Subgroups — C₇×#24809 #13202025

Order 1792.

830 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3068	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#472	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2955	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2069	C₂	1	2	yes	maximal, complemented
7	C₇×#2048	C₂	1	2	yes	maximal, complemented
8	C₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1903	C₂	1	2	yes	maximal, complemented
11	C₇×#1813	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#575	C₂²	1	1	yes	characteristic
13	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
15	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
16	C₇×#568	C₂²	1	2	yes	—
17	C₂×C₇×#131	C₂²	1	1	yes	characteristic
18	C₂×C₇×#131	C₂²	1	1	yes	characteristic
19	C₇×#472	—	2	4	no	—
20	C₇×#472	—	2	2	no	—
21	C₇×#472	—	2	2	no	—
22	C₇×#472	—	2	2	no	—
23	C₇×#472	—	2	2	no	—
24	C₇×#471	—	2	4	no	—
25	C₇×#444	—	2	2	no	—
26	C₇×#443	—	2	2	no	—
27	C₇×#430	—	2	2	no	—
28	C₇×#430	C₂²	1	2	yes	—
29	C₇×#430	C₂²	1	1	yes	characteristic
30	C₇×#430	C₂²	1	2	yes	—
31	C₂×C₇×#108	—	2	2	no	—
32	C₂×C₇×#108	C₂²	1	1	yes	characteristic
33	C₂×C₇×#108	C₂²	1	1	yes	characteristic
34	C₂×C₇×#108	C₂²	1	1	yes	characteristic
35	C₇×#421	C₂²	1	1	yes	characteristic
36	C₇×#421	C₂²	1	1	yes	characteristic
37	C₇×#418	C₂²	1	2	yes	—
38	C₇×#413	C₂²	1	2	yes	—
39	C₂×C₇×#100	C₂²	1	2	yes	—
40	C₂×C₇×#100	C₂²	1	2	yes	—
41	C₂×C₇×#100	C₂²	1	2	yes	—
42	C₇×#405	C₂²	1	1	yes	characteristic
43	C₇×#344	C₂²	1	2	yes	—
44	C₇×#338	C₂²	1	2	yes	—
45	#24809	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₇×#41	C₂³	1	1	yes	characteristic
47	C₂×C₇×#41	C₂³	1	1	yes	characteristic
48	C₂²×C₇×D₄	C₂³	1	2	yes	—
49	C₇×#137	—	2	1	no	—
50	C₇×#137	D₄	1	2	yes	—
51	C₇×AGL(1,ℤ/8)	—	2	1	no	—
52	C₇×AGL(1,ℤ/8)	—	2	4	no	—
53	C₇×AGL(1,ℤ/8)	—	2	4	no	—
54	C₇×AGL(1,ℤ/8)	—	2	4	no	—
55	C₇×AGL(1,ℤ/8)	—	4	2	no	—
56	C₇×AGL(1,ℤ/8)	D₄	1	2	yes	—
57	C₇×#135	—	2	2	no	—
58	C₂×C₇×SD₁₆	—	2	2	no	—
59	C₂×C₇×SD₁₆	—	2	2	no	—
60	C₂×C₇×D₈	—	2	2	no	—
61	C₂×C₇×D₈	—	2	2	no	—
62	C₂×C₇×D₈	—	2	1	no	—
63	C₂×C₇×D₈	—	2	1	no	—
64	C₂×C₇×D₈	—	2	1	no	—
65	C₂×C₇×D₈	—	2	1	no	—
66	C₂×C₇×D₈	—	2	1	no	—
67	C₂×C₇×D₈	—	2	1	no	—
68	C₇×#131	—	2	1	no	—
69	C₇×#131	—	2	1	no	—
70	C₇×#131	—	2	4	no	—
71	C₇×#131	—	2	1	no	—
72	C₇×#131	—	2	1	no	—
73	C₇×#131	—	2	1	no	—
74	C₇×#131	—	2	1	no	—
75	C₇×#131	—	4	2	no	—
76	C₇×#131	D₄	1	2	yes	—
77	C₇×#131	D₄	1	2	yes	—
