Subgroups — C₇×#60651 #13237867

Order 1792.

3138 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#578	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×#577	C₂	1	2	yes	maximal, complemented
3	C₇×#3500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3499	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#3156	C₂	1	6	yes	maximal, complemented
6	C₇×#3154	C₂	1	6	yes	maximal, complemented
7	C₇×#3153	C₂	1	3	yes	maximal, complemented
8	C₇×#3152	C₂	1	3	yes	maximal, complemented
9	C₇×#3150	C₂	1	3	yes	maximal, complemented
10	C₂×C₇×#497	C₂	1	3	yes	maximal, complemented
11	C₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
14	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
15	C₇×#578	—	2	16	no	—
16	C₇×#577	—	2	16	no	—
17	C₂×C₇×#137	C₂²	1	6	yes	—
18	C₂×C₇×#137	C₂²	1	6	yes	complemented
19	C₂×C₇×#137	C₂²	1	6	yes	complemented
20	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	—
21	C₂×C₇×#135	C₂²	1	6	yes	—
22	C₂×C₇×#135	C₂²	1	6	yes	complemented
23	C₂²×C₇×Q₁₆	C₂²	1	6	yes	complemented
24	C₂²×C₇×SD₁₆	C₂²	1	6	yes	—
25	C₂×C₇×#131	C₂²	1	2	yes	—
26	C₇×D₄×Q₈	—	2	2	no	—
27	C₇×#548	—	2	2	no	—
28	C₇×#547	—	2	6	no	—
29	C₇×#546	—	2	6	no	—
30	C₇×#536	C₂²	1	3	yes	—
31	C₇×#536	C₂²	1	6	yes	complemented
32	C₇×#535	C₂²	1	3	yes	—
33	C₇×#532	C₂²	1	3	yes	—
34	C₇×#532	C₂²	1	3	yes	complemented
35	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	complemented
36	C₂×C₇×#124	C₂²	1	3	yes	—
37	C₄×C₇×#41	C₂²	1	1	yes	characteristic
38	C₇×#497	C₂²	1	6	yes	complemented
39	C₇×#497	C₂²	1	12	yes	complemented
40	C₇×#497	C₂²	1	3	yes	complemented
41	C₇×#497	C₂²	1	6	yes	complemented
42	C₇×#497	C₂²	1	3	yes	complemented
43	C₇×#496	C₂²	1	1	yes	characteristic, complemented
44	C₇×#496	C₂²	1	3	yes	complemented
45	C₇×#496	C₂²	1	6	yes	complemented
46	C₇×#495	C₂²	1	12	yes	complemented
47	C₇×#494	C₂²	1	6	yes	complemented
48	C₇×#492	C₂²	1	6	yes	complemented
49	C₇×#405	C₂²	1	6	yes	complemented
50	C₄×C₇×M₁₆	C₂²	1	3	yes	complemented
51	C₂×C₇×#97	C₂²	1	3	yes	complemented
52	#60651	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₇×2₋¹⁺⁴	—	2	8	no	—
54	C₇×2₋¹⁺⁴	—	2	8	no	—
55	C₇×2₋¹⁺⁴	—	2	8	no	—
56	C₇×H(5,𝔽₂)	—	2	8	no	—
57	C₂×C₇×#41	—	2	1	no	—
58	C₂×C₇×#41	—	2	3	no	—
59	C₂×C₇×#41	—	2	6	no	—
60	C₂×C₇×#41	C₂³	1	3	yes	—
61	C₂×C₇×#41	C₂³	1	3	yes	—
62	C₂×C₇×#41	C₂³	1	6	yes	—
63	C₂×C₇×#41	C₂³	1	1	yes	characteristic
64	C₂²×C₇×Q₈	—	2	1	no	—
65	C₂²×C₇×Q₈	—	2	2	no	—
66	C₂²×C₇×Q₈	C₂³	1	3	yes	—
67	C₂²×C₇×Q₈	C₂³	1	6	yes	—
68	C₂²×C₇×D₄	—	2	3	no	—
69	C₂²×C₇×D₄	C₂³	1	3	yes	—
