Subgroups — C₇×#21538 #13198754

Order 1792.

1170 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇×#121	C₂	1	2	yes	maximal, complemented
5	C₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1941	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1934	C₂	1	2	yes	maximal, complemented
8	C₇×#1895	C₂	1	2	yes	maximal, complemented
9	C₇×#1809	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1776	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₇×#121	C₂²	1	2	yes	—
16	C₂×C₄×C₇×D₄	—	2	2	no	—
17	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
19	C₇×#469	C₄	1	8	yes	complemented
20	C₇×#442	—	2	4	no	—
21	C₂×C₇×#107	C₂²	1	1	yes	characteristic
22	C₂×C₇×#106	C₂²	1	1	yes	characteristic
23	C₂×C₇×#106	C₂²	1	1	yes	characteristic
24	C₂×C₇×#102	C₂²	1	2	yes	—
25	C₂×C₇×#102	C₂²	1	2	yes	—
26	C₂×C₇×#102	C₂²	1	2	yes	—
27	C₄×C₇×M₁₆	—	2	4	no	—
28	C₂×C₇×#97	C₂²	1	1	yes	characteristic
29	C₂×C₇×#97	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#390	C₂²	1	2	yes	—
33	C₇×#389	C₂²	1	2	yes	—
34	C₇×#388	—	2	2	no	—
35	C₇×#386	—	2	2	no	—
36	C₇×#380	C₂²	1	2	yes	—
37	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
38	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	#21538	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×D₄	—	2	2	no	—
44	C₂²×C₇×D₄	C₂³	1	2	yes	—
45	C₂³×C₄×C₇	—	2	2	no	—
46	C₂³×C₄×C₇	D₄	1	2	yes	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₇×D₈	—	2	2	no	—
49	C₂×C₇×D₈	—	2	2	no	—
50	C₂×C₇×D₈	C₂×C₄	1	4	yes	complemented
51	C₂×C₇×M₁₆	—	2	2	no	—
52	C₂×C₇×M₁₆	—	2	2	no	—
53	C₂×C₇×M₁₆	—	2	2	no	—
54	C₂×C₇×M₁₆	—	2	2	no	—
55	C₂×C₇×M₁₆	—	2	2	no	—
56	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	complemented
57	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₇×#121	—	2	4	no	—
62	C₇×#121	C₂×C₄	1	8	yes	complemented
63	C₄×C₇×D₄	—	2	4	no	—
64	C₄×C₇×D₄	—	2	4	no	—
65	C₄×C₇×D₄	—	4	8	no	—
66	C₂×C₇×C₄⋊C₄	—	2	2	no	—
67	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
68	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
73	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
74	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
75	C₂×C₄²×C₇	—	2	2	no	—
76	C₂×C₄²×C₇	—	2	2	no	—
77	C₂×C₄²×C₇	—	2	2	no	—
78	C₂×C₄²×C₇	D₄	1	2	yes	—
79	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
81	C₇×#107	—	2	2	no	—
82	C₇×#106	—	2	2	no	—
83	C₇×#106	C₂×C₄	1	4	yes	complemented
84	C₇×#102	—	2	4	no	—
85	C₇×#102	—	2	4	no	—
86	C₇×#102	C₂×C₄	1	8	yes	complemented
87	C₇×#97	—	2	2	no	—
88	C₇×#97	D₄	1	4	yes	complemented
89	C₄×C₇×C₈	—	2	2	no	—
90	C₄×C₇×C₈	—	2	2	no	—
91	C₇×#95	—	2	2	no	—
92	C₇×#95	—	2	2	no	—
93	C₇×#95	—	2	2	no	—
94	C₂×#469	C₂×C₇	1	1	yes	characteristic
95	C₂×#442	C₂×C₇	1	1	yes	characteristic, complemented
96	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
97	C₄×#121	C₂×C₇	1	2	yes	complemented
98	#1970	C₂×C₇	1	1	yes	characteristic, complemented
99	#1941	C₂×C₇	1	1	yes	characteristic, complemented
100	#1934	C₂×C₇	1	2	yes	complemented
101	#1895	C₂×C₇	1	2	yes	complemented
102	#1809	C₂×C₇	1	1	yes	characteristic, complemented
103	#1776	C₂×C₇	1	1	yes	characteristic, complemented
104	#1762	C₂×C₇	1	2	yes	complemented
105	C₂⁴×C₇	—	2	2	no	—
106	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇×D₄	—	2	8	no	—
108	C₂×C₇×D₄	—	2	2	no	—
109	C₂×C₇×D₄	—	2	2	no	—
110	C₂×C₇×D₄	—	2	2	no	—
111	C₂×C₇×D₄	—	2	2	no	—
112	C₂×C₇×D₄	—	2	2	no	—