78	C₂×C₇×M₁₆	—	2	1	no	—
79	C₂×C₇×M₁₆	—	2	1	no	—
80	C₂×C₇×M₁₆	—	2	4	no	—
81	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
82	C₂×C₇×M₁₆	C₂³	1	2	yes	—
83	C₂×C₇×M₁₆	C₂³	1	2	yes	—
84	C₂×C₇×M₁₆	C₂³	1	2	yes	—
85	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
86	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
87	C₂²×C₇×C₈	—	2	1	no	—
88	C₂²×C₇×C₈	—	2	1	no	—
89	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
90	C₇×#117	C₂³	1	1	yes	characteristic
91	C₇×#108	—	2	4	no	—
92	C₇×#108	—	2	4	no	—
93	C₇×#108	—	2	2	no	—
94	C₇×#108	—	2	1	no	—
95	C₇×#108	—	2	1	no	—
96	C₇×#108	—	2	1	no	—
97	C₇×#108	—	2	1	no	—
98	C₇×#108	D₄	1	2	yes	—
99	C₇×#108	D₄	1	2	yes	—
100	C₇×#104	—	2	2	no	—
101	C₇×#104	—	2	2	no	—
102	C₇×#102	—	4	2	no	—
103	C₇×#101	—	2	2	no	—
104	C₇×#100	—	2	2	no	—
105	C₇×#100	—	2	4	no	—
106	C₇×#100	—	2	4	no	—
107	C₇×#100	—	2	4	no	—
108	C₇×#97	—	2	1	no	—
109	C₄×C₇×C₈	—	2	1	no	—
110	#3068	C₂×C₇	1	2	yes	complemented
111	C₂×#472	C₂×C₇	1	1	yes	characteristic, complemented
112	C₂×#472	C₂×C₇	1	1	yes	characteristic, complemented
113	#2955	C₂×C₇	1	1	yes	characteristic, complemented
114	#2916	C₂×C₇	1	1	yes	characteristic, complemented
115	#2069	C₂×C₇	1	2	yes	complemented
116	#2048	C₂×C₇	1	2	yes	complemented
117	#1978	C₂×C₇	1	1	yes	characteristic, complemented
118	#1953	C₂×C₇	1	1	yes	characteristic, complemented
119	#1903	C₂×C₇	1	2	yes	complemented
120	#1813	C₂×C₇	1	1	yes	characteristic, complemented
121	C₂⁴×C₇	—	2	2	no	—
122	C₇×#41	—	2	1	no	—
123	C₇×#41	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
129	C₇×#41	C₂×D₄	1	2	yes	—
130	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₇×Q₈	#41	1	1	yes	characteristic
132	C₂×C₇×D₄	—	2	1	no	—
133	C₂×C₇×D₄	—	2	1	no	—
134	C₂×C₇×D₄	—	2	2	no	—
135	C₂×C₇×D₄	—	2	2	no	—
136	C₂×C₇×D₄	—	2	2	no	—
137	C₂×C₇×D₄	—	2	2	no	—
138	C₂×C₇×D₄	—	2	2	no	—
139	C₂×C₇×D₄	—	2	2	no	—
140	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₇×D₄	#41	1	1	yes	characteristic
142	C₂²×C₄×C₇	—	2	1	no	—
143	C₂²×C₄×C₇	—	2	1	no	—
144	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
145	C₇×Q₁₆	—	2	2	no	—
146	C₇×SD₁₆	—	2	2	no	—
147	C₇×SD₁₆	—	2	2	no	—
148	C₇×SD₁₆	—	4	4	no	—
149	C₇×SD₁₆	—	4	4	no	—
150	C₇×D₈	—	2	2	no	—
151	C₇×D₈	—	4	4	no	—
152	C₇×D₈	—	4	4	no	—
153	C₇×D₈	—	4	1	no	—
154	C₇×D₈	—	4	1	no	—
155	C₇×D₈	—	4	1	no	—
156	C₇×D₈	—	4	1	no	—
157	C₇×M₁₆	—	2	1	no	—
158	C₇×M₁₆	—	2	1	no	—
159	C₇×M₁₆	—	2	2	no	—
