Subgroups — C₇×#24800 #13202016

Order 1792.

842 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3046	C₂	1	2	yes	maximal
4	C₂×C₇×#444	C₂	1	1	yes	maximal, characteristic
5	C₇×#2912	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2064	C₂	1	2	yes	maximal, complemented
7	C₇×#2038	C₂	1	2	yes	maximal, complemented
8	C₇×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1882	C₂	1	2	yes	maximal, complemented
11	C₇×#1815	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#137	C₂²	1	2	yes	—
13	C₂×C₇×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₇×#131	C₂²	1	1	yes	characteristic
16	C₂×C₇×#131	C₂²	1	1	yes	characteristic
17	C₇×#536	C₂²	1	2	yes	—
18	C₇×#520	C₂²	1	2	yes	—
19	C₇×#473	—	2	4	no	—
20	C₇×#470	—	2	1	no	—
21	C₇×#470	—	2	1	no	—
22	C₇×#467	—	2	2	no	—
23	C₇×#464	—	2	2	no	—
24	C₇×#463	—	2	2	no	—
25	C₇×#462	—	2	2	no	—
26	C₇×#461	—	2	2	no	—
27	C₇×#444	—	2	4	no	—
28	C₂×C₇×#108	C₂²	1	1	yes	characteristic
29	C₂×C₇×#108	C₂²	1	1	yes	characteristic
30	C₇×#428	—	2	2	no	—
31	C₇×#428	C₂²	1	2	yes	—
32	C₇×#427	—	2	1	no	—
33	C₇×#427	C₂²	1	1	yes	characteristic
34	C₂×C₇×#106	—	2	1	no	—
35	C₂×C₇×#106	C₂²	1	1	yes	characteristic
36	C₇×#424	C₂²	1	2	yes	—
37	C₂×C₇×#104	C₂²	1	1	yes	characteristic
38	C₂×C₇×#104	C₂²	1	1	yes	characteristic
39	C₇×#419	C₂²	1	2	yes	—
40	C₇×#417	C₂²	1	2	yes	—
41	C₂×C₇×#103	C₂²	1	2	yes	—
42	C₂×C₇×#101	C₂²	1	2	yes	—
43	C₂×C₇×#97	C₂²	1	1	yes	characteristic
44	C₇×#391	C₂²	1	2	yes	—
45	C₇×#343	C₂²	1	2	yes	—
46	C₇×#339	C₂²	1	2	yes	—
47	#24800	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₇×#41	C₂³	1	1	yes	characteristic
49	C₂×C₇×#41	C₂³	1	1	yes	characteristic
50	C₂²×C₇×Q₈	C₂³	1	2	yes	—
51	C₇×#137	—	2	4	no	—
52	C₇×#137	—	2	4	no	—
53	C₇×#135	—	2	2	no	—
54	C₇×#135	—	2	2	no	—
55	C₂×C₇×Q₁₆	—	2	2	no	—
56	C₂×C₇×Q₁₆	—	2	1	no	—
57	C₂×C₇×Q₁₆	—	2	1	no	—
58	C₂×C₇×Q₁₆	—	2	2	no	—
59	C₂×C₇×Q₁₆	—	2	2	no	—
60	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
61	C₂×C₇×SD₁₆	—	2	2	no	—
62	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
63	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
64	C₂×C₇×D₈	D₄	1	1	yes	characteristic
65	C₇×#131	—	2	2	no	—
66	C₇×#131	—	2	2	no	—
67	C₇×#131	—	2	2	no	—
68	C₇×#131	D₄	1	4	yes	—
69	C₂×C₇×M₁₆	—	2	1	no	—
70	C₂×C₇×M₁₆	—	2	1	no	—
71	C₂×C₇×M₁₆	C₂³	1	2	yes	—
72	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
74	C₂²×C₇×C₈	—	2	1	no	—
75	C₂²×C₇×C₈	—	2	1	no	—
76	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₇×C₄⋊Q₈	—	2	2	no	—
79	C₇×#124	—	2	2	no	—
80	C₇×#123	—	4	2	no	—
81	C₇×C₂²⋊Q₈	—	2	2	no	—
82	C₇×C₂²⋊Q₈	—	2	2	no	—
83	C₄×C₇×Q₈	—	2	2	no	—
84	C₄×C₇×D₄	—	2	2	no	—
85	C₄×C₇×D₄	—	4	2	no	—
86	C₇×#117	—	2	2	no	—
87	C₇×#117	C₂³	1	2	yes	—
88	C₂×C₇×C₄⋊C₄	—	2	2	no	—
89	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
90	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
