Subgroups — C₇×#23181 #13200397

Order 1792.

914 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2935	C₂	1	2	yes	maximal, complemented
5	C₇×#1980	C₂	1	2	yes	maximal, complemented
6	C₇×#1975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1974	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₇×#1876	C₂	1	2	yes	maximal, complemented
10	C₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	1	yes	characteristic
13	C₂×C₇×#135	C₂²	1	1	yes	characteristic
14	C₂×C₇×#135	C₂²	1	2	yes	—
15	C₂×C₇×#131	C₂²	1	2	yes	—
16	C₂×C₇×#131	C₂²	1	2	yes	—
17	C₇×#532	C₂²	1	2	yes	—
18	C₇×#497	C₄	1	2	yes	complemented
19	C₇×#497	C₄	1	2	yes	complemented
20	C₇×#496	C₄	1	2	yes	complemented
21	C₇×#496	C₄	1	2	yes	complemented
22	C₇×#444	—	2	4	no	—
23	C₇×#443	—	2	4	no	—
24	C₇×#436	—	2	2	no	—
25	C₇×#435	—	2	2	no	—
26	C₇×#435	—	2	2	no	—
27	C₇×C₈×D₄	—	2	2	no	—
28	C₂×C₇×#108	C₂²	1	1	yes	characteristic
29	C₂×C₇×#108	C₂²	1	1	yes	characteristic
30	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₇×#104	C₂²	1	2	yes	—
32	C₂×C₇×#104	C₂²	1	2	yes	—
33	C₂×C₇×#104	C₂²	1	2	yes	—
34	C₂×C₇×#104	C₂²	1	2	yes	—
35	C₇×#408	C₂²	1	2	yes	—
36	C₇×#408	C₂²	1	2	yes	—
37	C₄×C₇×M₁₆	C₂²	1	2	yes	—
38	C₂×C₇×#97	C₂²	1	1	yes	characteristic
39	C₂×C₇×#97	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
42	C₇×#341	C₂²	1	2	yes	—
43	C₇×#336	C₂²	1	1	yes	characteristic
44	C₇×#336	C₂²	1	1	yes	characteristic
45	#23181	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₇×#41	C₂³	1	2	yes	—
47	C₂×C₇×#41	C₂³	1	2	yes	—
48	C₇×#135	—	2	2	no	—
49	C₇×#135	—	2	2	no	—
50	C₇×#135	—	2	2	no	—
51	C₇×#135	—	2	2	no	—
52	C₇×#135	—	4	2	no	—
53	C₇×#135	—	4	2	no	—
54	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
57	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
58	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
59	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
60	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
61	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
64	C₇×#131	—	2	4	no	—
65	C₇×#131	—	2	4	no	—
66	C₇×#131	—	4	2	no	—
67	C₇×#131	—	4	2	no	—
68	C₂×C₇×M₁₆	—	2	2	no	—
69	C₂×C₇×M₁₆	—	2	2	no	—
70	C₂×C₇×M₁₆	C₂³	1	2	yes	—
71	C₂×C₇×M₁₆	C₂³	1	2	yes	—
72	C₂²×C₇×C₈	—	2	2	no	—
73	C₂²×C₇×C₈	—	2	2	no	—
74	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
79	C₇×#127	C₂×C₄	1	2	yes	—
80	C₇×#124	C₂×C₄	1	2	yes	—
81	C₇×#124	C₂×C₄	1	2	yes	—
82	C₇×#121	—	2	2	no	—
83	C₇×#121	—	2	2	no	—
84	C₄×C₇×D₄	—	2	2	no	—
85	C₄×C₇×D₄	—	2	2	no	—
86	C₂×C₇×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
90	C₇×#108	—	2	2	no	—
91	C₇×#108	D₄	1	4	yes	complemented
92	C₇×C₄⋊C₈	D₄	1	2	yes	—
93	C₇×C₄⋊C₈	D₄	1	2	yes	—
94	C₇×#104	—	2	4	no	—
95	C₇×#104	—	2	4	no	—
96	C₇×#104	—	4	2	no	—
97	C₇×#104	—	4	2	no	—
98	C₇×C₂²⋊C₈	—	2	2	no	—
99	C₇×C₂²⋊C₈	—	2	2	no	—
100	C₇×C₂²⋊C₈	—	2	2	no	—
101	C₇×C₂²⋊C₈	—	2	2	no	—
102	C₇×#97	—	2	2	no	—
103	C₇×#97	—	2	2	no	—
