Subgroups — C₇×#33693 #13210909

Order 1792.

512 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2955	C₂	1	1	yes	maximal, characteristic
3	C₇×#2239	C₂	1	2	yes	maximal, complemented
4	C₇×#2231	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×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#575	C₂²	1	1	yes	characteristic
10	C₂×C₇×#131	—	2	1	no	—
11	C₇×#541	—	2	1	no	—
12	C₇×#510	C₂²	1	4	yes	—
13	C₇×#499	C₂²	1	1	yes	characteristic, complemented
14	C₇×#494	C₂²	1	1	yes	characteristic, complemented
15	C₇×#491	—	2	1	no	—
16	C₇×#470	C₂²	1	2	yes	—
17	C₇×#467	—	2	1	no	—
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	C₂²	1	2	yes	complemented
23	C₄×C₇×Q₁₆	—	2	1	no	—
24	C₇×#430	C₂²	1	1	yes	characteristic
25	C₇×#421	—	2	1	no	—
26	C₇×#405	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	#33693	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	1	no	—
44	C₇×#131	—	4	1	no	—
45	C₂×C₇×M₁₆	—	2	1	no	—
46	C₂×C₇×M₁₆	—	4	1	no	—
47	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
48	C₂²×C₇×C₈	—	4	1	no	—
49	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
50	C₇×C₄⋊Q₈	—	2	1	no	—
51	C₇×#126	—	4	2	no	—
52	C₇×#125	—	2	1	no	—
53	C₇×#125	—	4	2	no	—
54	C₇×C₂²⋊Q₈	—	2	2	no	—
55	C₄×C₇×Q₈	—	2	2	no	—
56	C₇×#117	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×#107	—	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	#3177	C₂×C₇	1	1	yes	characteristic, complemented
73	#2955	C₂×C₇	1	1	yes	characteristic
74	#2239	C₂×C₇	1	2	yes	complemented
75	#2231	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	#2166	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×D₄	—	2	1	no	—
87	C₂×C₇×D₄	—	4	1	no	—
88	C₂²×C₄×C₇	—	4	1	no	—
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₇×Q₁₆	—	2	1	no	—
91	C₇×Q₁₆	—	2	4	no	—
92	C₇×Q₁₆	—	4	1	no	—
93	C₇×Q₁₆	—	4	1	no	—
94	C₇×Q₁₆	#41	1	2	yes	—
95	C₇×SD₁₆	—	2	2	no	—
96	C₇×SD₁₆	—	2	2	no	—
97	C₇×D₈	—	2	1	no	—
98	C₇×D₈	#41	1	2	yes	—
99	C₇×M₁₆	—	2	2	no	—
100	C₇×M₁₆	—	2	2	no	—
101	C₇×M₁₆	—	4	2	no	—
102	C₇×M₁₆	—	4	1	no	—
103	C₇×M₁₆	—	4	2	no	—
104	C₂×C₇×C₈	—	2	1	no	—
105	C₂×C₇×C₈	—	2	1	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	1	no	—
114	C₇×C₄⋊C₄	—	2	2	no	—
115	C₇×C₄⋊C₄	—	4	2	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₇×C₄⋊C₄	—	4	2	no	—
118	C₇×C₄⋊C₄	—	4	2	no	—
119	C₇×C₂²⋊C₄	—	2	1	no	—
120	C₇×C₂²⋊C₄	—	4	2	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	C₂×#131	—	2	1	no	—
127	#541	—	2	1	no	—
128	#510	C₂²×C₇	1	4	yes	—
129	#499	C₂²×C₇	1	1	yes	characteristic, complemented
130	#494	C₂²×C₇	1	1	yes	characteristic, complemented
131	#491	—	2	1	no	—
132	#470	C₂²×C₇	1	2	yes	—
133	#467	—	2	1	no	—
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	C₂²×C₇	1	2	yes	complemented
139	C₄×Q₁₆	—	2	1	no	—
140	#430	C₂²×C₇	1	1	yes	characteristic
141	#421	—	2	1	no	—
142	#405	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×D₄	—	2	2	no	—
156	C₇×D₄	—	4	1	no	—
157	C₇×D₄	—	4	2	no	—
158	C₇×D₄	—	4	2	no	—
159	C₂×C₄×C₇	—	2	1	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	—	4	2	no	—
164	C₂×C₄×C₇	—	4	1	no	—
165	C₂×C₄×C₇	—	4	2	no	—
166	C₂×C₄×C₇	—	4	2	no	—
167	C₂×C₄×C₇	—	4	2	no	—
168	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₇×C₈	—	2	1	no	—
171	C₇×C₈	—	2	1	no	—
172	C₇×C₈	—	4	2	no	—
173	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₇×C₈	C₂×#41	1	2	yes	—
176	C₂×#41	—	2	1	no	—