160	C₇×M₁₆	—	2	2	no	—
161	C₇×M₁₆	—	2	2	no	—
162	C₇×M₁₆	—	2	1	no	—
163	C₇×M₁₆	—	2	2	no	—
164	C₇×M₁₆	—	2	2	no	—
165	C₇×M₁₆	—	2	2	no	—
166	C₇×M₁₆	—	2	1	no	—
167	C₇×M₁₆	—	2	2	no	—
168	C₇×M₁₆	—	4	4	no	—
169	C₇×M₁₆	—	4	1	no	—
170	C₇×M₁₆	—	4	1	no	—
171	C₇×M₁₆	C₂×D₄	1	2	yes	—
172	C₇×M₁₆	C₂×D₄	1	2	yes	—
173	C₇×M₁₆	C₂×D₄	1	4	yes	—
174	C₂×C₇×C₈	—	2	1	no	—
175	C₂×C₇×C₈	—	2	1	no	—
176	C₂×C₇×C₈	—	2	2	no	—
177	C₂×C₇×C₈	—	2	2	no	—
178	C₂×C₇×C₈	—	2	2	no	—
179	C₂×C₇×C₈	—	2	1	no	—
180	C₂×C₇×C₈	—	2	2	no	—
181	C₂×C₇×C₈	—	2	2	no	—
182	C₂×C₇×C₈	—	4	4	no	—
183	C₂×C₇×C₈	—	4	1	no	—
184	C₂×C₇×C₈	—	4	1	no	—
185	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
186	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
187	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
188	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
189	C₂×C₇×C₈	#41	1	1	yes	characteristic
190	C₂×C₇×C₈	#41	1	1	yes	characteristic
191	C₇×C₄⋊C₄	—	2	1	no	—
192	C₇×C₂²⋊C₄	—	2	1	no	—
193	C₄²×C₇	—	2	1	no	—
194	#575	C₂²×C₇	1	1	yes	characteristic
195	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
196	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
197	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
198	#568	C₂²×C₇	1	2	yes	—
199	C₂×#131	C₂²×C₇	1	1	yes	characteristic
200	C₂×#131	C₂²×C₇	1	1	yes	characteristic
201	#472	—	2	4	no	—
202	#472	—	2	2	no	—
203	#472	—	2	2	no	—
204	#472	—	2	2	no	—
205	#472	—	2	2	no	—
206	#471	—	2	4	no	—
207	#444	—	2	2	no	—
208	#443	—	2	2	no	—
209	#430	—	2	2	no	—
210	#430	C₂²×C₇	1	2	yes	—
211	#430	C₂²×C₇	1	1	yes	characteristic
212	#430	C₂²×C₇	1	2	yes	—
213	C₂×#108	—	2	2	no	—
214	C₂×#108	C₂²×C₇	1	1	yes	characteristic
215	C₂×#108	C₂²×C₇	1	1	yes	characteristic
216	C₂×#108	C₂²×C₇	1	1	yes	characteristic
217	#421	C₂²×C₇	1	1	yes	characteristic
218	#421	C₂²×C₇	1	1	yes	characteristic
219	#418	C₂²×C₇	1	2	yes	—
220	#413	C₂²×C₇	1	2	yes	—
221	C₂×#100	C₂²×C₇	1	2	yes	—
222	C₂×#100	C₂²×C₇	1	2	yes	—
223	C₂×#100	C₂²×C₇	1	2	yes	—
224	#405	C₂²×C₇	1	1	yes	characteristic
225	#344	C₂²×C₇	1	2	yes	—
226	#338	C₂²×C₇	1	2	yes	—
227	C₂³×C₇	—	2	1	no	—
228	C₂³×C₇	—	2	1	no	—
229	C₂³×C₇	—	4	2	no	—
230	C₂³×C₇	—	4	2	no	—
231	C₂³×C₇	—	4	2	no	—
232	C₂³×C₇	—	16	2	no	—
233	C₂³×C₇	2₋¹⁺⁴	1	1	yes	characteristic
234	C₇×Q₈	—	2	1	no	—
235	C₇×Q₈	—	2	1	no	—
236	C₇×Q₈	—	2	1	no	—
237	C₇×Q₈	#121	1	2	yes	—
238	C₇×D₄	—	2	2	no	—
239	C₇×D₄	—	2	2	no	—
240	C₇×D₄	—	2	1	no	—
241	C₇×D₄	—	2	1	no	—