104	C₇×#97	C₂×C₄	1	2	yes	—
105	C₇×#97	C₂×C₄	1	2	yes	—
106	C₄×C₇×C₈	—	2	2	no	—
107	C₄×C₇×C₈	—	2	2	no	—
108	C₇×#95	—	2	1	no	—
109	#3150	C₂×C₇	1	1	yes	characteristic
110	C₂×#444	C₂×C₇	1	1	yes	characteristic, complemented
111	C₂×#443	C₂×C₇	1	1	yes	characteristic, complemented
112	#2935	C₂×C₇	1	2	yes	complemented
113	#1980	C₂×C₇	1	2	yes	complemented
114	#1975	C₂×C₇	1	1	yes	characteristic, complemented
115	#1974	C₂×C₇	1	1	yes	characteristic, complemented
116	#1876	C₂×C₇	1	2	yes	complemented
117	#1876	C₂×C₇	1	2	yes	complemented
118	#1778	C₂×C₇	1	1	yes	characteristic, complemented
119	#1771	C₂×C₇	1	1	yes	characteristic, complemented
120	C₇×#41	—	2	2	no	—
121	C₇×#41	—	2	2	no	—
122	C₇×#41	—	2	2	no	—
123	C₇×#41	—	2	2	no	—
124	C₇×#41	—	2	2	no	—
125	C₇×#41	—	2	2	no	—
126	C₇×#41	—	2	2	no	—
127	C₇×#41	—	2	2	no	—
128	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
129	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
130	C₂×C₇×D₄	—	2	2	no	—
131	C₂×C₇×D₄	—	2	2	no	—
132	C₂×C₇×D₄	—	2	2	no	—
133	C₂×C₇×D₄	—	2	2	no	—
134	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
135	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₇	—	2	1	no	—
137	C₂²×C₄×C₇	—	2	2	no	—
138	C₂²×C₄×C₇	—	2	2	no	—
139	C₂²×C₄×C₇	—	2	1	no	—
140	C₂²×C₄×C₇	#41	1	1	yes	characteristic
141	C₂²×C₄×C₇	#41	1	1	yes	characteristic
142	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
143	C₇×Q₁₆	—	2	2	no	—
144	C₇×Q₁₆	—	2	2	no	—
145	C₇×Q₁₆	—	4	2	no	—
146	C₇×SD₁₆	—	2	4	no	—
147	C₇×SD₁₆	—	2	4	no	—
148	C₇×SD₁₆	—	4	2	no	—
149	C₇×SD₁₆	—	4	2	no	—
150	C₇×D₈	—	2	2	no	—
151	C₇×D₈	—	2	2	no	—
152	C₇×D₈	—	4	2	no	—
153	C₇×M₁₆	—	2	2	no	—
154	C₇×M₁₆	—	2	2	no	—
155	C₇×M₁₆	—	2	2	no	—
156	C₇×M₁₆	—	4	4	no	—
157	C₇×M₁₆	—	4	4	no	—
158	C₇×M₁₆	C₂×D₄	1	4	yes	—
159	C₂×C₇×C₈	—	2	1	no	—
160	C₂×C₇×C₈	—	2	1	no	—
161	C₂×C₇×C₈	—	2	2	no	—
162	C₂×C₇×C₈	—	2	1	no	—
163	C₂×C₇×C₈	—	2	1	no	—
164	C₂×C₇×C₈	—	2	1	no	—
165	C₂×C₇×C₈	—	2	1	no	—
166	C₂×C₇×C₈	—	2	2	no	—
167	C₂×C₇×C₈	—	4	4	no	—
168	C₂×C₇×C₈	—	4	4	no	—
169	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
170	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
174	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
176	C₂×C₇×C₈	#41	1	2	yes	—
177	C₇×C₄⋊C₄	—	2	2	no	—
178	C₇×C₄⋊C₄	—	2	2	no	—
179	C₇×C₄⋊C₄	—	4	2	no	—
180	C₇×C₂²⋊C₄	—	2	2	no	—
181	C₇×C₂²⋊C₄	—	2	2	no	—
182	C₇×C₂²⋊C₄	—	2	2	no	—
183	C₇×C₂²⋊C₄	—	2	2	no	—
184	C₄²×C₇	—	2	1	no	—
185	C₄²×C₇	—	2	1	no	—
186	C₄²×C₇	—	2	1	no	—
187	C₄²×C₇	—	2	1	no	—
188	C₄²×C₇	C₂²×C₄	1	2	yes	—
189	C₄²×C₇	C₂×D₄	1	2	yes	—
190	C₂×#135	C₂²×C₇	1	1	yes	characteristic
191	C₂×#135	C₂²×C₇	1	1	yes	characteristic
192	C₂×#135	C₂²×C₇	1	2	yes	—
193	C₂×#131	C₂²×C₇	1	2	yes	—
194	C₂×#131	C₂²×C₇	1	2	yes	—
195	#532	C₂²×C₇	1	2	yes	—
196	#497	C₄×C₇	1	2	yes	complemented
197	#497	C₄×C₇	1	2	yes	complemented
198	#496	C₄×C₇	1	2	yes	complemented
199	#496	C₄×C₇	1	2	yes	complemented
