Subgroups — C₇×#23003 #13200219

Order 1792.

1054 conjugacy classes of subgroups.

The subgroup lattice has 504 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	C₇×#2366	C₂	1	1	yes	maximal, characteristic
4	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1957	C₂	1	1	yes	maximal, characteristic
6	C₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1936	C₂	1	2	yes	maximal, complemented
8	C₇×#1880	C₂	1	2	yes	maximal, complemented
9	C₇×#1862	C₂	1	2	yes	maximal, complemented
10	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#103	C₂	1	1	yes	maximal, characteristic, 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	2	yes	—
15	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×D₄	—	2	1	no	—
17	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
18	C₇×#460	C₄	1	8	yes	complemented
19	C₇×#440	—	2	8	no	—
20	C₂×C₇×#107	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#103	C₂²	1	1	yes	characteristic
23	C₂×C₇×#103	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	C₂×C₇×#97	C₂²	1	2	yes	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#390	—	2	1	no	—
30	C₇×#390	C₂²	1	2	yes	—
31	C₇×#389	—	2	2	no	—
32	C₇×#389	C₂²	1	2	yes	—
33	C₇×#386	—	2	2	no	—
34	C₇×#384	—	2	1	no	—
35	C₇×#384	C₂²	1	1	yes	characteristic
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	C₇×#340	C₂²	1	1	yes	characteristic
40	C₇×#340	C₂²	1	1	yes	characteristic
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#336	C₂²	1	2	yes	—
43	#23003	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
45	C₂²×C₇×D₄	—	2	1	no	—
46	C₂²×C₇×D₄	—	2	1	no	—
47	C₂²×C₇×D₄	C₂³	1	2	yes	—
48	C₂³×C₄×C₇	—	2	2	no	—
49	C₂×C₇×SD₁₆	—	2	4	no	—
50	C₂×C₇×SD₁₆	—	2	4	no	—
51	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	complemented
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₇×C₈	C₂³	1	2	yes	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₇×#127	—	2	2	no	—
56	C₇×#127	C₂×C₄	1	4	yes	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₄×C₇×Q₈	—	2	4	no	—
59	C₄×C₇×Q₈	—	4	2	no	—
60	C₄×C₇×Q₈	C₂×C₄	1	4	yes	complemented
61	C₄×C₇×D₄	—	2	4	no	—
62	C₄×C₇×D₄	—	4	2	no	—
63	C₄×C₇×D₄	—	4	4	no	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₇×C₄⋊C₄	—	2	1	no	—
66	C₂×C₇×C₄⋊C₄	—	2	2	no	—
67	C₂×C₇×C₄⋊C₄	—	2	2	no	—
68	C₂×C₇×C₄⋊C₄	—	2	1	no	—
69	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
70	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
73	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
76	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₄²×C₇	—	2	1	no	—
79	C₂×C₄²×C₇	—	2	2	no	—
80	C₂×C₄²×C₇	—	2	1	no	—
81	C₂×C₄²×C₇	C₂³	1	2	yes	—
82	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
83	C₇×#107	—	2	4	no	—
84	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
85	C₇×#103	—	2	2	no	—
86	C₇×#103	D₄	1	4	yes	complemented
87	C₇×#102	—	2	4	no	—
88	C₇×#102	—	4	2	no	—
89	C₇×#102	C₂×C₄	1	8	yes	complemented
90	C₇×#97	—	2	4	no	—
91	C₄×C₇×C₈	—	2	2	no	—
92	C₇×#95	—	2	2	no	—
93	C₇×#95	—	2	1	no	—
94	C₇×#95	D₄	1	2	yes	—
95	C₂×#460	C₂×C₇	1	1	yes	characteristic
96	C₂×#440	C₂×C₇	1	2	yes	complemented
97	#2366	C₂×C₇	1	1	yes	characteristic
98	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
99	#1957	C₂×C₇	1	1	yes	characteristic
100	#1947	C₂×C₇	1	1	yes	characteristic, complemented
101	#1936	C₂×C₇	1	2	yes	complemented
102	#1880	C₂×C₇	1	2	yes	complemented
103	#1862	C₂×C₇	1	2	yes	complemented
104	#1770	C₂×C₇	1	1	yes	characteristic, complemented
105	C₄×#103	C₂×C₇	1	1	yes	characteristic, complemented
106	C₂⁴×C₇	—	2	2	no	—
107	C₂×C₇×Q₈	—	2	1	no	—
108	C₂×C₇×Q₈	—	2	1	no	—
109	C₂×C₇×Q₈	—	2	1	no	—
