Subgroups — C₇×#21070 #13198286

Order 1792.

1026 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#430	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#421	C₂	1	1	yes	maximal, characteristic
3	C₇×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1929	C₂	1	8	yes	maximal, complemented
7	C₇×#1733	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
10	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×#117	C₂²	1	1	yes	characteristic
15	C₂×C₇×#117	C₂²	1	1	yes	characteristic
16	C₇×#430	—	2	4	no	—
17	C₂×C₇×#108	C₂²	1	4	yes	complemented
18	C₇×#421	—	2	4	no	—
19	C₂×C₇×#104	C₂²	1	4	yes	—
20	C₇×#419	C₄	1	2	yes	complemented
21	C₇×#418	C₄	1	2	yes	complemented
22	C₇×#417	C₄	1	4	yes	complemented
23	C₇×#416	C₂²	1	4	yes	—
24	C₇×#408	—	2	1	no	—
25	C₇×#408	—	2	1	no	—
26	C₇×#408	C₂²	1	4	yes	—
27	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
28	C₇×#405	—	2	2	no	—
29	C₇×#405	C₂²	1	4	yes	complemented
30	C₂×C₇×#97	—	2	1	no	—
31	C₂×C₄×C₇×C₈	—	2	1	no	—
32	C₇×#339	C₂²	1	4	yes	complemented
33	C₇×#336	C₂²	1	4	yes	complemented
34	#21070	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₇×#41	—	2	1	no	—
36	C₂×C₇×#41	—	2	1	no	—
37	C₂×C₇×#41	—	2	2	no	—
38	C₂×C₇×#41	—	2	2	no	—
39	C₂×C₇×#41	—	2	2	no	—
40	C₂×C₇×#41	—	2	2	no	—
41	C₂×C₇×#41	C₂×C₄	1	2	yes	—
42	C₂×C₇×#41	C₂³	1	2	yes	—
43	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₇×D₄	—	2	1	no	—
45	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₇	—	2	1	no	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₇×M₁₆	—	2	1	no	—
49	C₂×C₇×M₁₆	—	2	2	no	—
50	C₂×C₇×M₁₆	—	2	2	no	—
51	C₂×C₇×M₁₆	—	2	2	no	—
52	C₂×C₇×M₁₆	—	2	1	no	—
53	C₂×C₇×M₁₆	—	2	2	no	—
54	C₂×C₇×M₁₆	—	2	2	no	—
55	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
56	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
57	C₂×C₇×M₁₆	D₄	1	4	yes	—
58	C₂×C₇×M₁₆	C₂³	1	2	yes	—
59	C₂×C₇×M₁₆	C₂³	1	2	yes	—
60	C₂²×C₇×C₈	—	2	1	no	—
61	C₂²×C₇×C₈	—	2	2	no	—
62	C₂²×C₇×C₈	—	2	2	no	—
63	C₂²×C₇×C₈	—	2	1	no	—
64	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
65	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
66	C₂²×C₇×C₈	D₄	1	2	yes	—
67	C₂²×C₇×C₈	C₂³	1	2	yes	—
68	C₂²×C₇×C₈	C₂³	1	2	yes	—
69	C₇×#117	—	2	1	no	—
70	C₇×#117	—	2	1	no	—
71	C₇×#117	—	2	2	no	—
72	C₇×#117	—	2	2	no	—
73	C₇×#117	C₂×C₄	1	2	yes	—
74	C₇×#117	C₂³	1	2	yes	—
75	C₂×C₇×C₄⋊C₄	—	2	1	no	—
76	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
77	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
78	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
79	C₂×C₄²×C₇	—	2	1	no	—
80	C₂×C₄²×C₇	C₂³	1	2	yes	—
81	C₇×#108	—	2	8	no	—
82	C₇×#104	—	2	8	no	—
83	C₇×#103	C₂×C₄	1	8	yes	—
84	C₇×#102	C₂×C₄	1	8	yes	—
85	C₇×C₂²⋊C₈	—	2	4	no	—
86	C₇×C₂²⋊C₈	—	2	4	no	—
87	C₇×#97	—	2	4	no	—
88	C₄×C₇×C₈	—	2	4	no	—