242	C₇×D₄	—	2	1	no	—
243	C₇×D₄	—	4	1	no	—
244	C₇×D₄	—	4	1	no	—
245	C₇×D₄	—	4	2	no	—
246	C₇×D₄	—	4	2	no	—
247	C₇×D₄	—	4	2	no	—
248	C₇×D₄	—	4	2	no	—
249	C₇×D₄	#121	1	2	yes	—
250	C₂×C₄×C₇	—	2	1	no	—
251	C₂×C₄×C₇	—	2	2	no	—
252	C₂×C₄×C₇	—	2	2	no	—
253	C₂×C₄×C₇	—	2	1	no	—
254	C₂×C₄×C₇	—	4	1	no	—
255	C₂×C₄×C₇	—	4	1	no	—
256	C₂×C₄×C₇	—	4	1	no	—
257	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
263	C₇×C₈	—	2	2	no	—
264	C₇×C₈	—	2	2	no	—
265	C₇×C₈	—	2	4	no	—
266	C₇×C₈	—	4	1	no	—
267	C₇×C₈	—	4	2	no	—
268	C₇×C₈	—	4	2	no	—
269	C₇×C₈	#121	1	1	yes	characteristic
270	C₇×C₈	#121	1	1	yes	characteristic
271	C₇×C₈	#121	1	1	yes	characteristic
272	C₇×C₈	#121	1	1	yes	characteristic
273	C₂×#41	C₂³×C₇	1	1	yes	characteristic
274	C₂×#41	C₂³×C₇	1	1	yes	characteristic
275	C₂²×D₄	C₂³×C₇	1	2	yes	—
276	#137	—	2	1	no	—
277	#137	C₇×D₄	1	2	yes	—
278	AGL(1,ℤ/8)	—	2	1	no	—
279	AGL(1,ℤ/8)	—	2	4	no	—
280	AGL(1,ℤ/8)	—	2	4	no	—
281	AGL(1,ℤ/8)	—	2	4	no	—
282	AGL(1,ℤ/8)	—	4	2	no	—
283	AGL(1,ℤ/8)	C₇×D₄	1	2	yes	—
284	#135	—	2	2	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	—	2	1	no	—
290	C₂×D₈	—	2	1	no	—
291	C₂×D₈	—	2	1	no	—
292	C₂×D₈	—	2	1	no	—
293	C₂×D₈	—	2	1	no	—
294	C₂×D₈	—	2	1	no	—
295	#131	—	2	1	no	—
296	#131	—	2	1	no	—
297	#131	—	2	4	no	—
298	#131	—	2	1	no	—
299	#131	—	2	1	no	—
300	#131	—	2	1	no	—
301	#131	—	2	1	no	—
302	#131	—	4	2	no	—
303	#131	C₇×D₄	1	2	yes	—
304	#131	C₇×D₄	1	2	yes	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	2	1	no	—
307	C₂×M₁₆	—	2	4	no	—
308	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
309	C₂×M₁₆	C₂³×C₇	1	2	yes	—
310	C₂×M₁₆	C₂³×C₇	1	2	yes	—
311	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
312	C₂×M₁₆	C₂³×C₇	1	2	yes	—
313	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
314	C₂²×C₈	—	2	1	no	—
315	C₂²×C₈	—	2	1	no	—
316	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
317	#117	C₂³×C₇	1	1	yes	characteristic
318	#108	—	2	4	no	—
319	#108	—	2	2	no	—
320	#108	—	2	4	no	—
321	#108	—	2	1	no	—
322	#108	—	2	1	no	—
323	#108	—	2	1	no	—
324	#108	—	2	1	no	—
325	#108	C₇×D₄	1	2	yes	—
326	#108	C₇×D₄	1	2	yes	—
327	#104	—	2	2	no	—
328	#104	—	2	2	no	—
329	#102	—	4	2	no	—
330	#101	—	2	2	no	—
331	#100	—	2	2	no	—
332	#100	—	2	4	no	—
333	#100	—	2	4	no	—
334	#100	—	2	4	no	—
335	#97	—	2	1	no	—
336	C₄×C₈	—	2	1	no	—
337	C₂²×C₇	—	2	2	no	—
338	C₂²×C₇	—	4	1	no	—
339	C₂²×C₇	—	4	1	no	—
340	C₂²×C₇	—	8	1	no	—
341	C₂²×C₇	—	8	1	no	—