70	C₇×#137	—	2	24	no	—
71	C₇×#137	—	2	24	no	—
72	C₇×#137	—	2	24	no	—
73	C₇×AGL(1,ℤ/8)	—	2	24	no	—
74	C₇×#135	—	2	24	no	—
75	C₇×#135	—	2	24	no	—
76	C₂×C₇×Q₁₆	—	2	24	no	—
77	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
78	C₂×C₇×Q₁₆	C₂³	1	12	yes	—
79	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
80	C₂×C₇×SD₁₆	—	2	24	no	—
81	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
82	C₂×C₇×SD₁₆	C₂³	1	12	yes	—
83	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
84	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
85	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
86	C₂×C₇×D₈	C₂³	1	6	yes	—
87	C₂×C₇×D₈	C₂³	1	2	yes	—
88	C₇×#131	D₄	1	16	yes	complemented
89	C₂×C₇×M₁₆	C₂³	1	6	yes	—
90	C₂²×C₇×C₈	C₂³	1	6	yes	—
91	C₇×C₄⋊Q₈	—	2	6	no	—
92	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
93	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
94	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
95	C₇×#127	C₂³	1	3	yes	—
96	C₇×#127	C₂³	1	1	yes	characteristic
97	C₇×#124	—	2	6	no	—
98	C₇×#124	C₂³	1	6	yes	—
99	C₇×#124	C₂³	1	6	yes	—
100	C₇×#124	C₂³	1	3	yes	—
101	C₇×#124	C₂³	1	1	yes	characteristic
102	C₇×#123	—	2	6	no	—
103	C₇×#123	—	2	12	no	—
104	C₇×C₂²⋊Q₈	—	2	6	no	—
105	C₇×C₂²⋊Q₈	—	2	12	no	—
106	C₇×#121	—	2	6	no	—
107	C₇×#121	—	2	6	no	—
108	C₇×#121	—	2	6	no	—
109	C₇×#120	—	2	6	no	—
110	C₄×C₇×Q₈	D₄	1	2	yes	—
111	C₄×C₇×D₄	—	2	6	no	—
112	C₄×C₇×D₄	—	2	6	no	—
113	C₄×C₇×D₄	D₄	1	6	yes	—
114	C₇×#117	C₂³	1	3	yes	—
115	C₂×C₇×C₄⋊C₄	—	2	6	no	—
116	C₂×C₇×C₂²⋊C₄	—	2	6	no	—
117	C₂×C₄²×C₇	C₂³	1	3	yes	—
118	C₇×#97	C₂³	1	1	yes	characteristic
119	C₇×#97	C₂³	1	3	yes	—
120	C₇×#97	C₂³	1	6	yes	—
121	C₄×C₇×C₈	C₂³	1	6	yes	—
122	C₂×#578	C₂×C₇	1	2	yes	complemented
123	C₂×#577	C₂×C₇	1	2	yes	complemented
124	#3500	C₂×C₇	1	1	yes	characteristic, complemented
125	#3499	C₂×C₇	1	1	yes	characteristic, complemented
126	#3156	C₂×C₇	1	6	yes	complemented
127	#3154	C₂×C₇	1	6	yes	complemented
128	#3153	C₂×C₇	1	3	yes	complemented
129	#3152	C₂×C₇	1	3	yes	complemented
130	#3150	C₂×C₇	1	3	yes	complemented
131	C₂×#497	C₂×C₇	1	3	yes	complemented
132	#2875	C₂×C₇	1	1	yes	characteristic, complemented
133	C₂⁴×C₇	—	2	3	no	—
134	C₇×#41	—	2	12	no	—
135	C₇×#41	—	2	12	no	—
136	C₇×#41	—	2	12	no	—
137	C₇×#41	—	2	12	no	—
138	C₇×#41	—	4	4	no	—
139	C₇×#41	—	4	12	no	—
140	C₇×#41	—	4	24	no	—
141	C₇×#41	C₂×D₄	1	8	yes	—
142	C₂×C₇×Q₈	—	2	3	no	—
143	C₂×C₇×Q₈	—	2	6	no	—
144	C₂×C₇×Q₈	—	2	12	no	—
145	C₂×C₇×Q₈	—	2	12	no	—
146	C₂×C₇×Q₈	—	2	12	no	—
147	C₂×C₇×Q₈	—	4	4	no	—