89	C₂×#430	C₂×C₇	1	1	yes	characteristic, complemented
90	C₂×#421	C₂×C₇	1	1	yes	characteristic
91	#2896	C₂×C₇	1	1	yes	characteristic
92	C₂×#408	C₂×C₇	1	1	yes	characteristic
93	C₂×#405	C₂×C₇	1	1	yes	characteristic, complemented
94	#1929	C₂×C₇	1	8	yes	complemented
95	#1733	C₂×C₇	1	1	yes	characteristic, complemented
96	#1729	C₂×C₇	1	1	yes	characteristic, complemented
97	C₂⁴×C₇	—	2	1	no	—
98	C₂⁴×C₇	#41	1	1	yes	characteristic
99	C₇×#41	—	2	4	no	—
100	C₇×#41	—	2	4	no	—
101	C₇×#41	—	2	2	no	—
102	C₇×#41	—	2	2	no	—
103	C₇×#41	—	2	2	no	—
104	C₇×#41	—	2	2	no	—
105	C₇×#41	—	4	2	no	—
106	C₇×#41	—	4	2	no	—
107	C₇×#41	—	4	2	no	—
108	C₇×#41	—	4	2	no	—
109	C₂×C₇×Q₈	—	2	1	no	—
110	C₂×C₇×Q₈	—	2	2	no	—
111	C₂×C₇×Q₈	—	2	2	no	—
112	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
113	C₂×C₇×D₄	—	2	2	no	—
114	C₂×C₇×D₄	—	2	2	no	—
115	C₂×C₇×D₄	—	2	2	no	—
116	C₂×C₇×D₄	—	2	2	no	—
117	C₂×C₇×D₄	—	2	1	no	—
118	C₂×C₇×D₄	—	4	2	no	—
119	C₂×C₇×D₄	—	4	2	no	—
120	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₇	—	2	2	no	—
122	C₂²×C₄×C₇	—	2	1	no	—
123	C₂²×C₄×C₇	—	2	2	no	—
124	C₂²×C₄×C₇	—	2	2	no	—
125	C₂²×C₄×C₇	—	2	2	no	—
126	C₂²×C₄×C₇	—	4	2	no	—
127	C₂²×C₄×C₇	—	4	2	no	—
128	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₇	#41	1	2	yes	—
134	C₂²×C₄×C₇	#41	1	2	yes	—
135	C₂²×C₄×C₇	#41	1	2	yes	—
136	C₂²×C₄×C₇	#41	1	1	yes	characteristic
137	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
138	C₇×M₁₆	—	2	4	no	—
139	C₇×M₁₆	—	2	4	no	—
140	C₇×M₁₆	—	2	4	no	—
141	C₇×M₁₆	—	2	2	no	—
142	C₇×M₁₆	—	2	2	no	—
143	C₇×M₁₆	—	2	2	no	—
144	C₇×M₁₆	—	4	2	no	—
145	C₇×M₁₆	C₂×D₄	1	4	yes	—
146	C₂×C₇×C₈	—	2	4	no	—
147	C₂×C₇×C₈	—	2	4	no	—
148	C₂×C₇×C₈	—	2	2	no	—
149	C₂×C₇×C₈	—	2	2	no	—
150	C₂×C₇×C₈	—	2	2	no	—
151	C₂×C₇×C₈	—	2	2	no	—
152	C₂×C₇×C₈	—	2	2	no	—
153	C₂×C₇×C₈	—	2	2	no	—
154	C₂×C₇×C₈	—	2	2	no	—
155	C₂×C₇×C₈	—	2	2	no	—
156	C₂×C₇×C₈	—	4	2	no	—
157	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
158	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
159	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
160	C₇×C₄⋊C₄	—	2	2	no	—
161	C₇×C₄⋊C₄	—	4	2	no	—
162	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
163	C₇×C₂²⋊C₄	—	2	2	no	—
164	C₇×C₂²⋊C₄	—	2	2	no	—
165	C₇×C₂²⋊C₄	—	4	2	no	—
166	C₄²×C₇	—	2	2	no	—
167	C₄²×C₇	—	2	2	no	—
168	C₄²×C₇	—	2	2	no	—
169	C₄²×C₇	—	2	2	no	—
170	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
171	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
173	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
174	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
175	C₂×#117	C₂²×C₇	1	1	yes	characteristic
176	C₂×#117	C₂²×C₇	1	1	yes	characteristic
177	#430	—	2	4	no	—
178	C₂×#108	C₂²×C₇	1	4	yes	complemented
179	#421	—	2	4	no	—
180	C₂×#104	C₂²×C₇	1	4	yes	—