200	#444	—	2	4	no	—
201	#443	—	2	4	no	—
202	#436	—	2	2	no	—
203	#435	—	2	2	no	—
204	#435	—	2	2	no	—
205	C₈×D₄	—	2	2	no	—
206	C₂×#108	C₂²×C₇	1	1	yes	characteristic
207	C₂×#108	C₂²×C₇	1	1	yes	characteristic
208	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
209	C₂×#104	C₂²×C₇	1	2	yes	—
210	C₂×#104	C₂²×C₇	1	2	yes	—
211	C₂×#104	C₂²×C₇	1	2	yes	—
212	C₂×#104	C₂²×C₇	1	2	yes	—
213	#408	C₂²×C₇	1	2	yes	—
214	#408	C₂²×C₇	1	2	yes	—
215	C₄×M₁₆	C₂²×C₇	1	2	yes	—
216	C₂×#97	C₂²×C₇	1	1	yes	characteristic
217	C₂×#97	C₂²×C₇	1	1	yes	characteristic
218	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
219	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
220	#341	C₂²×C₇	1	2	yes	—
221	#336	C₂²×C₇	1	1	yes	characteristic
222	#336	C₂²×C₇	1	1	yes	characteristic
223	C₂³×C₇	—	2	2	no	—
224	C₂³×C₇	—	2	2	no	—
225	C₂³×C₇	C₂×#41	1	1	yes	characteristic
226	C₇×Q₈	—	2	2	no	—
227	C₇×Q₈	—	2	2	no	—
228	C₇×Q₈	—	2	2	no	—
229	C₇×Q₈	—	2	2	no	—
230	C₇×D₄	—	2	2	no	—
231	C₇×D₄	—	2	2	no	—
232	C₇×D₄	—	2	2	no	—
233	C₇×D₄	—	2	2	no	—
234	C₇×D₄	—	4	4	no	—
235	C₇×D₄	—	4	2	no	—
236	C₇×D₄	—	4	2	no	—
237	C₇×D₄	—	4	2	no	—
238	C₇×D₄	—	4	2	no	—
239	C₇×D₄	—	8	2	no	—
240	C₂×C₄×C₇	—	2	2	no	—
241	C₂×C₄×C₇	—	2	2	no	—
242	C₂×C₄×C₇	—	2	2	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	—	4	1	no	—
245	C₂×C₄×C₇	—	4	1	no	—
246	C₂×C₄×C₇	—	4	1	no	—
247	C₂×C₄×C₇	—	4	2	no	—
248	C₂×C₄×C₇	—	4	2	no	—
249	C₂×C₄×C₇	—	4	2	no	—
250	C₂×C₄×C₇	—	4	2	no	—
251	C₂×C₄×C₇	—	4	1	no	—
252	C₂×C₄×C₇	—	4	1	no	—
253	C₂×C₄×C₇	—	4	1	no	—
254	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
255	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
256	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
262	C₇×C₈	—	2	2	no	—
263	C₇×C₈	—	2	1	no	—
264	C₇×C₈	—	2	2	no	—
265	C₇×C₈	—	2	1	no	—
266	C₇×C₈	—	2	4	no	—
267	C₇×C₈	—	4	2	no	—
268	C₇×C₈	C₄×D₄	1	2	yes	—
269	C₇×C₈	C₄×D₄	1	2	yes	—
270	C₂×#41	C₂³×C₇	1	2	yes	—
271	C₂×#41	C₂³×C₇	1	2	yes	—
272	#135	—	2	2	no	—
273	#135	—	2	2	no	—
274	#135	—	2	2	no	—
275	#135	—	2	2	no	—
276	#135	—	4	2	no	—
277	#135	—	4	2	no	—
278	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
279	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
280	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
281	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
282	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
283	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
284	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
285	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
286	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
287	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
288	#131	—	2	4	no	—
289	#131	—	2	4	no	—
290	#131	—	4	2	no	—
291	#131	—	4	2	no	—
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	C₂³×C₇	1	2	yes	—
295	C₂×M₁₆	C₂³×C₇	1	2	yes	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	—	2	2	no	—