110	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
111	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
112	C₂×C₇×Q₈	#41	1	2	yes	—
113	C₂×C₇×D₄	—	2	4	no	—
114	C₂×C₇×D₄	—	2	4	no	—
115	C₂×C₇×D₄	—	2	2	no	—
116	C₂×C₇×D₄	—	2	2	no	—
117	C₂×C₇×D₄	—	2	2	no	—
118	C₂×C₇×D₄	—	2	2	no	—
119	C₂×C₇×D₄	—	2	2	no	—
120	C₂×C₇×D₄	—	4	2	no	—
121	C₂×C₇×D₄	—	4	2	no	—
122	C₂×C₇×D₄	—	8	1	no	—
123	C₂×C₇×D₄	—	8	1	no	—
124	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₇	—	2	1	no	—
126	C₂²×C₄×C₇	—	2	1	no	—
127	C₂²×C₄×C₇	—	2	1	no	—
128	C₂²×C₄×C₇	—	2	1	no	—
129	C₂²×C₄×C₇	—	2	2	no	—
130	C₂²×C₄×C₇	—	4	4	no	—
131	C₂²×C₄×C₇	—	4	4	no	—
132	C₂²×C₄×C₇	—	8	2	no	—
133	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
138	C₇×SD₁₆	—	2	16	no	—
139	C₂×C₇×C₈	—	2	2	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	2	no	—
142	C₂×C₇×C₈	—	2	1	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	1	no	—
145	C₂×C₇×C₈	—	2	1	no	—
146	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
147	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
148	C₇×C₄⋊C₄	—	2	4	no	—
149	C₇×C₄⋊C₄	—	2	2	no	—
150	C₇×C₄⋊C₄	—	2	2	no	—
151	C₇×C₄⋊C₄	—	2	1	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	2	2	no	—
154	C₇×C₄⋊C₄	—	2	1	no	—
155	C₇×C₄⋊C₄	—	2	2	no	—
156	C₇×C₄⋊C₄	—	2	1	no	—
157	C₇×C₄⋊C₄	—	4	4	no	—
158	C₇×C₄⋊C₄	—	4	2	no	—
159	C₇×C₄⋊C₄	—	8	1	no	—
160	C₇×C₄⋊C₄	—	8	2	no	—
161	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
162	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
163	C₇×C₂²⋊C₄	—	4	4	no	—
164	C₇×C₂²⋊C₄	—	4	4	no	—
165	C₇×C₂²⋊C₄	—	8	2	no	—
166	C₄²×C₇	—	2	4	no	—
167	C₄²×C₇	—	2	2	no	—
168	C₄²×C₇	—	2	2	no	—
169	C₄²×C₇	—	4	4	no	—
170	C₄²×C₇	—	4	2	no	—
171	C₄²×C₇	C₂²×C₄	1	2	yes	—
172	C₄²×C₇	#41	1	2	yes	—
173	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
174	C₂×#127	C₂²×C₇	1	1	yes	characteristic
175	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
176	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
177	C₂×C₄×D₄	—	2	1	no	—
178	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
179	#460	C₄×C₇	1	8	yes	complemented
180	#440	—	2	8	no	—
181	C₂×#107	C₂²×C₇	1	2	yes	—
182	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
183	C₂×#103	C₂²×C₇	1	1	yes	characteristic
184	C₂×#103	C₂²×C₇	1	1	yes	characteristic
185	C₂×#102	C₂²×C₇	1	2	yes	—
186	C₂×#102	C₂²×C₇	1	2	yes	—
187	C₂×#102	C₂²×C₇	1	2	yes	—
188	C₂×#97	C₂²×C₇	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
190	#390	—	2	1	no	—
191	#390	C₂²×C₇	1	2	yes	—
192	#389	—	2	2	no	—
193	#389	C₂²×C₇	1	2	yes	—
194	#386	—	2	2	no	—
195	#384	—	2	1	no	—
196	#384	C₂²×C₇	1	1	yes	characteristic
197	C₄×C₄⋊C₄	—	2	1	no	—
198	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
199	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
200	#340	C₂²×C₇	1	1	yes	characteristic
201	#340	C₂²×C₇	1	2	yes	—
202	#340	C₂²×C₇	1	1	yes	characteristic
203	#336	C₂²×C₇	1	2	yes	—
204	C₂³×C₇	—	2	4	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	—	4	2	no	—
210	C₂³×C₇	—	4	2	no	—
211	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
212	C₇×Q₈	—	2	2	no	—
213	C₇×Q₈	—	2	2	no	—
214	C₇×Q₈	—	2	2	no	—
215	C₇×Q₈	C₄×D₄	1	4	yes	complemented
216	C₇×D₄	—	2	4	no	—
217	C₇×D₄	—	2	4	no	—
218	C₇×D₄	—	4	2	no	—
219	C₇×D₄	—	4	2	no	—
220	C₇×D₄	—	4	8	no	—
221	C₇×D₄	—	8	4	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	1	no	—
226	C₂×C₄×C₇	—	2	1	no	—
227	C₂×C₄×C₇	—	2	2	no	—
228	C₂×C₄×C₇	—	2	2	no	—
229	C₂×C₄×C₇	—	2	2	no	—
230	C₂×C₄×C₇	—	2	2	no	—
231	C₂×C₄×C₇	—	2	2	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	2	no	—
234	C₂×C₄×C₇	—	2	2	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	1	no	—