181	#419	C₄×C₇	1	2	yes	complemented
182	#418	C₄×C₇	1	2	yes	complemented
183	#417	C₄×C₇	1	4	yes	complemented
184	#416	C₂²×C₇	1	4	yes	—
185	#408	—	2	1	no	—
186	#408	—	2	1	no	—
187	#408	C₂²×C₇	1	4	yes	—
188	C₂×C₂²⋊C₈	—	2	2	no	—
189	#405	—	2	2	no	—
190	#405	C₂²×C₇	1	4	yes	complemented
191	C₂×#97	—	2	1	no	—
192	C₂×C₄×C₈	—	2	1	no	—
193	#339	C₂²×C₇	1	4	yes	complemented
194	#336	C₂²×C₇	1	4	yes	complemented
195	C₂³×C₇	—	2	2	no	—
196	C₂³×C₇	—	4	2	no	—
197	C₂³×C₇	—	4	4	no	—
198	C₂³×C₇	—	4	2	no	—
199	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
200	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
201	C₂³×C₇	#124	1	2	yes	—
202	C₂³×C₇	#126	1	2	yes	—
203	C₂³×C₇	C₂×#41	1	1	yes	characteristic
204	C₇×Q₈	—	2	2	no	—
205	C₇×Q₈	—	2	2	no	—
206	C₇×Q₈	—	4	2	no	—
207	C₇×D₄	—	2	2	no	—
208	C₇×D₄	—	2	2	no	—
209	C₇×D₄	—	4	2	no	—
210	C₇×D₄	—	4	2	no	—
211	C₇×D₄	—	4	4	no	—
212	C₇×D₄	—	4	2	no	—
213	C₂×C₄×C₇	—	2	4	no	—
214	C₂×C₄×C₇	—	2	2	no	—
215	C₂×C₄×C₇	—	4	4	no	—
216	C₂×C₄×C₇	—	4	2	no	—
217	C₂×C₄×C₇	—	4	2	no	—
218	C₂×C₄×C₇	—	4	2	no	—
219	C₂×C₄×C₇	—	4	4	no	—
220	C₂×C₄×C₇	—	4	4	no	—
221	C₂×C₄×C₇	—	4	2	no	—
222	C₂×C₄×C₇	#117	1	2	yes	—
223	C₂×C₄×C₇	#117	1	2	yes	—
224	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
225	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
226	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
227	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
228	C₂×C₄×C₇	#121	1	2	yes	—
229	C₂×C₄×C₇	#121	1	2	yes	—
230	C₂×C₄×C₇	#123	1	2	yes	—
231	C₂×C₄×C₇	#123	1	2	yes	—
232	C₂×C₄×C₇	#124	1	2	yes	—
233	C₂×C₄×C₇	#126	1	2	yes	—
234	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
240	C₇×C₈	—	2	4	no	—
241	C₇×C₈	—	2	4	no	—
242	C₇×C₈	—	2	2	no	—
243	C₇×C₈	—	2	2	no	—
244	C₇×C₈	—	4	2	no	—
245	C₂×#41	—	2	1	no	—
246	C₂×#41	—	2	1	no	—
247	C₂×#41	—	2	2	no	—
248	C₂×#41	—	2	2	no	—
249	C₂×#41	—	2	2	no	—
250	C₂×#41	—	2	2	no	—
251	C₂×#41	C₂×C₄×C₇	1	2	yes	—
252	C₂×#41	C₂³×C₇	1	2	yes	—
253	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
254	C₂²×D₄	—	2	1	no	—
255	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
256	C₂³×C₄	—	2	1	no	—
257	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
258	C₂×M₁₆	—	2	1	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	1	no	—
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	—	2	2	no	—
265	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
266	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
267	C₂×M₁₆	C₇×D₄	1	4	yes	—
268	C₂×M₁₆	C₂³×C₇	1	2	yes	—
269	C₂×M₁₆	C₂³×C₇	1	2	yes	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	—	2	2	no	—
272	C₂²×C₈	—	2	2	no	—
273	C₂²×C₈	—	2	1	no	—
274	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
275	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
276	C₂²×C₈	C₇×D₄	1	2	yes	—
