Subgroups — C₇×#21505 #13198721

Order 1792.

1140 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#460	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#440	C₂	1	2	yes	maximal, complemented
3	C₄×C₇×#127	C₂	1	1	yes	maximal, characteristic
4	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1925	C₂	1	2	yes	maximal, complemented
8	C₇×#1880	C₂	1	2	yes	maximal, complemented
9	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₇×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×D₄	—	2	1	no	—
18	C₂×C₄×C₇×D₄	—	2	1	no	—
19	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
20	C₇×#460	C₄	1	8	yes	complemented
21	C₇×#440	—	2	8	no	—
22	C₂×C₇×#107	C₂²	1	2	yes	—
23	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₇×#103	C₂²	1	2	yes	—
26	C₂×C₇×#102	C₂²	1	2	yes	—
27	C₂×C₇×#102	C₂²	1	2	yes	—
28	C₂×C₇×#102	C₂²	1	2	yes	—
29	C₂×C₇×#97	C₂²	1	2	yes	—
30	C₇×#390	—	2	1	no	—
31	C₇×#390	—	2	1	no	—
32	C₇×#390	C₂²	1	2	yes	—
33	C₇×#384	C₂²	1	1	yes	characteristic
34	C₄×C₇×C₄⋊C₄	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	—	2	1	no	—
36	C₄×C₇×C₄⋊C₄	—	2	1	no	—
37	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₄³×C₇	—	2	1	no	—
41	C₄³×C₇	C₂²	1	1	yes	characteristic
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#340	C₂²	1	2	yes	—
44	C₇×#336	C₂²	1	2	yes	—
45	#21505	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇×D₄	—	2	1	no	—
48	C₂²×C₇×D₄	—	2	1	no	—
49	C₂²×C₇×D₄	C₂³	1	2	yes	—
50	C₂³×C₄×C₇	—	2	2	no	—
51	C₂×C₇×SD₁₆	—	2	4	no	—
52	C₂×C₇×SD₁₆	—	2	4	no	—
53	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	complemented
54	C₂²×C₇×C₈	C₂³	1	2	yes	—
55	C₂²×C₇×C₈	C₂³	1	2	yes	—
56	C₇×#127	—	2	2	no	—
57	C₇×#127	C₂×C₄	1	4	yes	—
58	C₄×C₇×Q₈	—	2	2	no	—
59	C₄×C₇×Q₈	—	2	2	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×Q₈	—	2	2	no	—
62	C₄×C₇×Q₈	C₂×C₄	1	4	yes	complemented
63	C₄×C₇×Q₈	D₄	1	4	yes	complemented
64	C₄×C₇×D₄	—	2	4	no	—
65	C₄×C₇×D₄	—	2	4	no	—
66	C₄×C₇×D₄	—	4	4	no	—
67	C₄×C₇×D₄	—	4	4	no	—
68	C₂×C₇×C₄⋊C₄	—	2	1	no	—
69	C₂×C₇×C₄⋊C₄	—	2	1	no	—
70	C₂×C₇×C₄⋊C₄	—	2	1	no	—
71	C₂×C₇×C₄⋊C₄	—	2	1	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
74	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
80	C₂×C₄²×C₇	—	2	1	no	—
81	C₂×C₄²×C₇	—	2	1	no	—
82	C₂×C₄²×C₇	—	2	1	no	—
83	C₂×C₄²×C₇	—	2	1	no	—
84	C₂×C₄²×C₇	—	2	1	no	—
85	C₂×C₄²×C₇	—	2	1	no	—
86	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
93	C₇×#107	—	2	4	no	—
94	C₇×C₄⋊C₈	—	2	2	no	—
95	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
96	C₇×#103	—	2	4	no	—
97	C₇×#102	—	2	4	no	—
98	C₇×#102	—	2	4	no	—
99	C₇×#102	C₂×C₄	1	8	yes	complemented
100	C₇×#97	—	2	4	no	—
101	C₇×#95	—	2	1	no	—
102	C₇×#95	—	2	1	no	—
103	C₇×#95	—	2	1	no	—
104	C₂×#460	C₂×C₇	1	1	yes	characteristic
105	C₂×#440	C₂×C₇	1	2	yes	complemented
106	C₄×#127	C₂×C₇	1	1	yes	characteristic
107	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
108	#1982	C₂×C₇	1	1	yes	characteristic