91	C₇×#108	—	2	2	no	—
92	C₇×#108	—	2	2	no	—
93	C₇×#107	—	2	2	no	—
94	C₇×#107	D₄	1	2	yes	—
95	C₇×#106	—	2	1	no	—
96	C₇×#106	—	2	1	no	—
97	C₇×#106	—	2	2	no	—
98	C₇×#106	D₄	1	1	yes	characteristic
99	C₇×#106	D₄	1	1	yes	characteristic
100	C₇×C₄⋊C₈	—	2	2	no	—
101	C₇×C₄⋊C₈	—	2	2	no	—
102	C₇×#104	—	2	2	no	—
103	C₇×#104	—	2	2	no	—
104	C₇×#103	—	2	2	no	—
105	C₇×#103	—	2	2	no	—
106	C₇×#103	—	2	2	no	—
107	C₇×#103	—	2	2	no	—
108	C₇×#103	—	2	2	no	—
109	C₇×#102	—	2	2	no	—
110	C₇×#101	—	2	4	no	—
111	C₇×#97	—	2	2	no	—
112	C₇×#95	—	4	2	no	—
113	C₂×#473	C₂×C₇	1	1	yes	characteristic, complemented
114	#3059	C₂×C₇	1	1	yes	characteristic, complemented
115	#3046	C₂×C₇	1	2	yes	—
116	C₂×#444	C₂×C₇	1	1	yes	characteristic
117	#2912	C₂×C₇	1	1	yes	characteristic, complemented
118	#2064	C₂×C₇	1	2	yes	complemented
119	#2038	C₂×C₇	1	2	yes	complemented
120	#1971	C₂×C₇	1	1	yes	characteristic, complemented
121	#1946	C₂×C₇	1	1	yes	characteristic, complemented
122	#1882	C₂×C₇	1	2	yes	complemented
123	#1815	C₂×C₇	1	1	yes	characteristic, complemented
124	C₇×#41	—	2	1	no	—
125	C₇×#41	—	2	1	no	—
126	C₇×#41	—	2	1	no	—
127	C₇×#41	—	2	1	no	—
128	C₇×#41	—	2	1	no	—
129	C₇×#41	—	2	1	no	—
130	C₇×#41	C₂×D₄	1	2	yes	—
131	C₇×#41	#41	1	2	yes	—
132	C₂×C₇×Q₈	—	2	2	no	—
133	C₂×C₇×Q₈	—	2	2	no	—
134	C₂×C₇×Q₈	—	2	2	no	—
135	C₂×C₇×Q₈	—	2	2	no	—
136	C₂×C₇×Q₈	—	2	2	no	—
137	C₂×C₇×Q₈	—	2	2	no	—
138	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
140	C₂×C₇×D₄	—	2	1	no	—
141	C₂×C₇×D₄	—	2	1	no	—
142	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
143	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₇	—	2	1	no	—
145	C₂²×C₄×C₇	—	2	1	no	—
146	C₂²×C₄×C₇	—	2	1	no	—
147	C₂²×C₄×C₇	—	2	2	no	—
148	C₂²×C₄×C₇	—	2	2	no	—
149	C₂²×C₄×C₇	—	4	2	no	—
150	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
151	C₇×Q₁₆	—	2	2	no	—
152	C₇×Q₁₆	—	4	4	no	—
153	C₇×Q₁₆	—	4	2	no	—
154	C₇×Q₁₆	—	4	2	no	—
155	C₇×SD₁₆	—	2	2	no	—
156	C₇×SD₁₆	—	2	2	no	—
157	C₇×SD₁₆	—	4	4	no	—
158	C₇×D₈	—	2	2	no	—
159	C₇×M₁₆	—	2	1	no	—
160	C₇×M₁₆	—	2	4	no	—
161	C₇×M₁₆	—	2	2	no	—
162	C₇×M₁₆	—	2	1	no	—
163	C₇×M₁₆	—	2	1	no	—
164	C₇×M₁₆	—	2	2	no	—
165	C₇×M₁₆	—	4	2	no	—
166	C₇×M₁₆	C₂×D₄	1	2	yes	—
167	C₂×C₇×C₈	—	2	1	no	—
168	C₂×C₇×C₈	—	2	1	no	—
169	C₂×C₇×C₈	—	2	1	no	—
170	C₂×C₇×C₈	—	2	4	no	—
171	C₂×C₇×C₈	—	2	2	no	—
172	C₂×C₇×C₈	—	2	1	no	—
173	C₂×C₇×C₈	—	4	2	no	—
174	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
176	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
177	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
178	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₇×C₈	#41	1	2	yes	—
180	C₇×C₄⋊C₄	—	2	2	no	—
181	C₇×C₄⋊C₄	—	2	2	no	—
182	C₇×C₄⋊C₄	—	4	2	no	—
183	C₇×C₄⋊C₄	—	4	2	no	—
184	C₇×C₄⋊C₄	—	4	2	no	—
185	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
186	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