277	C₂²×C₈	C₂³×C₇	1	2	yes	—
278	C₂²×C₈	C₂³×C₇	1	2	yes	—
279	#117	—	2	1	no	—
280	#117	—	2	1	no	—
281	#117	—	2	2	no	—
282	#117	—	2	2	no	—
283	#117	C₂×C₄×C₇	1	2	yes	—
284	#117	C₂³×C₇	1	2	yes	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	C₂³×C₇	1	2	yes	—
291	#108	—	2	8	no	—
292	#104	—	2	8	no	—
293	#103	C₂×C₄×C₇	1	8	yes	—
294	#102	C₂×C₄×C₇	1	8	yes	—
295	C₂²⋊C₈	—	2	4	no	—
296	C₂²⋊C₈	—	2	4	no	—
297	#97	—	2	4	no	—
298	C₄×C₈	—	2	4	no	—
299	C₂²×C₇	—	2	2	no	—
300	C₂²×C₇	—	2	2	no	—
301	C₂²×C₇	—	2	2	no	—
302	C₂²×C₇	—	4	2	no	—
303	C₂²×C₇	—	4	2	no	—
304	C₂²×C₇	—	4	4	no	—
305	C₂²×C₇	—	4	2	no	—
306	C₂²×C₇	—	8	4	no	—
307	C₂²×C₇	#388	1	2	yes	—
308	C₂²×C₇	#388	1	2	yes	—
309	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
310	C₂²×C₇	C₂×#124	1	1	yes	characteristic
311	C₂²×C₇	C₂×#126	1	1	yes	characteristic
312	C₄×C₇	—	4	4	no	—
313	C₄×C₇	—	4	2	no	—
314	C₄×C₇	—	8	2	no	—
315	C₄×C₇	#388	1	2	yes	—
316	C₄×C₇	#388	1	2	yes	—
317	C₄×C₇	C₂×#117	1	1	yes	characteristic
318	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
319	C₄×C₇	C₂×#121	1	1	yes	characteristic
320	C₄×C₇	C₂×#123	1	1	yes	characteristic
321	C₂⁴	—	2	1	no	—
322	C₂⁴	C₇×#41	1	1	yes	characteristic
323	#41	—	2	4	no	—
324	#41	—	2	4	no	—
325	#41	—	2	2	no	—
326	#41	—	2	2	no	—
327	#41	—	2	2	no	—
328	#41	—	2	2	no	—
329	#41	—	4	2	no	—
330	#41	—	4	2	no	—
331	#41	—	4	2	no	—
332	#41	—	4	2	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	—	4	2	no	—
344	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
353	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
355	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
356	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
357	C₂²×C₄	C₇×#41	1	2	yes	—
358	C₂²×C₄	C₇×#41	1	2	yes	—
359	C₂²×C₄	C₇×#41	1	2	yes	—
360	C₂²×C₄	C₇×#41	1	1	yes	characteristic
361	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
362	M₁₆	—	2	4	no	—
363	M₁₆	—	2	4	no	—
364	M₁₆	—	2	4	no	—
365	M₁₆	—	2	2	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	4	2	no	—
369	M₁₆	C₂×C₇×D₄	1	4	yes	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	—	2	4	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	4	2	no	—
381	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
382	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
383	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₂×C₇	—	2	1	no	—
395	C₂×C₇	—	2	2	no	—
396	C₂×C₇	—	2	1	no	—
397	C₂×C₇	—	2	2	no	—
398	C₂×C₇	—	8	2	no	—
399	C₂×C₇	C₂×#388	1	1	yes	characteristic
400	C₂×C₇	#2955	1	2	yes	—
401	C₂³	—	2	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	4	no	—
404	C₂³	—	4	2	no	—
405	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
406	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
407	C₂³	C₇×#124	1	2	yes	—
408	C₂³	C₇×#126	1	2	yes	—