148	C₂×C₇×Q₈	—	4	8	no	—
149	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
150	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
151	C₂×C₇×Q₈	C₂⁴	1	6	yes	—
152	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
153	C₂×C₇×D₄	—	2	3	no	—
154	C₂×C₇×D₄	—	2	3	no	—
155	C₂×C₇×D₄	—	2	6	no	—
156	C₂×C₇×D₄	—	2	12	no	—
157	C₂×C₇×D₄	—	2	2	no	—
158	C₂×C₇×D₄	—	2	3	no	—
159	C₂×C₇×D₄	—	2	6	no	—
160	C₂×C₇×D₄	—	4	12	no	—
161	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
162	C₂×C₇×D₄	C₂⁴	1	2	yes	—
163	C₂×C₇×D₄	C₂⁴	1	3	yes	—
164	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₇	—	2	3	no	—
166	C₂²×C₄×C₇	—	2	6	no	—
167	C₂²×C₄×C₇	—	2	3	no	—
168	C₂²×C₄×C₇	—	2	3	no	—
169	C₂²×C₄×C₇	—	2	6	no	—
170	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
171	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
172	C₇×Q₁₆	—	2	12	no	—
173	C₇×Q₁₆	—	2	24	no	—
174	C₇×Q₁₆	—	2	12	no	—
175	C₇×SD₁₆	—	2	12	no	—
176	C₇×SD₁₆	—	2	24	no	—
177	C₇×SD₁₆	—	2	4	no	—
178	C₇×SD₁₆	—	2	12	no	—
179	C₇×SD₁₆	—	2	12	no	—
180	C₇×D₈	—	2	4	no	—
181	C₇×D₈	—	2	12	no	—
182	C₇×M₁₆	C₂×D₄	1	24	yes	—
183	C₂×C₇×C₈	C₂×D₄	1	24	yes	—
184	C₂×C₇×C₈	C₂⁴	1	6	yes	—
185	C₂×C₇×C₈	C₂⁴	1	2	yes	—
186	C₇×C₄⋊C₄	—	2	6	no	—
187	C₇×C₄⋊C₄	—	2	12	no	—
188	C₇×C₄⋊C₄	—	2	6	no	—
189	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
190	C₇×C₂²⋊C₄	—	2	6	no	—
191	C₇×C₂²⋊C₄	—	2	12	no	—
192	C₇×C₂²⋊C₄	—	2	2	no	—
193	C₇×C₂²⋊C₄	—	2	12	no	—
194	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
195	C₄²×C₇	C₂×D₄	1	6	yes	—
196	C₄²×C₇	C₂⁴	1	3	yes	—
197	C₄²×C₇	C₂⁴	1	1	yes	characteristic
198	C₂×2₋¹⁺⁴	C₂²×C₇	1	2	yes	—
199	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
200	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
201	#578	—	2	16	no	—
202	#577	—	2	16	no	—
203	C₂×#137	C₂²×C₇	1	6	yes	—
204	C₂×#137	C₂²×C₇	1	6	yes	complemented
205	C₂×#137	C₂²×C₇	1	6	yes	complemented
206	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	6	yes	—
207	C₂×#135	C₂²×C₇	1	6	yes	—
208	C₂×#135	C₂²×C₇	1	6	yes	complemented
209	C₂²×Q₁₆	C₂²×C₇	1	6	yes	complemented
210	C₂²×SD₁₆	C₂²×C₇	1	6	yes	—
211	C₂×#131	C₂²×C₇	1	2	yes	—
212	D₄×Q₈	—	2	2	no	—
213	#548	—	2	2	no	—
214	#547	—	2	6	no	—
215	#546	—	2	6	no	—
216	#536	C₂²×C₇	1	3	yes	—
217	#536	C₂²×C₇	1	6	yes	complemented
218	#535	C₂²×C₇	1	3	yes	—
219	#532	C₂²×C₇	1	3	yes	—
220	#532	C₂²×C₇	1	3	yes	complemented
221	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	complemented
222	C₂×#124	C₂²×C₇	1	3	yes	—
223	C₄×#41	C₂²×C₇	1	1	yes	characteristic