109	#1947	C₂×C₇	1	1	yes	characteristic, complemented
110	#1925	C₂×C₇	1	2	yes	complemented
111	#1880	C₂×C₇	1	2	yes	complemented
112	#1803	C₂×C₇	1	1	yes	characteristic, complemented
113	#1767	C₂×C₇	1	1	yes	characteristic, complemented
114	#1762	C₂×C₇	1	2	yes	complemented
115	C₂⁴×C₇	—	2	2	no	—
116	C₂×C₇×Q₈	—	2	1	no	—
117	C₂×C₇×Q₈	—	2	1	no	—
118	C₂×C₇×Q₈	—	2	1	no	—
119	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
120	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
121	C₂×C₇×Q₈	#41	1	2	yes	—
122	C₂×C₇×D₄	—	2	4	no	—
123	C₂×C₇×D₄	—	2	4	no	—
124	C₂×C₇×D₄	—	2	2	no	—
125	C₂×C₇×D₄	—	2	2	no	—
126	C₂×C₇×D₄	—	2	2	no	—
127	C₂×C₇×D₄	—	2	2	no	—
128	C₂×C₇×D₄	—	2	2	no	—
129	C₂×C₇×D₄	—	4	2	no	—
130	C₂×C₇×D₄	—	4	2	no	—
131	C₂×C₇×D₄	—	4	2	no	—
132	C₂×C₇×D₄	—	4	2	no	—
133	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₇	—	2	1	no	—
135	C₂²×C₄×C₇	—	2	1	no	—
136	C₂²×C₄×C₇	—	2	1	no	—
137	C₂²×C₄×C₇	—	2	1	no	—
138	C₂²×C₄×C₇	—	2	2	no	—
139	C₂²×C₄×C₇	—	4	4	no	—
140	C₂²×C₄×C₇	—	4	4	no	—
141	C₂²×C₄×C₇	—	4	4	no	—
142	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
149	C₇×SD₁₆	—	2	16	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	C₂²×C₄	1	4	yes	—
156	C₇×C₄⋊C₄	—	2	4	no	—
157	C₇×C₄⋊C₄	—	2	4	no	—
158	C₇×C₄⋊C₄	—	2	4	no	—
159	C₇×C₄⋊C₄	—	2	1	no	—
160	C₇×C₄⋊C₄	—	2	1	no	—
161	C₇×C₄⋊C₄	—	2	1	no	—
162	C₇×C₄⋊C₄	—	2	2	no	—
163	C₇×C₄⋊C₄	—	2	2	no	—
164	C₇×C₄⋊C₄	—	4	2	no	—
165	C₇×C₄⋊C₄	—	4	2	no	—
166	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
167	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
168	C₇×C₄⋊C₄	#41	1	2	yes	—
169	C₇×C₂²⋊C₄	—	4	4	no	—
170	C₇×C₂²⋊C₄	—	4	4	no	—
171	C₇×C₂²⋊C₄	—	4	4	no	—
172	C₄²×C₇	—	2	4	no	—
173	C₄²×C₇	—	2	4	no	—
174	C₄²×C₇	—	2	4	no	—
175	C₄²×C₇	—	2	2	no	—
176	C₄²×C₇	—	2	2	no	—
177	C₄²×C₇	—	2	2	no	—
178	C₄²×C₇	—	2	1	no	—
179	C₄²×C₇	—	2	1	no	—
180	C₄²×C₇	—	4	1	no	—
181	C₄²×C₇	—	4	2	no	—
182	C₄²×C₇	—	4	2	no	—
183	C₄²×C₇	—	4	2	no	—
184	C₄²×C₇	C₂²×C₄	1	2	yes	—
185	C₄²×C₇	C₂×D₄	1	2	yes	—
186	C₄²×C₇	#41	1	2	yes	—
187	C₄²×C₇	#41	1	2	yes	—
188	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
189	C₂×#127	C₂²×C₇	1	1	yes	characteristic
190	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
191	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
192	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
193	C₂×C₄×D₄	—	2	1	no	—
194	C₂×C₄×D₄	—	2	1	no	—
195	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
196	#460	C₄×C₇	1	8	yes	complemented
197	#440	—	2	8	no	—
198	C₂×#107	C₂²×C₇	1	2	yes	—
199	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
201	C₂×#103	C₂²×C₇	1	2	yes	—
202	C₂×#102	C₂²×C₇	1	2	yes	—
203	C₂×#102	C₂²×C₇	1	2	yes	—
204	C₂×#102	C₂²×C₇	1	2	yes	—
205	C₂×#97	C₂²×C₇	1	2	yes	—
206	#390	—	2	1	no	—
207	#390	—	2	1	no	—
208	#390	C₂²×C₇	1	2	yes	—
209	#384	C₂²×C₇	1	1	yes	characteristic
210	C₄×C₄⋊C₄	—	2	1	no	—
211	C₄×C₄⋊C₄	—	2	1	no	—
212	C₄×C₄⋊C₄	—	2	1	no	—
213	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