113	C₂×C₇×D₄	—	4	4	no	—
114	C₂×C₇×D₄	—	4	4	no	—
115	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	1	no	—
118	C₂²×C₄×C₇	—	2	1	no	—
119	C₂²×C₄×C₇	—	2	1	no	—
120	C₂²×C₄×C₇	—	2	2	no	—
121	C₂²×C₄×C₇	—	2	4	no	—
122	C₂²×C₄×C₇	—	2	4	no	—
123	C₂²×C₄×C₇	—	2	2	no	—
124	C₂²×C₄×C₇	—	2	2	no	—
125	C₂²×C₄×C₇	—	4	2	no	—
126	C₂²×C₄×C₇	—	4	4	no	—
127	C₂²×C₄×C₇	—	4	4	no	—
128	C₂²×C₄×C₇	—	4	4	no	—
129	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₇	#41	1	2	yes	—
134	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
135	C₇×D₈	—	2	8	no	—
136	C₇×M₁₆	—	2	8	no	—
137	C₇×M₁₆	—	2	8	no	—
138	C₂×C₇×C₈	—	2	1	no	—
139	C₂×C₇×C₈	—	2	1	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	1	no	—
142	C₂×C₇×C₈	—	2	1	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
145	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
146	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
147	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
148	C₂×C₇×C₈	#41	1	2	yes	—
149	C₂×C₇×C₈	#41	1	2	yes	—
150	C₇×C₄⋊C₄	—	2	2	no	—
151	C₇×C₄⋊C₄	—	2	2	no	—
152	C₇×C₄⋊C₄	—	2	2	no	—
153	C₇×C₄⋊C₄	—	2	2	no	—
154	C₇×C₄⋊C₄	—	2	2	no	—
155	C₇×C₄⋊C₄	—	4	4	no	—
156	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
157	C₇×C₂²⋊C₄	—	2	8	no	—
158	C₇×C₂²⋊C₄	—	4	4	no	—
159	C₇×C₂²⋊C₄	—	4	4	no	—
160	C₇×C₂²⋊C₄	—	4	4	no	—
161	C₇×C₂²⋊C₄	—	4	4	no	—
162	C₄²×C₇	—	2	4	no	—
163	C₄²×C₇	—	2	1	no	—
164	C₄²×C₇	—	2	1	no	—
165	C₄²×C₇	—	4	4	no	—
166	C₄²×C₇	C₂×D₄	1	2	yes	—
167	C₄²×C₇	#41	1	2	yes	—
168	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
169	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
170	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
171	C₂×#121	C₂²×C₇	1	2	yes	—
172	C₂×C₄×D₄	—	2	2	no	—
173	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
174	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
175	#469	C₄×C₇	1	8	yes	complemented
176	#442	—	2	4	no	—
177	C₂×#107	C₂²×C₇	1	1	yes	characteristic
178	C₂×#106	C₂²×C₇	1	1	yes	characteristic
179	C₂×#106	C₂²×C₇	1	1	yes	characteristic
180	C₂×#102	C₂²×C₇	1	2	yes	—
181	C₂×#102	C₂²×C₇	1	2	yes	—
182	C₂×#102	C₂²×C₇	1	2	yes	—
183	C₄×M₁₆	—	2	4	no	—
184	C₂×#97	C₂²×C₇	1	1	yes	characteristic
185	C₂×#97	C₂²×C₇	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
188	#390	C₂²×C₇	1	2	yes	—
189	#389	C₂²×C₇	1	2	yes	—
190	#388	—	2	2	no	—
191	#386	—	2	2	no	—
192	#380	C₂²×C₇	1	2	yes	—
193	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
194	C₄×C₂²⋊C₄	—	2	2	no	—
195	#340	C₂²×C₇	1	2	yes	—
196	#340	C₂²×C₇	1	2	yes	—
197	#340	C₂²×C₇	1	2	yes	—
198	C₂³×C₇	—	2	1	no	—
199	C₂³×C₇	—	2	1	no	—
200	C₂³×C₇	—	2	2	no	—
201	C₂³×C₇	—	2	2	no	—
202	C₂³×C₇	—	2	4	no	—
203	C₂³×C₇	—	4	2	no	—
204	C₂³×C₇	—	4	2	no	—
205	C₂³×C₇	—	4	2	no	—
206	C₂³×C₇	—	4	2	no	—
207	C₂³×C₇	—	4	2	no	—
208	C₂³×C₇	—	4	2	no	—
209	C₂³×C₇	C₄×D₄	1	2	yes	—
210	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
211	C₇×D₄	—	2	4	no	—
212	C₇×D₄	—	2	4	no	—
213	C₇×D₄	—	4	16	no	—
214	C₇×D₄	—	4	2	no	—
215	C₇×D₄	—	4	2	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	2	1	no	—
218	C₂×C₄×C₇	—	2	1	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	2	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	4	no	—
227	C₂×C₄×C₇	—	4	8	no	—
228	C₂×C₄×C₇	—	4	2	no	—
229	C₂×C₄×C₇	—	4	2	no	—
230	C₂×C₄×C₇	—	4	2	no	—
231	C₂×C₄×C₇	—	4	2	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	2	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	—	4	2	no	—