342	C₂²×C₇	—	8	2	no	—
343	C₂²×C₇	—	16	2	no	—
344	C₂²×C₇	—	16	2	no	—
345	C₂²×C₇	—	16	2	no	—
346	C₂²×C₇	#537	1	1	yes	characteristic
347	C₂²×C₇	#547	1	1	yes	characteristic
348	C₂²×C₇	#548	1	1	yes	characteristic
349	C₄×C₇	—	2	2	no	—
350	C₄×C₇	—	4	1	no	—
351	C₄×C₇	—	8	1	no	—
352	C₄×C₇	C₂×#121	1	1	yes	characteristic
353	C₄×C₇	C₂×#121	1	1	yes	characteristic
354	C₄×C₇	D₄²	1	1	yes	characteristic
355	C₄×C₇	#553	1	1	yes	characteristic
356	C₂⁴	—	2	2	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	2	1	no	—
360	#41	—	2	1	no	—
361	#41	—	2	1	no	—
362	#41	—	2	1	no	—
363	#41	C₂×C₇×D₄	1	2	yes	—
364	#41	C₂×C₇×D₄	1	2	yes	—
365	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
366	C₂×Q₈	C₇×#41	1	1	yes	characteristic
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
376	C₂×D₄	C₇×#41	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
380	Q₁₆	—	2	2	no	—
381	SD₁₆	—	2	2	no	—
382	SD₁₆	—	2	2	no	—
383	SD₁₆	—	4	4	no	—
384	SD₁₆	—	4	4	no	—
385	D₈	—	2	2	no	—
386	D₈	—	4	4	no	—
387	D₈	—	4	4	no	—
388	D₈	—	4	1	no	—
389	D₈	—	4	1	no	—
390	D₈	—	4	1	no	—
391	D₈	—	4	1	no	—
392	M₁₆	—	2	1	no	—
393	M₁₆	—	2	1	no	—
394	M₁₆	—	2	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	2	2	no	—
397	M₁₆	—	2	1	no	—
398	M₁₆	—	2	2	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	2	2	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	4	4	no	—
404	M₁₆	—	4	1	no	—
405	M₁₆	—	4	1	no	—
406	M₁₆	C₂×C₇×D₄	1	2	yes	—
407	M₁₆	C₂×C₇×D₄	1	2	yes	—
408	M₁₆	C₂×C₇×D₄	1	4	yes	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	4	4	no	—
418	C₂×C₈	—	4	1	no	—
419	C₂×C₈	—	4	1	no	—
420	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
421	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
422	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
423	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
424	C₂×C₈	C₇×#41	1	1	yes	characteristic
425	C₂×C₈	C₇×#41	1	1	yes	characteristic
426	C₄⋊C₄	—	2	1	no	—
427	C₂²⋊C₄	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₂×C₇	—	2	1	no	—
430	C₂×C₇	—	2	1	no	—
431	C₂×C₇	—	2	1	no	—
432	C₂×C₇	—	4	2	no	—
433	C₂×C₇	—	8	1	no	—
434	C₂×C₇	—	16	2	no	—
435	C₂×C₇	#2416	1	1	yes	characteristic
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	16	2	no	—
442	C₂³	C₇×2₋¹⁺⁴	1	1	yes	characteristic
443	Q₈	—	2	1	no	—
444	Q₈	—	2	1	no	—
445	Q₈	—	2	1	no	—
446	Q₈	C₇×#121	1	2	yes	—
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	—	2	1	no	—