224	#497	C₂²×C₇	1	3	yes	complemented
225	#497	C₂²×C₇	1	6	yes	complemented
226	#497	C₂²×C₇	1	12	yes	complemented
227	#497	C₂²×C₇	1	3	yes	complemented
228	#497	C₂²×C₇	1	6	yes	complemented
229	#496	C₂²×C₇	1	1	yes	characteristic, complemented
230	#496	C₂²×C₇	1	3	yes	complemented
231	#496	C₂²×C₇	1	6	yes	complemented
232	#495	C₂²×C₇	1	12	yes	complemented
233	#494	C₂²×C₇	1	6	yes	complemented
234	#492	C₂²×C₇	1	6	yes	complemented
235	#405	C₂²×C₇	1	6	yes	complemented
236	C₄×M₁₆	C₂²×C₇	1	3	yes	complemented
237	C₂×#97	C₂²×C₇	1	3	yes	complemented
238	C₂³×C₇	—	2	2	no	—
239	C₂³×C₇	—	2	3	no	—
240	C₂³×C₇	—	2	1	no	—
241	C₂³×C₇	—	4	6	no	—
242	C₂³×C₇	—	8	6	no	—
243	C₂³×C₇	C₂²×D₄	1	3	yes	—
244	C₇×Q₈	—	2	6	no	—
245	C₇×Q₈	—	2	6	no	—
246	C₇×Q₈	—	2	6	no	—
247	C₇×Q₈	—	2	2	no	—
248	C₇×Q₈	—	4	6	no	—
249	C₇×Q₈	—	4	12	no	—
250	C₇×Q₈	#127	1	4	yes	—
251	C₇×D₄	—	2	2	no	—
252	C₇×D₄	—	2	6	no	—
253	C₇×D₄	—	2	2	no	—
254	C₇×D₄	—	2	2	no	—
255	C₇×D₄	—	4	4	no	—
256	C₇×D₄	—	4	6	no	—
257	C₇×D₄	—	4	6	no	—
258	C₇×D₄	—	4	12	no	—
259	C₇×D₄	—	4	6	no	—
260	C₇×D₄	—	8	6	no	—
261	C₇×D₄	#127	1	12	yes	—
262	C₂×C₄×C₇	—	2	3	no	—
263	C₂×C₄×C₇	—	2	1	no	—
264	C₂×C₄×C₇	—	2	6	no	—
265	C₂×C₄×C₇	—	4	6	no	—
266	C₂×C₄×C₇	—	4	6	no	—
267	C₂×C₄×C₇	—	4	12	no	—
268	C₂×C₄×C₇	—	4	6	no	—
269	C₂×C₄×C₇	—	4	6	no	—
270	C₂×C₄×C₇	—	4	12	no	—
271	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
273	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
274	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
275	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
276	C₇×C₈	C₂²×D₄	1	4	yes	—
277	C₇×C₈	C₂²×D₄	1	12	yes	—
278	2₋¹⁺⁴	—	2	8	no	—
279	2₋¹⁺⁴	—	2	8	no	—
280	2₋¹⁺⁴	—	2	8	no	—
281	H(5,𝔽₂)	—	2	8	no	—
282	C₂×#41	—	2	1	no	—
283	C₂×#41	—	2	3	no	—
284	C₂×#41	—	2	6	no	—
285	C₂×#41	C₂³×C₇	1	3	yes	—
286	C₂×#41	C₂³×C₇	1	3	yes	—
287	C₂×#41	C₂³×C₇	1	6	yes	—
288	C₂×#41	C₂³×C₇	1	1	yes	characteristic
289	C₂²×Q₈	—	2	1	no	—
290	C₂²×Q₈	—	2	2	no	—
291	C₂²×Q₈	C₂³×C₇	1	3	yes	—
292	C₂²×Q₈	C₂³×C₇	1	6	yes	—
293	C₂²×D₄	—	2	3	no	—
294	C₂²×D₄	C₂³×C₇	1	3	yes	—
295	#137	—	2	24	no	—
296	#137	—	2	24	no	—
297	#137	—	2	24	no	—
298	AGL(1,ℤ/8)	—	2	24	no	—
299	#135	—	2	24	no	—
300	#135	—	2	24	no	—
301	C₂×Q₁₆	—	2	24	no	—
302	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
303	C₂×Q₁₆	C₂³×C₇	1	12	yes	—
304	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
305	C₂×SD₁₆	—	2	24	no	—
306	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
307	C₂×SD₁₆	C₂³×C₇	1	12	yes	—
308	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