236	C₂×C₄×C₇	—	8	8	no	—
237	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
238	C₂×C₄×C₇	#121	1	4	yes	—
239	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
240	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
241	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
246	C₇×C₈	—	2	2	no	—
247	C₇×C₈	—	2	2	no	—
248	C₇×C₈	—	2	2	no	—
249	C₇×C₈	C₄×D₄	1	4	yes	complemented
250	C₂²×D₄	—	2	2	no	—
251	C₂²×D₄	C₂³×C₇	1	2	yes	—
252	C₂³×C₄	—	2	2	no	—
253	C₂³×C₄	C₇×D₄	1	2	yes	—
254	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
255	C₂×D₈	—	2	2	no	—
256	C₂×D₈	—	2	2	no	—
257	C₂×D₈	C₂×C₄×C₇	1	4	yes	complemented
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	complemented
264	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
268	#121	—	2	4	no	—
269	#121	C₂×C₄×C₇	1	8	yes	complemented
270	C₄×D₄	—	2	4	no	—
271	C₄×D₄	—	2	4	no	—
272	C₄×D₄	—	4	8	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	C₇×D₄	1	2	yes	—
286	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
288	#107	—	2	2	no	—
289	#106	—	2	2	no	—
290	#106	C₂×C₄×C₇	1	4	yes	complemented
291	#102	—	2	4	no	—
292	#102	—	2	4	no	—
293	#102	C₂×C₄×C₇	1	8	yes	complemented
294	#97	—	2	2	no	—
295	#97	C₇×D₄	1	4	yes	complemented
296	C₄×C₈	—	2	2	no	—
297	C₄×C₈	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	C₂²×C₇	—	2	4	no	—
302	C₂²×C₇	—	2	2	no	—
303	C₂²×C₇	—	4	1	no	—
304	C₂²×C₇	—	4	1	no	—
305	C₂²×C₇	—	4	2	no	—
306	C₂²×C₇	—	4	4	no	—
307	C₂²×C₇	—	4	4	no	—
308	C₂²×C₇	—	4	4	no	—
309	C₂²×C₇	—	8	2	no	—
310	C₂²×C₇	—	8	2	no	—
311	C₂²×C₇	—	8	2	no	—
312	C₂²×C₇	—	8	2	no	—
313	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
314	C₂²×C₇	C₂×#121	1	1	yes	characteristic
315	C₂²×C₇	#532	1	1	yes	characteristic
316	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
317	C₂²×C₇	#575	1	2	yes	—
318	C₄×C₇	—	2	4	no	—
319	C₄×C₇	—	2	4	no	—
320	C₄×C₇	—	2	2	no	—
321	C₄×C₇	—	4	4	no	—
322	C₄×C₇	—	8	2	no	—
323	C₄×C₇	—	8	2	no	—
324	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
325	C₄×C₇	C₄×#41	1	1	yes	characteristic
326	C₄×C₇	#525	1	1	yes	characteristic
327	C₄×C₇	#529	1	1	yes	characteristic
328	C₂⁴	—	2	2	no	—
329	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	4	4	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
353	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
354	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
355	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
356	C₂²×C₄	C₇×#41	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
358	D₈	—	2	8	no	—
359	M₁₆	—	2	8	no	—
360	M₁₆	—	2	8	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
368	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
369	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
370	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
371	C₂×C₈	C₇×#41	1	2	yes	—
372	C₂×C₈	C₇×#41	1	2	yes	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	4	4	no	—
379	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
380	C₂²⋊C₄	—	2	8	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	4	4	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₄²	—	2	4	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	4	4	no	—
389	C₄²	C₂×C₇×D₄	1	2	yes	—
390	C₄²	C₇×#41	1	2	yes	—
391	C₂×C₇	—	4	2	no	—
392	C₂×C₇	—	8	4	no	—
393	C₂×C₇	C₄×#121	1	1	yes	characteristic