237	C₂×C₄×C₇	—	4	1	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	2	no	—
244	C₂×C₄×C₇	—	4	1	no	—
245	C₂×C₄×C₇	—	8	8	no	—
246	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
247	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
248	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
249	C₂×C₄×C₇	#137	1	2	yes	—
250	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
255	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
256	C₇×C₈	—	2	4	no	—
257	C₇×C₈	—	2	2	no	—
258	C₇×C₈	—	4	1	no	—
259	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
260	C₂²×D₄	—	2	1	no	—
261	C₂²×D₄	—	2	1	no	—
262	C₂²×D₄	C₂³×C₇	1	2	yes	—
263	C₂³×C₄	—	2	2	no	—
264	C₂×SD₁₆	—	2	4	no	—
265	C₂×SD₁₆	—	2	4	no	—
266	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	complemented
267	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₇	1	2	yes	—
269	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
270	#127	—	2	2	no	—
271	#127	C₂×C₄×C₇	1	4	yes	—
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	4	no	—
274	C₄×Q₈	—	4	2	no	—
275	C₄×Q₈	C₂×C₄×C₇	1	4	yes	complemented
276	C₄×D₄	—	2	4	no	—
277	C₄×D₄	—	4	2	no	—
278	C₄×D₄	—	4	4	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	—	2	2	no	—
296	C₂×C₄²	C₂³×C₇	1	2	yes	—
297	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
298	#107	—	2	4	no	—
299	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
300	#103	—	2	2	no	—
301	#103	C₇×D₄	1	4	yes	complemented
302	#102	—	2	4	no	—
303	#102	—	4	2	no	—
304	#102	C₂×C₄×C₇	1	8	yes	complemented
305	#97	—	2	4	no	—
306	C₄×C₈	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	C₇×D₄	1	2	yes	—
310	C₂²×C₇	—	4	4	no	—
311	C₂²×C₇	—	4	4	no	—
312	C₂²×C₇	—	4	4	no	—
313	C₂²×C₇	—	8	2	no	—
314	C₂²×C₇	—	8	2	no	—
315	C₂²×C₇	—	8	2	no	—
316	C₂²×C₇	—	8	2	no	—
317	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
318	C₂²×C₇	D₄²	1	1	yes	characteristic
319	C₂²×C₇	#548	1	1	yes	characteristic
320	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₂²×C₇	C₂×#137	1	1	yes	characteristic
322	C₂²×C₇	#576	1	1	yes	characteristic
323	C₂²×C₇	#577	1	1	yes	characteristic
324	C₄×C₇	—	2	4	no	—
325	C₄×C₇	—	2	2	no	—
326	C₄×C₇	—	4	2	no	—
327	C₄×C₇	—	4	2	no	—
328	C₄×C₇	—	4	2	no	—
329	C₄×C₇	—	4	2	no	—
330	C₄×C₇	—	8	1	no	—
331	C₄×C₇	—	8	2	no	—
332	C₄×C₇	#440	1	4	yes	complemented
333	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
334	C₄×C₇	#520	1	1	yes	characteristic
335	C₄×C₇	#547	1	1	yes	characteristic
336	C₄×C₇	#553	1	1	yes	characteristic
337	C₂⁴	—	2	2	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
342	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
343	C₂×Q₈	C₇×#41	1	2	yes	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	—	8	1	no	—
354	C₂×D₄	—	8	1	no	—
355	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	—	4	4	no	—
363	C₂²×C₄	—	8	2	no	—
364	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
365	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
366	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
369	SD₁₆	—	2	16	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
378	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
379	C₄⋊C₄	—	2	4	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	4	4	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	8	1	no	—
391	C₄⋊C₄	—	8	2	no	—
392	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
393	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
394	C₂²⋊C₄	—	4	4	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₂²⋊C₄	—	8	2	no	—
397	C₄²	—	2	4	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	4	4	no	—
401	C₄²	—	4	2	no	—