309	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
310	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
311	C₂×D₈	C₂³×C₇	1	6	yes	—
312	C₂×D₈	C₂³×C₇	1	2	yes	—
313	#131	C₇×D₄	1	16	yes	complemented
314	C₂×M₁₆	C₂³×C₇	1	6	yes	—
315	C₂²×C₈	C₂³×C₇	1	6	yes	—
316	C₄⋊Q₈	—	2	6	no	—
317	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
318	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
319	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
320	#127	C₂³×C₇	1	3	yes	—
321	#127	C₂³×C₇	1	1	yes	characteristic
322	#124	—	2	6	no	—
323	#124	C₂³×C₇	1	3	yes	—
324	#124	C₂³×C₇	1	6	yes	—
325	#124	C₂³×C₇	1	1	yes	characteristic
326	#124	C₂³×C₇	1	6	yes	—
327	#123	—	2	6	no	—
328	#123	—	2	12	no	—
329	C₂²⋊Q₈	—	2	6	no	—
330	C₂²⋊Q₈	—	2	12	no	—
331	#121	—	2	6	no	—
332	#121	—	2	6	no	—
333	#121	—	2	6	no	—
334	#120	—	2	6	no	—
335	C₄×Q₈	C₇×D₄	1	2	yes	—
336	C₄×D₄	—	2	6	no	—
337	C₄×D₄	—	2	6	no	—
338	C₄×D₄	C₇×D₄	1	6	yes	—
339	#117	C₂³×C₇	1	3	yes	—
340	C₂×C₄⋊C₄	—	2	6	no	—
341	C₂×C₂²⋊C₄	—	2	6	no	—
342	C₂×C₄²	C₂³×C₇	1	3	yes	—
343	#97	C₂³×C₇	1	1	yes	characteristic
344	#97	C₂³×C₇	1	3	yes	—
345	#97	C₂³×C₇	1	6	yes	—
346	C₄×C₈	C₂³×C₇	1	6	yes	—
347	C₂²×C₇	—	2	6	no	—
348	C₂²×C₇	—	4	4	no	—
349	C₂²×C₇	—	4	2	no	—
350	C₂²×C₇	—	4	6	no	—
351	C₂²×C₇	—	4	2	no	—
352	C₂²×C₇	—	4	1	no	—
353	C₂²×C₇	—	4	3	no	—
354	C₂²×C₇	—	16	6	no	—
355	C₂²×C₇	C₂×#127	1	6	yes	—
356	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
357	C₄×C₇	—	2	2	no	—
358	C₄×C₇	—	2	6	no	—
359	C₄×C₇	—	4	1	no	—
360	C₄×C₇	—	4	3	no	—
361	C₄×C₇	—	4	6	no	—
362	C₄×C₇	C₂×#127	1	6	yes	—
363	C₄×C₇	C₂³×D₄	1	2	yes	—
364	C₂⁴	—	2	3	no	—
365	#41	—	2	12	no	—
366	#41	—	2	12	no	—
367	#41	—	2	12	no	—
368	#41	—	2	12	no	—
369	#41	—	4	4	no	—
370	#41	—	4	12	no	—
371	#41	—	4	24	no	—
372	#41	C₂×C₇×D₄	1	8	yes	—
373	C₂×Q₈	—	2	3	no	—
374	C₂×Q₈	—	2	6	no	—
375	C₂×Q₈	—	2	12	no	—
376	C₂×Q₈	—	2	12	no	—
377	C₂×Q₈	—	2	12	no	—
378	C₂×Q₈	—	4	4	no	—
379	C₂×Q₈	—	4	8	no	—
380	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
381	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
382	C₂×Q₈	C₂⁴×C₇	1	6	yes	—
383	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
384	C₂×D₄	—	2	3	no	—
385	C₂×D₄	—	2	3	no	—
386	C₂×D₄	—	2	6	no	—
387	C₂×D₄	—	2	12	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	—	2	6	no	—
391	C₂×D₄	—	4	12	no	—
392	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
393	C₂×D₄	C₂⁴×C₇	1	2	yes	—
394	C₂×D₄	C₂⁴×C₇	1	3	yes	—
395	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
396	C₂²×C₄	—	2	3	no	—