187	C₇×C₂²⋊C₄	—	2	2	no	—
188	C₇×C₂²⋊C₄	—	4	2	no	—
189	C₇×C₂²⋊C₄	—	4	2	no	—
190	C₇×C₂²⋊C₄	—	4	2	no	—
191	C₄²×C₇	—	2	2	no	—
192	C₄²×C₇	—	2	1	no	—
193	C₄²×C₇	—	2	1	no	—
194	C₄²×C₇	—	4	2	no	—
195	C₂×#137	C₂²×C₇	1	2	yes	—
196	C₂×#135	C₂²×C₇	1	1	yes	characteristic
197	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
198	C₂×#131	C₂²×C₇	1	1	yes	characteristic
199	C₂×#131	C₂²×C₇	1	1	yes	characteristic
200	#536	C₂²×C₇	1	2	yes	—
201	#520	C₂²×C₇	1	2	yes	—
202	#473	—	2	4	no	—
203	#470	—	2	1	no	—
204	#470	—	2	1	no	—
205	#467	—	2	2	no	—
206	#464	—	2	2	no	—
207	#463	—	2	2	no	—
208	#462	—	2	2	no	—
209	#461	—	2	2	no	—
210	#444	—	2	4	no	—
211	C₂×#108	C₂²×C₇	1	1	yes	characteristic
212	C₂×#108	C₂²×C₇	1	1	yes	characteristic
213	#428	—	2	2	no	—
214	#428	C₂²×C₇	1	2	yes	—
215	#427	—	2	1	no	—
216	#427	C₂²×C₇	1	1	yes	characteristic
217	C₂×#106	—	2	1	no	—
218	C₂×#106	C₂²×C₇	1	1	yes	characteristic
219	#424	C₂²×C₇	1	2	yes	—
220	C₂×#104	C₂²×C₇	1	1	yes	characteristic
221	C₂×#104	C₂²×C₇	1	1	yes	characteristic
222	#419	C₂²×C₇	1	2	yes	—
223	#417	C₂²×C₇	1	2	yes	—
224	C₂×#103	C₂²×C₇	1	2	yes	—
225	C₂×#101	C₂²×C₇	1	2	yes	—
226	C₂×#97	C₂²×C₇	1	1	yes	characteristic
227	#391	C₂²×C₇	1	2	yes	—
228	#343	C₂²×C₇	1	2	yes	—
229	#339	C₂²×C₇	1	2	yes	—
230	C₂³×C₇	—	2	1	no	—
231	C₂³×C₇	—	2	1	no	—
232	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
233	C₇×Q₈	—	2	1	no	—
234	C₇×Q₈	—	2	1	no	—
235	C₇×Q₈	—	2	1	no	—
236	C₇×Q₈	—	4	2	no	—
237	C₇×Q₈	—	4	2	no	—
238	C₇×Q₈	—	4	2	no	—
239	C₇×Q₈	—	4	2	no	—
240	C₇×Q₈	#121	1	2	yes	—
241	C₇×D₄	—	2	2	no	—
242	C₇×D₄	—	2	2	no	—
243	C₇×D₄	—	2	1	no	—
244	C₇×D₄	—	2	1	no	—
245	C₇×D₄	—	2	1	no	—
246	C₇×D₄	—	4	1	no	—
247	C₇×D₄	—	4	1	no	—
248	C₇×D₄	#121	1	2	yes	—
249	C₂×C₄×C₇	—	2	1	no	—
250	C₂×C₄×C₇	—	2	2	no	—
251	C₂×C₄×C₇	—	2	2	no	—
252	C₂×C₄×C₇	—	2	2	no	—
253	C₂×C₄×C₇	—	2	2	no	—
254	C₂×C₄×C₇	—	2	1	no	—
255	C₂×C₄×C₇	—	4	1	no	—
256	C₂×C₄×C₇	—	4	1	no	—
257	C₂×C₄×C₇	—	4	2	no	—
258	C₂×C₄×C₇	—	4	1	no	—
259	C₂×C₄×C₇	—	4	2	no	—
260	C₂×C₄×C₇	—	4	2	no	—
261	C₂×C₄×C₇	—	4	2	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	—	8	2	no	—
264	C₂×C₄×C₇	—	16	2	no	—
265	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
271	C₇×C₈	—	2	2	no	—
272	C₇×C₈	—	2	1	no	—
273	C₇×C₈	—	2	1	no	—
274	C₇×C₈	—	4	1	no	—
275	C₇×C₈	—	4	2	no	—
276	C₇×C₈	#121	1	2	yes	—
277	C₇×C₈	#121	1	2	yes	—
278	C₂×#41	C₂³×C₇	1	1	yes	characteristic
279	C₂×#41	C₂³×C₇	1	1	yes	characteristic
280	C₂²×Q₈	C₂³×C₇	1	2	yes	—
281	#137	—	2	4	no	—
282	#137	—	2	4	no	—
283	#135	—	2	2	no	—
284	#135	—	2	2	no	—
285	C₂×Q₁₆	—	2	2	no	—
286	C₂×Q₁₆	—	2	1	no	—
287	C₂×Q₁₆	—	2	1	no	—
288	C₂×Q₁₆	—	2	2	no	—
289	C₂×Q₁₆	—	2	2	no	—
290	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
293	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
294	C₂×D₈	C₇×D₄	1	1	yes	characteristic
295	#131	—	2	2	no	—