409	C₂³	C₂×C₇×#41	1	1	yes	characteristic
410	Q₈	—	2	2	no	—
411	Q₈	—	2	2	no	—
412	Q₈	—	4	2	no	—
413	D₄	—	2	2	no	—
414	D₄	—	2	2	no	—
415	D₄	—	4	4	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	C₇×#117	1	2	yes	—
429	C₂×C₄	C₇×#117	1	2	yes	—
430	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
431	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
432	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
433	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
434	C₂×C₄	C₇×#121	1	2	yes	—
435	C₂×C₄	C₇×#121	1	2	yes	—
436	C₂×C₄	C₇×#123	1	2	yes	—
437	C₂×C₄	C₇×#123	1	2	yes	—
438	C₂×C₄	C₇×#124	1	2	yes	—
439	C₂×C₄	C₇×#126	1	2	yes	—
440	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
441	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
442	C₂×C₄	C₂²×C₇×D₄	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	4	no	—
447	C₈	—	2	4	no	—
448	C₈	—	2	2	no	—
449	C₈	—	2	2	no	—
450	C₈	—	4	2	no	—
451	C₇	#21070	1	1	yes	characteristic, direct product factor, Sylow
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	8	4	no	—
460	C₂²	C₇×#388	1	2	yes	—
461	C₂²	C₇×#388	1	2	yes	—
462	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
463	C₂²	C₂×C₇×#124	1	1	yes	characteristic
464	C₂²	C₂×C₇×#126	1	1	yes	characteristic
465	C₄	—	4	4	no	—
466	C₄	—	4	2	no	—
467	C₄	—	8	2	no	—
468	C₄	C₇×#388	1	2	yes	—
469	C₄	C₇×#388	1	2	yes	—
470	C₄	C₂×C₇×#117	1	1	yes	characteristic
471	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
472	C₄	C₂×C₇×#121	1	1	yes	characteristic
473	C₄	C₂×C₇×#123	1	1	yes	characteristic
474	C₂	—	2	1	no	—
475	C₂	—	2	2	no	—
476	C₂	—	2	2	no	—
477	C₂	—	2	1	no	—
478	C₂	—	8	2	no	—
479	C₂	C₂×C₇×#388	1	1	yes	characteristic
480	C₂	C₇×#2955	1	2	yes	—

	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	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#430	C₂	semidirect
5. C₂×C₇×#405	C₂	semidirect
6. C₇×#1929	C₂	semidirect
7. C₇×#1733	C₂	semidirect
8. C₇×#1729	C₂	semidirect
17. C₂×C₇×#108	C₂²	semidirect
20. C₇×#419	C₄	semidirect
21. C₇×#418	C₄	semidirect
22. C₇×#417	C₄	semidirect
29. C₇×#405	C₂²	semidirect
32. C₇×#339	C₂²	semidirect
33. C₇×#336	C₂²	semidirect
34. #21070	C₇	direct (N × H)
89. C₂×#430	C₂×C₇	semidirect
93. C₂×#405	C₂×C₇	semidirect
94. #1929	C₂×C₇	semidirect
95. #1733	C₂×C₇	semidirect
96. #1729	C₂×C₇	semidirect
178. C₂×#108	C₂²×C₇	semidirect
181. #419	C₄×C₇	semidirect
182. #418	C₄×C₇	semidirect
183. #417	C₄×C₇	semidirect
190. #405	C₂²×C₇	semidirect
193. #339	C₂²×C₇	semidirect
194. #336	C₂²×C₇	semidirect
451. C₇	#21070	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×#430	2
C₂×C₇×#421	2
C₇×#2896	2
C₂×C₇×#408	2
C₂×C₇×#405	2
C₇×#1929	2
C₇×#1733	2
C₇×#1729	2
#21070	7

Subgroups — C7×#21070 #13198286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21070 #13198286