
178	#137	—	2	2	no	—
179	AGL(1,ℤ/8)	—	2	2	no	—
180	#135	—	2	1	no	—
181	#135	C₂³×C₇	1	2	yes	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
184	#131	—	2	2	no	—
185	#131	—	2	2	no	—
186	#131	—	4	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	4	2	no	—
189	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
190	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	#125	—	4	1	no	—
194	C₂²⋊Q₈	—	2	2	no	—
195	C₂²⋊Q₈	—	4	1	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	—	4	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
200	Q₃₂	—	2	4	no	—
201	SD₃₂	—	2	4	no	—
202	M₃₂	C₂³×C₇	1	4	yes	—
203	#108	—	2	1	no	—
204	#108	C₂³×C₇	1	2	yes	—
205	#106	—	2	1	no	—
206	#106	C₂³×C₇	1	2	yes	—
207	#104	—	2	2	no	—
208	#104	—	2	2	no	—
209	#103	—	2	2	no	—
210	#103	—	2	2	no	—
211	#97	C₇×D₄	1	1	yes	characteristic
212	#97	C₂³×C₇	1	1	yes	characteristic
213	C₄×C₈	C₇×D₄	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
215	C₂²×C₇	—	2	1	no	—
216	C₂²×C₇	—	4	1	no	—
217	C₂²×C₇	—	4	2	no	—
218	C₂²×C₇	—	16	1	no	—
219	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₄×C₇	—	2	2	no	—
221	C₄×C₇	—	2	1	no	—
222	C₄×C₇	—	4	1	no	—
223	C₄×C₇	—	4	2	no	—
224	C₄×C₇	—	8	2	no	—
225	C₄×C₇	—	8	2	no	—
226	C₄×C₇	#525	1	1	yes	characteristic
227	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	#41	—	4	1	no	—
231	#41	—	4	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
240	Q₁₆	—	2	1	no	—
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	2	no	—
243	Q₁₆	C₇×#41	1	2	yes	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	2	2	no	—
246	D₈	—	2	1	no	—
247	D₈	C₇×#41	1	2	yes	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	4	2	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
258	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
259	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
260	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
271	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
272	C₁₆	—	2	4	no	—
273	C₂×C₇	—	2	1	no	—
274	C₂×C₇	—	4	1	no	—
275	C₂×C₇	—	8	2	no	—
276	C₂×C₇	#3109	1	1	yes	characteristic
277	C₂³	—	4	1	no	—
278	C₂³	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	2	no	—
303	C₈	C₇×AGL(1,ℤ/8)	1	2	yes	—
304	C₈	C₂×C₇×#41	1	2	yes	—
305	C₇	#33689	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	16	1	no	—
310	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₇×#525	1	1	yes	characteristic
318	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	4	1	no	—
321	C₂	—	8	2	no	—
322	C₂	C₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₇×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3185	C₂	semidirect
3. C₇×#2239	C₂	semidirect
4. C₇×#2230	C₂	semidirect
5. C₇×#2214	C₂	semidirect
7. C₇×#2180	C₂	semidirect
8. C₇×#2165	C₂	semidirect
13. C₇×#498	C₂²	semidirect
14. C₇×#494	C₂²	semidirect
23. C₇×#441	C₂²	semidirect
27. C₇×#365	C₂²	semidirect
28. C₇×#364	C₂²	semidirect
29. C₇×#362	C₂²	semidirect
32. C₇×#347	C₂²	semidirect
33. #33689	C₇	direct (N × H)
72. #3185	C₂×C₇	semidirect
74. #2239	C₂×C₇	semidirect
75. #2230	C₂×C₇	semidirect
76. #2214	C₂×C₇	semidirect
78. #2180	C₂×C₇	semidirect
79. #2165	C₂×C₇	semidirect
129. #498	C₂²×C₇	semidirect
130. #494	C₂²×C₇	semidirect
139. #441	C₂²×C₇	semidirect
143. #365	C₂²×C₇	semidirect
144. #364	C₂²×C₇	semidirect
145. #362	C₂²×C₇	semidirect
148. #347	C₂²×C₇	semidirect
305. C₇	#33689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3185	2
C₇×#2956	2
C₇×#2239	2
C₇×#2230	2
C₇×#2214	2
C₇×#2191	2
C₇×#2180	2
C₇×#2165	2
#33689	7

Subgroups — C7×#33689 #13210905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33689 #13210905