216	C₄³	—	2	1	no	—
217	C₄³	C₂²×C₇	1	1	yes	characteristic
218	#340	C₂²×C₇	1	2	yes	—
219	#340	C₂²×C₇	1	2	yes	—
220	#336	C₂²×C₇	1	2	yes	—
221	C₂³×C₇	—	2	4	no	—
222	C₂³×C₇	—	4	2	no	—
223	C₂³×C₇	—	4	2	no	—
224	C₂³×C₇	—	4	2	no	—
225	C₂³×C₇	—	4	2	no	—
226	C₂³×C₇	—	4	2	no	—
227	C₂³×C₇	—	4	2	no	—
228	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
229	C₇×Q₈	—	2	2	no	—
230	C₇×Q₈	—	2	2	no	—
231	C₇×Q₈	—	2	2	no	—
232	C₇×Q₈	C₄×D₄	1	4	yes	complemented
233	C₇×D₄	—	2	4	no	—
234	C₇×D₄	—	2	4	no	—
235	C₇×D₄	—	4	8	no	—
236	C₇×D₄	—	4	2	no	—
237	C₇×D₄	—	4	2	no	—
238	C₇×D₄	—	4	8	no	—
239	C₂×C₄×C₇	—	2	2	no	—
240	C₂×C₄×C₇	—	2	2	no	—
241	C₂×C₄×C₇	—	2	1	no	—
242	C₂×C₄×C₇	—	2	2	no	—
243	C₂×C₄×C₇	—	2	1	no	—
244	C₂×C₄×C₇	—	2	1	no	—
245	C₂×C₄×C₇	—	2	1	no	—
246	C₂×C₄×C₇	—	2	1	no	—
247	C₂×C₄×C₇	—	2	1	no	—
248	C₂×C₄×C₇	—	2	1	no	—
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	1	no	—
253	C₂×C₄×C₇	—	2	2	no	—
254	C₂×C₄×C₇	—	2	1	no	—
255	C₂×C₄×C₇	—	2	1	no	—
256	C₂×C₄×C₇	—	4	1	no	—
257	C₂×C₄×C₇	—	4	1	no	—
258	C₂×C₄×C₇	—	4	1	no	—
259	C₂×C₄×C₇	—	4	1	no	—
260	C₂×C₄×C₇	—	4	1	no	—
261	C₂×C₄×C₇	—	4	1	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	—	4	2	no	—
264	C₂×C₄×C₇	—	4	2	no	—
265	C₂×C₄×C₇	—	8	8	no	—
266	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
267	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
268	C₂×C₄×C₇	#121	1	2	yes	—
269	C₂×C₄×C₇	#121	1	2	yes	—
270	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
271	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₄×C₇	#137	1	2	yes	—
273	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
276	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
279	C₇×C₈	—	2	4	no	—
280	C₇×C₈	—	4	2	no	—
281	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	C₂³×C₇	1	2	yes	—
285	C₂³×C₄	—	2	2	no	—
286	C₂×SD₁₆	—	2	4	no	—
287	C₂×SD₁₆	—	2	4	no	—
288	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	complemented
289	C₂²×C₈	C₂³×C₇	1	2	yes	—
290	C₂²×C₈	C₂³×C₇	1	2	yes	—
291	#127	—	2	2	no	—
292	#127	C₂×C₄×C₇	1	4	yes	—
293	C₄×Q₈	—	2	2	no	—
294	C₄×Q₈	—	2	2	no	—
295	C₄×Q₈	—	2	2	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×Q₈	C₂×C₄×C₇	1	4	yes	complemented
298	C₄×Q₈	C₇×D₄	1	4	yes	complemented
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	2	4	no	—
301	C₄×D₄	—	4	4	no	—
302	C₄×D₄	—	4	4	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
322	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
323	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
324	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
328	#107	—	2	4	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
331	#103	—	2	4	no	—
332	#102	—	2	4	no	—
333	#102	—	2	4	no	—
334	#102	C₂×C₄×C₇	1	8	yes	complemented
335	#97	—	2	4	no	—
336	#95	—	2	1	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	C₂²×C₇	—	4	4	no	—
340	C₂²×C₇	—	4	4	no	—
341	C₂²×C₇	—	4	4	no	—
342	C₂²×C₇	—	8	2	no	—
343	C₂²×C₇	—	8	2	no	—