298	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
302	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
303	#127	C₂×C₄×C₇	1	2	yes	—
304	#124	C₂×C₄×C₇	1	2	yes	—
305	#124	C₂×C₄×C₇	1	2	yes	—
306	#121	—	2	2	no	—
307	#121	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
314	#108	—	2	2	no	—
315	#108	C₇×D₄	1	4	yes	complemented
316	C₄⋊C₈	C₇×D₄	1	2	yes	—
317	C₄⋊C₈	C₇×D₄	1	2	yes	—
318	#104	—	2	4	no	—
319	#104	—	2	4	no	—
320	#104	—	4	2	no	—
321	#104	—	4	2	no	—
322	C₂²⋊C₈	—	2	2	no	—
323	C₂²⋊C₈	—	2	2	no	—
324	C₂²⋊C₈	—	2	2	no	—
325	C₂²⋊C₈	—	2	2	no	—
326	#97	—	2	2	no	—
327	#97	—	2	2	no	—
328	#97	C₂×C₄×C₇	1	2	yes	—
329	#97	C₂×C₄×C₇	1	2	yes	—
330	C₄×C₈	—	2	2	no	—
331	C₄×C₈	—	2	2	no	—
332	#95	—	2	1	no	—
333	C₂²×C₇	—	2	1	no	—
334	C₂²×C₇	—	2	1	no	—
335	C₂²×C₇	—	4	2	no	—
336	C₂²×C₇	—	4	2	no	—
337	C₂²×C₇	—	4	2	no	—
338	C₂²×C₇	—	4	2	no	—
339	C₂²×C₇	—	4	2	no	—
340	C₂²×C₇	—	4	2	no	—
341	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
342	C₂²×C₇	#550	1	1	yes	characteristic
343	C₂²×C₇	#559	1	1	yes	characteristic
344	C₄×C₇	—	2	2	no	—
345	C₄×C₇	—	2	2	no	—
346	C₄×C₇	—	4	2	no	—
347	C₄×C₇	—	4	2	no	—
348	C₄×C₇	—	4	2	no	—
349	C₄×C₇	—	8	1	no	—
350	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
351	C₄×C₇	#518	1	1	yes	characteristic
352	C₄×C₇	D₄²	1	1	yes	characteristic
353	C₄×C₇	#550	1	1	yes	characteristic
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	#41	—	2	2	no	—
361	#41	—	2	2	no	—
362	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
363	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
369	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	C₇×#41	1	1	yes	characteristic
375	C₂²×C₄	C₇×#41	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
377	Q₁₆	—	2	2	no	—
378	Q₁₆	—	2	2	no	—
379	Q₁₆	—	4	2	no	—
380	SD₁₆	—	2	4	no	—
381	SD₁₆	—	2	4	no	—
382	SD₁₆	—	4	2	no	—
383	SD₁₆	—	4	2	no	—
384	D₈	—	2	2	no	—
385	D₈	—	2	2	no	—
386	D₈	—	4	2	no	—
387	M₁₆	—	2	2	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	—	4	4	no	—
391	M₁₆	—	4	4	no	—
392	M₁₆	C₂×C₇×D₄	1	4	yes	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	4	no	—
402	C₂×C₈	—	4	4	no	—
403	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
404	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
405	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
406	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
407	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
408	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
409	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
410	C₂×C₈	C₇×#41	1	2	yes	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₄²	C₂²×C₄×C₇	1	2	yes	—
423	C₄²	C₂×C₇×D₄	1	2	yes	—
424	C₂×C₇	—	2	1	no	—
425	C₂×C₇	—	2	1	no	—
426	C₂×C₇	—	8	2	no	—
427	C₂×C₇	—	8	2	no	—
428	C₂×C₇	#2366	1	1	yes	characteristic
429	C₂×C₇	#2955	1	1	yes	characteristic