296	#131	—	2	2	no	—
297	#131	—	2	2	no	—
298	#131	C₇×D₄	1	4	yes	—
299	C₂×M₁₆	—	2	1	no	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
302	C₂×M₁₆	C₂³×C₇	1	2	yes	—
303	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	2	1	no	—
306	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
308	C₄⋊Q₈	—	2	2	no	—
309	#124	—	2	2	no	—
310	#123	—	4	2	no	—
311	C₂²⋊Q₈	—	2	2	no	—
312	C₂²⋊Q₈	—	2	2	no	—
313	C₄×Q₈	—	2	2	no	—
314	C₄×D₄	—	2	2	no	—
315	C₄×D₄	—	4	2	no	—
316	#117	—	2	2	no	—
317	#117	C₂³×C₇	1	2	yes	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
320	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
321	#108	—	2	2	no	—
322	#108	—	2	2	no	—
323	#107	—	2	2	no	—
324	#107	C₇×D₄	1	2	yes	—
325	#106	—	2	1	no	—
326	#106	—	2	1	no	—
327	#106	—	2	2	no	—
328	#106	C₇×D₄	1	1	yes	characteristic
329	#106	C₇×D₄	1	1	yes	characteristic
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	2	no	—
332	#104	—	2	2	no	—
333	#104	—	2	2	no	—
334	#103	—	2	2	no	—
335	#103	—	2	2	no	—
336	#103	—	2	2	no	—
337	#103	—	2	2	no	—
338	#103	—	2	2	no	—
339	#102	—	2	2	no	—
340	#101	—	2	4	no	—
341	#97	—	2	2	no	—
342	#95	—	4	2	no	—
343	C₂²×C₇	—	2	2	no	—
344	C₂²×C₇	—	2	2	no	—
345	C₂²×C₇	—	4	2	no	—
346	C₂²×C₇	—	4	1	no	—
347	C₂²×C₇	—	4	2	no	—
348	C₂²×C₇	#537	1	1	yes	characteristic
349	C₂²×C₇	D₄²	1	1	yes	characteristic
350	C₂²×C₇	#553	1	1	yes	characteristic
351	C₄×C₇	—	2	2	no	—
352	C₄×C₇	—	4	1	no	—
353	C₄×C₇	—	4	2	no	—
354	C₄×C₇	—	4	2	no	—
355	C₄×C₇	—	8	1	no	—
356	C₄×C₇	—	8	2	no	—
357	C₄×C₇	—	8	2	no	—
358	C₄×C₇	C₂×#121	1	1	yes	characteristic
359	C₄×C₇	C₂×#121	1	1	yes	characteristic
360	C₄×C₇	#547	1	1	yes	characteristic
361	C₄×C₇	#548	1	1	yes	characteristic
362	#41	—	2	1	no	—
363	#41	—	2	1	no	—
364	#41	—	2	1	no	—
365	#41	—	2	1	no	—
366	#41	—	2	1	no	—
367	#41	—	2	1	no	—
368	#41	C₂×C₇×D₄	1	2	yes	—
369	#41	C₇×#41	1	2	yes	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
377	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
381	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
389	Q₁₆	—	2	2	no	—
390	Q₁₆	—	4	4	no	—
391	Q₁₆	—	4	2	no	—
392	Q₁₆	—	4	2	no	—
393	SD₁₆	—	2	2	no	—
394	SD₁₆	—	2	2	no	—
395	SD₁₆	—	4	4	no	—
396	D₈	—	2	2	no	—
397	M₁₆	—	2	1	no	—
398	M₁₆	—	2	4	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	2	1	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	4	2	no	—
404	M₁₆	C₂×C₇×D₄	1	2	yes	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	4	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	4	2	no	—
412	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
413	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
414	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
415	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
416	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
417	C₂×C₈	C₇×#41	1	2	yes	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