177	#137	—	2	2	no	—
178	AGL(1,ℤ/8)	—	2	2	no	—
179	#135	—	2	1	no	—
180	#135	C₂³×C₇	1	2	yes	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	#131	—	4	1	no	—
186	#131	—	4	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	4	1	no	—
189	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
190	C₂²×C₈	—	4	1	no	—
191	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
192	C₄⋊Q₈	—	2	1	no	—
193	#126	—	4	2	no	—
194	#125	—	2	1	no	—
195	#125	—	4	2	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	#117	C₇×D₄	1	1	yes	characteristic
199	Q₃₂	—	2	4	no	—
200	SD₃₂	—	2	4	no	—
201	M₃₂	C₂³×C₇	1	4	yes	—
202	#108	—	2	1	no	—
203	#108	C₂³×C₇	1	2	yes	—
204	#107	—	2	1	no	—
205	#106	C₂³×C₇	1	2	yes	—
206	#104	—	2	2	no	—
207	#104	—	2	2	no	—
208	#103	—	2	2	no	—
209	#103	—	2	2	no	—
210	#97	C₇×D₄	1	1	yes	characteristic
211	#97	C₂³×C₇	1	1	yes	characteristic
212	C₄×C₈	C₇×D₄	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	4	1	no	—
216	C₂²×C₇	—	4	2	no	—
217	C₂²×C₇	—	8	2	no	—
218	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₄×C₇	—	2	1	no	—
220	C₄×C₇	—	4	1	no	—
221	C₄×C₇	—	4	1	no	—
222	C₄×C₇	—	4	2	no	—
223	C₄×C₇	—	8	2	no	—
224	C₄×C₇	—	8	2	no	—
225	C₄×C₇	#525	1	1	yes	characteristic
226	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
237	Q₁₆	—	2	1	no	—
238	Q₁₆	—	2	4	no	—
239	Q₁₆	—	4	1	no	—
240	Q₁₆	—	4	1	no	—
241	Q₁₆	C₇×#41	1	2	yes	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	2	2	no	—
244	D₈	—	2	1	no	—
245	D₈	C₇×#41	1	2	yes	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
258	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
259	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
270	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
271	C₁₆	—	2	4	no	—
272	C₂×C₇	—	2	1	no	—
273	C₂×C₇	—	4	1	no	—
274	C₂×C₇	—	8	2	no	—
275	C₂×C₇	#3109	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	Q₈	—	4	1	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	D₄	—	2	2	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	4	2	no	—
300	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₈	C₂×C₇×#41	1	2	yes	—
303	C₇	#33693	1	1	yes	characteristic, direct product factor, Sylow
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₇×#525	1	1	yes	characteristic
316	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	4	1	no	—
319	C₂	—	8	2	no	—
320	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 ◁ C₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3177	C₂	semidirect
3. C₇×#2239	C₂	semidirect
4. C₇×#2231	C₂	semidirect
5. C₇×#2214	C₂	semidirect
7. C₇×#2180	C₂	semidirect
8. C₇×#2166	C₂	semidirect
13. C₇×#499	C₂²	semidirect
14. C₇×#494	C₂²	semidirect
22. 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. #33693	C₇	direct (N × H)
72. #3177	C₂×C₇	semidirect
74. #2239	C₂×C₇	semidirect
75. #2231	C₂×C₇	semidirect
76. #2214	C₂×C₇	semidirect
78. #2180	C₂×C₇	semidirect
79. #2166	C₂×C₇	semidirect
129. #499	C₂²×C₇	semidirect
130. #494	C₂²×C₇	semidirect
138. #441	C₂²×C₇	semidirect
143. #365	C₂²×C₇	semidirect
144. #364	C₂²×C₇	semidirect
145. #362	C₂²×C₇	semidirect
148. #347	C₂²×C₇	semidirect
303. C₇	#33693	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×#3177	2
C₇×#2955	2
C₇×#2239	2
C₇×#2231	2
C₇×#2214	2
C₇×#2191	2
C₇×#2180	2
C₇×#2166	2
#33693	7

Subgroups — C7×#33693 #13210909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33693 #13210909