344	C₂²×C₇	—	8	2	no	—
345	C₂²×C₇	—	8	2	no	—
346	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
347	C₂²×C₇	C₂×#121	1	1	yes	characteristic
348	C₂²×C₇	#525	1	1	yes	characteristic
349	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
351	C₂²×C₇	C₂×#137	1	1	yes	characteristic
352	C₂²×C₇	#575	1	1	yes	characteristic
353	C₄×C₇	—	2	2	no	—
354	C₄×C₇	—	2	2	no	—
355	C₄×C₇	—	2	2	no	—
356	C₄×C₇	—	2	2	no	—
357	C₄×C₇	—	2	4	no	—
358	C₄×C₇	—	4	1	no	—
359	C₄×C₇	—	4	2	no	—
360	C₄×C₇	—	4	2	no	—
361	C₄×C₇	—	4	2	no	—
362	C₄×C₇	—	8	2	no	—
363	C₄×C₇	#440	1	4	yes	complemented
364	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
365	C₄×C₇	C₄×#41	1	1	yes	characteristic
366	C₄×C₇	#529	1	1	yes	characteristic
367	C₄×C₇	#532	1	1	yes	characteristic
368	C₂⁴	—	2	2	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
373	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
374	C₂×Q₈	C₇×#41	1	2	yes	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	4	4	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
396	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
397	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
398	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
399	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
400	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
402	SD₁₆	—	2	16	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	—	4	2	no	—
419	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
420	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
421	C₄⋊C₄	C₇×#41	1	2	yes	—
422	C₂²⋊C₄	—	4	4	no	—
423	C₂²⋊C₄	—	4	4	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₄²	—	2	4	no	—
426	C₄²	—	2	4	no	—
427	C₄²	—	2	4	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	4	1	no	—
434	C₄²	—	4	2	no	—
435	C₄²	—	4	2	no	—
436	C₄²	—	4	2	no	—
437	C₄²	C₂²×C₄×C₇	1	2	yes	—
438	C₄²	C₂×C₇×D₄	1	2	yes	—
439	C₄²	C₇×#41	1	2	yes	—
440	C₄²	C₇×#41	1	2	yes	—
441	C₂×C₇	—	8	4	no	—
442	C₂×C₇	C₄×#121	1	1	yes	characteristic
443	C₂×C₇	C₂×#440	1	1	yes	characteristic
444	C₂×C₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
445	C₂×C₇	#3037	1	1	yes	characteristic
446	C₂×C₇	#3038	1	1	yes	characteristic
447	C₂×C₇	#3112	1	1	yes	characteristic
448	C₂×C₇	#3115	1	1	yes	characteristic
449	C₂³	—	2	4	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
457	Q₈	—	2	2	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	2	2	no	—
460	Q₈	C₄×C₇×D₄	1	4	yes	complemented
461	D₄	—	2	4	no	—
462	D₄	—	2	4	no	—
463	D₄	—	4	8	no	—
464	D₄	—	4	2	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	8	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	8	8	no	—
494	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
495	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
496	C₂×C₄	C₇×#121	1	2	yes	—
497	C₂×C₄	C₇×#121	1	2	yes	—
498	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
499	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
500	C₂×C₄	C₇×#137	1	2	yes	—