424	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	4	2	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	1	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	4	2	no	—
433	C₂×C₇	—	2	1	no	—
434	C₂×C₇	—	2	1	no	—
435	C₂×C₇	—	4	2	no	—
436	C₂×C₇	—	8	1	no	—
437	C₂×C₇	#2416	1	1	yes	characteristic
438	C₂×C₇	#3289	1	2	yes	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
442	Q₈	—	2	1	no	—
443	Q₈	—	2	1	no	—
444	Q₈	—	2	1	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	4	2	no	—
448	Q₈	—	4	2	no	—
449	Q₈	C₇×#121	1	2	yes	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	1	no	—
453	D₄	—	2	1	no	—
454	D₄	—	2	1	no	—
455	D₄	—	4	1	no	—
456	D₄	—	4	1	no	—
457	D₄	C₇×#121	1	2	yes	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	8	2	no	—
473	C₂×C₄	—	16	2	no	—
474	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
475	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
476	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
477	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
478	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
479	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	1	no	—
482	C₈	—	2	1	no	—
483	C₈	—	4	1	no	—
484	C₈	—	4	2	no	—
485	C₈	C₇×#121	1	2	yes	—
486	C₈	C₇×#121	1	2	yes	—
487	C₇	#24800	1	1	yes	characteristic, direct product factor, Sylow
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	4	2	no	—
493	C₂²	C₇×#537	1	1	yes	characteristic
494	C₂²	C₇×D₄²	1	1	yes	characteristic
495	C₂²	C₇×#553	1	1	yes	characteristic
496	C₄	—	2	2	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	8	1	no	—
501	C₄	—	8	2	no	—
502	C₄	—	8	2	no	—
503	C₄	C₂×C₇×#121	1	1	yes	characteristic
504	C₄	C₂×C₇×#121	1	1	yes	characteristic
505	C₄	C₇×#547	1	1	yes	characteristic
506	C₄	C₇×#548	1	1	yes	characteristic
507	C₂	—	2	1	no	—
508	C₂	—	2	1	no	—
509	C₂	—	4	2	no	—
510	C₂	—	8	1	no	—
511	C₂	C₇×#2416	1	1	yes	characteristic
512	C₂	C₇×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#473	C₂	semidirect
2. C₇×#3059	C₂	semidirect
5. C₇×#2912	C₂	semidirect
6. C₇×#2064	C₂	semidirect
7. C₇×#2038	C₂	semidirect
8. C₇×#1971	C₂	semidirect
9. C₇×#1946	C₂	semidirect
10. C₇×#1882	C₂	semidirect
11. C₇×#1815	C₂	semidirect
47. #24800	C₇	direct (N × H)
113. C₂×#473	C₂×C₇	semidirect
114. #3059	C₂×C₇	semidirect
117. #2912	C₂×C₇	semidirect
118. #2064	C₂×C₇	semidirect
119. #2038	C₂×C₇	semidirect
120. #1971	C₂×C₇	semidirect
121. #1946	C₂×C₇	semidirect
122. #1882	C₂×C₇	semidirect
123. #1815	C₂×C₇	semidirect
487. C₇	#24800	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×#473	2
C₇×#3059	2
C₇×#3046	2
C₂×C₇×#444	2
C₇×#2912	2
C₇×#2064	2
C₇×#2038	2
C₇×#1971	2
C₇×#1946	2
C₇×#1882	2
C₇×#1815	2
#24800	7

Subgroups — C7×#24800 #13202016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24800 #13202016