450	D₄	—	2	1	no	—
451	D₄	—	2	1	no	—
452	D₄	—	4	1	no	—
453	D₄	—	4	1	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	C₇×#121	1	2	yes	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
467	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
468	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
469	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
470	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
471	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	2	4	no	—
475	C₈	—	4	1	no	—
476	C₈	—	4	2	no	—
477	C₈	—	4	2	no	—
478	C₈	C₇×#121	1	1	yes	characteristic
479	C₈	C₇×#121	1	1	yes	characteristic
480	C₈	C₇×#121	1	1	yes	characteristic
481	C₈	C₇×#121	1	1	yes	characteristic
482	C₇	#24809	1	1	yes	characteristic, direct product factor, Sylow
483	C₂²	—	2	2	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	16	2	no	—
490	C₂²	—	16	2	no	—
491	C₂²	—	16	2	no	—
492	C₂²	C₇×#537	1	1	yes	characteristic
493	C₂²	C₇×#547	1	1	yes	characteristic
494	C₂²	C₇×#548	1	1	yes	characteristic
495	C₄	—	2	2	no	—
496	C₄	—	4	1	no	—
497	C₄	—	8	1	no	—
498	C₄	C₂×C₇×#121	1	1	yes	characteristic
499	C₄	C₂×C₇×#121	1	1	yes	characteristic
500	C₄	C₇×D₄²	1	1	yes	characteristic
501	C₄	C₇×#553	1	1	yes	characteristic
502	C₂	—	2	1	no	—
503	C₂	—	2	1	no	—
504	C₂	—	2	1	no	—
505	C₂	—	4	2	no	—
506	C₂	—	8	1	no	—
507	C₂	—	16	2	no	—
508	C₂	C₇×#2416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#2416
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3068	C₂	semidirect
2. C₂×C₇×#472	C₂	semidirect
3. C₂×C₇×#472	C₂	semidirect
4. C₇×#2955	C₂	semidirect
5. C₇×#2916	C₂	semidirect
6. C₇×#2069	C₂	semidirect
7. C₇×#2048	C₂	semidirect
8. C₇×#1978	C₂	semidirect
9. C₇×#1953	C₂	semidirect
10. C₇×#1903	C₂	semidirect
11. C₇×#1813	C₂	semidirect
45. #24809	C₇	direct (N × H)
110. #3068	C₂×C₇	semidirect
111. C₂×#472	C₂×C₇	semidirect
112. C₂×#472	C₂×C₇	semidirect
113. #2955	C₂×C₇	semidirect
114. #2916	C₂×C₇	semidirect
115. #2069	C₂×C₇	semidirect
116. #2048	C₂×C₇	semidirect
117. #1978	C₂×C₇	semidirect
118. #1953	C₂×C₇	semidirect
119. #1903	C₂×C₇	semidirect
120. #1813	C₂×C₇	semidirect
482. C₇	#24809	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×#3068	2
C₂×C₇×#472	2
C₇×#2955	2
C₇×#2916	2
C₇×#2069	2
C₇×#2048	2
C₇×#1978	2
C₇×#1953	2
C₇×#1903	2
C₇×#1813	2
#24809	7

Subgroups — C7×#24809 #13202025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24809 #13202025