501	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
502	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
503	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
504	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
505	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
506	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
507	C₈	—	2	4	no	—
508	C₈	—	4	2	no	—
509	C₇	#21505	1	1	yes	characteristic, direct product factor, Sylow
510	C₂²	—	4	4	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	8	2	no	—
514	C₂²	—	8	2	no	—
515	C₂²	—	8	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
518	C₂²	C₂×C₇×#121	1	1	yes	characteristic
519	C₂²	C₇×#525	1	1	yes	characteristic
520	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
521	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
522	C₂²	C₂×C₇×#137	1	1	yes	characteristic
523	C₂²	C₇×#575	1	1	yes	characteristic
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	2	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	2	no	—
533	C₄	—	8	2	no	—
534	C₄	C₇×#440	1	4	yes	complemented
535	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
536	C₄	C₄×C₇×#41	1	1	yes	characteristic
537	C₄	C₇×#529	1	1	yes	characteristic
538	C₄	C₇×#532	1	1	yes	characteristic
539	C₂	—	8	4	no	—
540	C₂	C₄×C₇×#121	1	1	yes	characteristic
541	C₂	C₂×C₇×#440	1	1	yes	characteristic
542	C₂	C₄×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
543	C₂	C₇×#3037	1	1	yes	characteristic
544	C₂	C₇×#3038	1	1	yes	characteristic
545	C₂	C₇×#3112	1	1	yes	characteristic
546	C₂	C₇×#3115	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×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#440	C₂	semidirect
4. C₄²×C₇×Q₈	C₂	semidirect
6. C₇×#1947	C₂	semidirect
7. C₇×#1925	C₂	semidirect
8. C₇×#1880	C₂	semidirect
9. C₇×#1803	C₂	semidirect
10. C₇×#1767	C₂	semidirect
11. C₇×#1762	C₂	semidirect
20. C₇×#460	C₄	semidirect
45. #21505	C₇	direct (N × H)
53. C₂×C₇×SD₁₆	C₂×C₄	semidirect
62. C₄×C₇×Q₈	C₂×C₄	semidirect
63. C₄×C₇×Q₈	D₄	semidirect
95. C₇×C₄⋊C₈	C₂×C₄	semidirect
99. C₇×#102	C₂×C₄	semidirect
105. C₂×#440	C₂×C₇	semidirect
107. C₄²×Q₈	C₂×C₇	semidirect
109. #1947	C₂×C₇	semidirect
110. #1925	C₂×C₇	semidirect
111. #1880	C₂×C₇	semidirect
112. #1803	C₂×C₇	semidirect
113. #1767	C₂×C₇	semidirect
114. #1762	C₂×C₇	semidirect
196. #460	C₄×C₇	semidirect
232. C₇×Q₈	C₄×D₄	semidirect
288. C₂×SD₁₆	C₂×C₄×C₇	semidirect
297. C₄×Q₈	C₂×C₄×C₇	semidirect
298. C₄×Q₈	C₇×D₄	semidirect
330. C₄⋊C₈	C₂×C₄×C₇	semidirect
334. #102	C₂×C₄×C₇	semidirect
363. C₄×C₇	#440	semidirect
460. Q₈	C₄×C₇×D₄	semidirect
509. C₇	#21505	direct (N × H)
534. C₄	C₇×#440	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#460	2
C₂×C₇×#440	2
C₄×C₇×#127	2
C₄²×C₇×Q₈	2
C₇×#1982	2
C₇×#1947	2
C₇×#1925	2
C₇×#1880	2
C₇×#1803	2
C₇×#1767	2
C₇×#1762	2
#21505	7

Subgroups — C7×#21505 #13198721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21505 #13198721