394	C₂×C₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
395	C₂×C₇	#3055	1	2	yes	—
396	C₂×C₇	#3183	1	1	yes	characteristic
397	C₂×C₇	#3184	1	1	yes	characteristic
398	C₂³	—	2	1	no	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	4	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	C₄×C₇×D₄	1	2	yes	—
410	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
411	D₄	—	2	4	no	—
412	D₄	—	2	4	no	—
413	D₄	—	4	16	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	4	8	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	8	8	no	—
437	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
438	C₂×C₄	C₇×#121	1	4	yes	—
439	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
440	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
441	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
442	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
443	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
444	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
445	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
446	C₈	—	2	2	no	—
447	C₈	—	2	2	no	—
448	C₈	—	2	2	no	—
449	C₈	C₄×C₇×D₄	1	4	yes	complemented
450	C₇	#21538	1	1	yes	characteristic, direct product factor, Sylow
451	C₂²	—	2	4	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
464	C₂²	C₂×C₇×#121	1	1	yes	characteristic
465	C₂²	C₇×#532	1	1	yes	characteristic
466	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
467	C₂²	C₇×#575	1	2	yes	—
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	4	no	—
472	C₄	—	8	2	no	—
473	C₄	—	8	2	no	—
474	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
475	C₄	C₄×C₇×#41	1	1	yes	characteristic
476	C₄	C₇×#525	1	1	yes	characteristic
477	C₄	C₇×#529	1	1	yes	characteristic
478	C₂	—	4	2	no	—
479	C₂	—	8	4	no	—
480	C₂	C₄×C₇×#121	1	1	yes	characteristic
481	C₂	C₄×C₇×AGL(1,ℤ/8)	1	2	yes	—
482	C₂	C₇×#3055	1	2	yes	—
483	C₂	C₇×#3183	1	1	yes	characteristic
484	C₂	C₇×#3184	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#442	C₂	semidirect
3. C₂×C₄×C₇×M₁₆	C₂	semidirect
4. C₄×C₇×#121	C₂	semidirect
5. C₇×#1970	C₂	semidirect
6. C₇×#1941	C₂	semidirect
7. C₇×#1934	C₂	semidirect
8. C₇×#1895	C₂	semidirect
9. C₇×#1809	C₂	semidirect
10. C₇×#1776	C₂	semidirect
11. C₇×#1762	C₂	semidirect
19. C₇×#469	C₄	semidirect
42. #21538	C₇	direct (N × H)
50. C₂×C₇×D₈	C₂×C₄	semidirect
56. C₂×C₇×M₁₆	C₂×C₄	semidirect
62. C₇×#121	C₂×C₄	semidirect
83. C₇×#106	C₂×C₄	semidirect
86. C₇×#102	C₂×C₄	semidirect
88. C₇×#97	D₄	semidirect
95. C₂×#442	C₂×C₇	semidirect
96. C₂×C₄×M₁₆	C₂×C₇	semidirect
97. C₄×#121	C₂×C₇	semidirect
98. #1970	C₂×C₇	semidirect
99. #1941	C₂×C₇	semidirect
100. #1934	C₂×C₇	semidirect
101. #1895	C₂×C₇	semidirect
102. #1809	C₂×C₇	semidirect
103. #1776	C₂×C₇	semidirect
104. #1762	C₂×C₇	semidirect
175. #469	C₄×C₇	semidirect
249. C₇×C₈	C₄×D₄	semidirect
257. C₂×D₈	C₂×C₄×C₇	semidirect
263. C₂×M₁₆	C₂×C₄×C₇	semidirect
269. #121	C₂×C₄×C₇	semidirect
290. #106	C₂×C₄×C₇	semidirect
293. #102	C₂×C₄×C₇	semidirect
295. #97	C₇×D₄	semidirect
449. C₈	C₄×C₇×D₄	semidirect
450. C₇	#21538	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×#469	2
C₂×C₇×#442	2
C₂×C₄×C₇×M₁₆	2
C₄×C₇×#121	2
C₇×#1970	2
C₇×#1941	2
C₇×#1934	2
C₇×#1895	2
C₇×#1809	2
C₇×#1776	2
C₇×#1762	2
#21538	7

Subgroups — C7×#21538 #13198754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21538 #13198754