397	C₂²×C₄	—	2	6	no	—
398	C₂²×C₄	—	2	3	no	—
399	C₂²×C₄	—	2	3	no	—
400	C₂²×C₄	—	2	6	no	—
401	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
402	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
403	Q₁₆	—	2	12	no	—
404	Q₁₆	—	2	24	no	—
405	Q₁₆	—	2	12	no	—
406	SD₁₆	—	2	12	no	—
407	SD₁₆	—	2	24	no	—
408	SD₁₆	—	2	4	no	—
409	SD₁₆	—	2	12	no	—
410	SD₁₆	—	2	12	no	—
411	D₈	—	2	4	no	—
412	D₈	—	2	12	no	—
413	M₁₆	C₂×C₇×D₄	1	24	yes	—
414	C₂×C₈	C₂×C₇×D₄	1	24	yes	—
415	C₂×C₈	C₂⁴×C₇	1	6	yes	—
416	C₂×C₈	C₂⁴×C₇	1	2	yes	—
417	C₄⋊C₄	—	2	6	no	—
418	C₄⋊C₄	—	2	12	no	—
419	C₄⋊C₄	—	2	6	no	—
420	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
421	C₂²⋊C₄	—	2	6	no	—
422	C₂²⋊C₄	—	2	12	no	—
423	C₂²⋊C₄	—	2	2	no	—
424	C₂²⋊C₄	—	2	12	no	—
425	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
426	C₄²	C₂×C₇×D₄	1	6	yes	—
427	C₄²	C₂⁴×C₇	1	3	yes	—
428	C₄²	C₂⁴×C₇	1	1	yes	characteristic
429	C₂×C₇	—	2	6	no	—
430	C₂×C₇	—	8	2	no	—
431	C₂×C₇	—	8	1	no	—
432	C₂×C₇	—	8	3	no	—
433	C₂×C₇	C₂²×#127	1	1	yes	characteristic
434	C₂×C₇	#3586	1	2	yes	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	3	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	4	6	no	—
439	C₂³	—	8	6	no	—
440	C₂³	C₂²×C₇×D₄	1	3	yes	—
441	Q₈	—	2	6	no	—
442	Q₈	—	2	6	no	—
443	Q₈	—	2	6	no	—
444	Q₈	—	2	2	no	—
445	Q₈	—	4	6	no	—
446	Q₈	—	4	12	no	—
447	Q₈	C₇×#127	1	4	yes	—
448	D₄	—	2	2	no	—
449	D₄	—	2	6	no	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	4	4	no	—
453	D₄	—	4	6	no	—
454	D₄	—	4	6	no	—
455	D₄	—	4	12	no	—
456	D₄	—	4	6	no	—
457	D₄	—	8	6	no	—
458	D₄	C₇×#127	1	12	yes	—
459	C₂×C₄	—	2	3	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	6	no	—
462	C₂×C₄	—	4	6	no	—
463	C₂×C₄	—	4	6	no	—
464	C₂×C₄	—	4	12	no	—
465	C₂×C₄	—	4	6	no	—
466	C₂×C₄	—	4	6	no	—
467	C₂×C₄	—	4	12	no	—
468	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
469	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
470	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
471	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
472	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
473	C₈	C₂²×C₇×D₄	1	4	yes	—
474	C₈	C₂²×C₇×D₄	1	12	yes	—
475	C₇	#60651	1	1	yes	characteristic, direct product factor, Sylow
476	C₂²	—	2	6	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	6	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	3	no	—
483	C₂²	—	16	6	no	—
484	C₂²	C₂×C₇×#127	1	6	yes	—
485	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	2	6	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	3	no	—
490	C₄	—	4	6	no	—