402	C₄²	C₂²×C₄×C₇	1	2	yes	—
403	C₄²	C₇×#41	1	2	yes	—
404	C₂×C₇	—	8	4	no	—
405	C₂×C₇	#2361	1	1	yes	characteristic
406	C₂×C₇	C₂×#440	1	1	yes	characteristic
407	C₂×C₇	#2946	1	1	yes	characteristic
408	C₂×C₇	#3274	1	1	yes	characteristic
409	C₂×C₇	#3278	1	1	yes	characteristic
410	C₂×C₇	#3368	1	1	yes	characteristic
411	C₂×C₇	#3375	1	1	yes	characteristic
412	C₂³	—	2	4	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
420	Q₈	—	2	2	no	—
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	C₄×C₇×D₄	1	4	yes	complemented
424	D₄	—	2	4	no	—
425	D₄	—	2	4	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	4	8	no	—
429	D₄	—	8	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	8	8	no	—
454	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
455	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
456	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
457	C₂×C₄	C₇×#137	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
460	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
461	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
462	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
463	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
464	C₈	—	2	4	no	—
465	C₈	—	2	2	no	—
466	C₈	—	4	1	no	—
467	C₇	#23003	1	1	yes	characteristic, direct product factor, Sylow
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
476	C₂²	C₇×D₄²	1	1	yes	characteristic
477	C₂²	C₇×#548	1	1	yes	characteristic
478	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
479	C₂²	C₂×C₇×#137	1	1	yes	characteristic
480	C₂²	C₇×#576	1	1	yes	characteristic
481	C₂²	C₇×#577	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	2	4	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	8	2	no	—
489	C₄	—	8	1	no	—
490	C₄	C₇×#440	1	4	yes	complemented
491	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
492	C₄	C₇×#520	1	1	yes	characteristic
493	C₄	C₇×#547	1	1	yes	characteristic
494	C₄	C₇×#553	1	1	yes	characteristic
495	C₂	—	8	4	no	—
496	C₂	C₇×#2361	1	1	yes	characteristic
497	C₂	C₂×C₇×#440	1	1	yes	characteristic
498	C₂	C₇×#2946	1	1	yes	characteristic
499	C₂	C₇×#3274	1	1	yes	characteristic
500	C₂	C₇×#3278	1	1	yes	characteristic
501	C₂	C₇×#3368	1	1	yes	characteristic
502	C₂	C₇×#3375	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 ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#440	C₂	semidirect
4. C₇×Q₈×C₄⋊C₄	C₂	semidirect
6. C₇×#1947	C₂	semidirect
7. C₇×#1936	C₂	semidirect
8. C₇×#1880	C₂	semidirect
9. C₇×#1862	C₂	semidirect
10. C₇×#1770	C₂	semidirect
11. C₄×C₇×#103	C₂	semidirect
18. C₇×#460	C₄	semidirect
43. #23003	C₇	direct (N × H)
51. C₂×C₇×SD₁₆	C₂×C₄	semidirect
60. C₄×C₇×Q₈	C₂×C₄	semidirect
84. C₇×C₄⋊C₈	C₂×C₄	semidirect
86. C₇×#103	D₄	semidirect
89. C₇×#102	C₂×C₄	semidirect
96. C₂×#440	C₂×C₇	semidirect
98. Q₈×C₄⋊C₄	C₂×C₇	semidirect
100. #1947	C₂×C₇	semidirect
101. #1936	C₂×C₇	semidirect
102. #1880	C₂×C₇	semidirect
103. #1862	C₂×C₇	semidirect
104. #1770	C₂×C₇	semidirect
105. C₄×#103	C₂×C₇	semidirect
179. #460	C₄×C₇	semidirect
215. C₇×Q₈	C₄×D₄	semidirect
266. C₂×SD₁₆	C₂×C₄×C₇	semidirect
275. C₄×Q₈	C₂×C₄×C₇	semidirect
299. C₄⋊C₈	C₂×C₄×C₇	semidirect
301. #103	C₇×D₄	semidirect
304. #102	C₂×C₄×C₇	semidirect
332. C₄×C₇	#440	semidirect
423. Q₈	C₄×C₇×D₄	semidirect
467. C₇	#23003	direct (N × H)
490. 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
C₇×#2366	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1957	2
C₇×#1947	2
C₇×#1936	2
C₇×#1880	2
C₇×#1862	2
C₇×#1770	2
C₄×C₇×#103	2
#23003	7

Subgroups — C7×#23003 #13200219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23003 #13200219