430	C₂×C₇	#2956	1	1	yes	characteristic
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	C₂×C₇×#41	1	1	yes	characteristic
434	Q₈	—	2	2	no	—
435	Q₈	—	2	2	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	2	2	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	2	no	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	4	4	no	—
443	D₄	—	4	2	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	8	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
463	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
464	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
465	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
466	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
467	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
468	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
469	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
470	C₈	—	2	2	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	1	no	—
474	C₈	—	2	4	no	—
475	C₈	—	4	2	no	—
476	C₈	C₄×C₇×D₄	1	2	yes	—
477	C₈	C₄×C₇×D₄	1	2	yes	—
478	C₇	#23181	1	1	yes	characteristic, direct product factor, Sylow
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
488	C₂²	C₇×#550	1	1	yes	characteristic
489	C₂²	C₇×#559	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	4	2	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	8	1	no	—
496	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
497	C₄	C₇×#518	1	1	yes	characteristic
498	C₄	C₇×D₄²	1	1	yes	characteristic
499	C₄	C₇×#550	1	1	yes	characteristic
500	C₂	—	2	1	no	—
501	C₂	—	2	1	no	—
502	C₂	—	8	2	no	—
503	C₂	—	8	2	no	—
504	C₂	C₇×#2366	1	1	yes	characteristic
505	C₂	C₇×#2955	1	1	yes	characteristic
506	C₂	C₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#444	C₂	semidirect
3. C₂×C₇×#443	C₂	semidirect
4. C₇×#2935	C₂	semidirect
5. C₇×#1980	C₂	semidirect
6. C₇×#1975	C₂	semidirect
7. C₇×#1974	C₂	semidirect
8. C₇×#1876	C₂	semidirect
9. C₇×#1876	C₂	semidirect
10. C₇×#1778	C₂	semidirect
11. C₇×#1771	C₂	semidirect
18. C₇×#497	C₄	semidirect
19. C₇×#497	C₄	semidirect
20. C₇×#496	C₄	semidirect
21. C₇×#496	C₄	semidirect
45. #23181	C₇	direct (N × H)
91. C₇×#108	D₄	semidirect
110. C₂×#444	C₂×C₇	semidirect
111. C₂×#443	C₂×C₇	semidirect
112. #2935	C₂×C₇	semidirect
113. #1980	C₂×C₇	semidirect
114. #1975	C₂×C₇	semidirect
115. #1974	C₂×C₇	semidirect
116. #1876	C₂×C₇	semidirect
117. #1876	C₂×C₇	semidirect
118. #1778	C₂×C₇	semidirect
119. #1771	C₂×C₇	semidirect
196. #497	C₄×C₇	semidirect
197. #497	C₄×C₇	semidirect
198. #496	C₄×C₇	semidirect
199. #496	C₄×C₇	semidirect
315. #108	C₇×D₄	semidirect
478. C₇	#23181	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×#3150	2
C₂×C₇×#444	2
C₂×C₇×#443	2
C₇×#2935	2
C₇×#1980	2
C₇×#1975	2
C₇×#1974	2
C₇×#1876	2
C₇×#1778	2
C₇×#1771	2
#23181	7

Subgroups — C7×#23181 #13200397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23181 #13200397