491	C₄	C₂×C₇×#127	1	6	yes	—
492	C₄	C₂³×C₇×D₄	1	2	yes	—
493	C₂	—	2	6	no	—
494	C₂	—	8	3	no	—
495	C₂	—	8	2	no	—
496	C₂	—	8	1	no	—
497	C₂	C₂²×C₇×#127	1	1	yes	characteristic
498	C₂	C₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#578	C₂	semidirect
2. C₂×C₇×#577	C₂	semidirect
3. C₇×#3500	C₂	semidirect
4. C₇×#3499	C₂	semidirect
5. C₇×#3156	C₂	semidirect
6. C₇×#3154	C₂	semidirect
7. C₇×#3153	C₂	semidirect
8. C₇×#3152	C₂	semidirect
9. C₇×#3150	C₂	semidirect
10. C₂×C₇×#497	C₂	semidirect
11. C₇×#2875	C₂	semidirect
18. C₂×C₇×#137	C₂²	semidirect
19. C₂×C₇×#137	C₂²	semidirect
22. C₂×C₇×#135	C₂²	semidirect
23. C₂²×C₇×Q₁₆	C₂²	semidirect
31. C₇×#536	C₂²	semidirect
34. C₇×#532	C₂²	semidirect
35. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
38. C₇×#497	C₂²	semidirect
39. C₇×#497	C₂²	semidirect
40. C₇×#497	C₂²	semidirect
41. C₇×#497	C₂²	semidirect
42. C₇×#497	C₂²	semidirect
43. C₇×#496	C₂²	semidirect
44. C₇×#496	C₂²	semidirect
45. C₇×#496	C₂²	semidirect
46. C₇×#495	C₂²	semidirect
47. C₇×#494	C₂²	semidirect
48. C₇×#492	C₂²	semidirect
49. C₇×#405	C₂²	semidirect
50. C₄×C₇×M₁₆	C₂²	semidirect
51. C₂×C₇×#97	C₂²	semidirect
52. #60651	C₇	direct (N × H)
88. C₇×#131	D₄	semidirect
122. C₂×#578	C₂×C₇	semidirect
123. C₂×#577	C₂×C₇	semidirect
124. #3500	C₂×C₇	semidirect
125. #3499	C₂×C₇	semidirect
126. #3156	C₂×C₇	semidirect
127. #3154	C₂×C₇	semidirect
128. #3153	C₂×C₇	semidirect
129. #3152	C₂×C₇	semidirect
130. #3150	C₂×C₇	semidirect
131. C₂×#497	C₂×C₇	semidirect
132. #2875	C₂×C₇	semidirect
204. C₂×#137	C₂²×C₇	semidirect
205. C₂×#137	C₂²×C₇	semidirect
208. C₂×#135	C₂²×C₇	semidirect
209. C₂²×Q₁₆	C₂²×C₇	semidirect
217. #536	C₂²×C₇	semidirect
220. #532	C₂²×C₇	semidirect
221. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
224. #497	C₂²×C₇	semidirect
225. #497	C₂²×C₇	semidirect
226. #497	C₂²×C₇	semidirect
227. #497	C₂²×C₇	semidirect
228. #497	C₂²×C₇	semidirect
229. #496	C₂²×C₇	semidirect
230. #496	C₂²×C₇	semidirect
231. #496	C₂²×C₇	semidirect
232. #495	C₂²×C₇	semidirect
233. #494	C₂²×C₇	semidirect
234. #492	C₂²×C₇	semidirect
235. #405	C₂²×C₇	semidirect
236. C₄×M₁₆	C₂²×C₇	semidirect
237. C₂×#97	C₂²×C₇	semidirect
313. #131	C₇×D₄	semidirect
475. C₇	#60651	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#578	2
C₂×C₇×#577	2
C₇×#3500	2
C₇×#3499	2
C₇×#3156	2
C₇×#3154	2
C₇×#3153	2
C₇×#3152	2
C₇×#3150	2
C₂×C₇×#497	2
C₇×#2875	2
#60651	7

Subgroups — C7×#60651 #13237867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60651 #13237867