Subgroups — C₇×#23220 #13200436

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#486	C₂	1	1	yes	maximal, characteristic
2	C₇×#2366	C₂	1	1	yes	maximal, characteristic
3	C₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇×#1936	C₂	1	2	yes	maximal, complemented
7	C₇×#1926	C₂	1	2	yes	maximal, complemented
8	C₇×#1864	C₂	1	2	yes	maximal, complemented
9	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1762	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	—	2	1	no	—
15	C₂×C₄×C₇×D₄	—	2	1	no	—
16	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
17	C₇×#486	C₄	1	8	yes	complemented
18	C₂×C₇×#106	C₂²	1	1	yes	characteristic
19	C₂×C₇×#106	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#102	C₂²	1	2	yes	—
22	C₂×C₇×#102	C₂²	1	2	yes	—
23	C₂×C₇×#102	C₂²	1	2	yes	—
24	C₂×C₇×#102	C₂²	1	2	yes	—
25	C₂×C₇×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	C₇×#391	—	2	1	no	—
28	C₇×#390	—	2	1	no	—
29	C₇×#390	C₂²	1	2	yes	—
30	C₇×#389	C₂²	1	2	yes	—
31	C₇×#387	—	2	2	no	—
32	C₇×#386	—	2	2	no	—
33	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₇×#340	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	1	yes	characteristic
40	C₇×#340	C₂²	1	1	yes	characteristic
41	C₇×#336	C₂²	1	2	yes	—
42	#23220	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×Q₈	—	2	1	no	—
44	C₂²×C₇×D₄	—	2	1	no	—
45	C₂²×C₇×D₄	—	2	1	no	—
46	C₂²×C₇×D₄	C₂³	1	2	yes	—
47	C₂³×C₄×C₇	—	2	2	no	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₇×C₄⋊Q₈	—	2	2	no	—
52	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
53	C₇×#127	—	2	2	no	—
54	C₇×#127	C₂×C₄	1	4	yes	—
55	C₄×C₇×Q₈	—	4	4	no	—
56	C₄×C₇×D₄	—	2	4	no	—
57	C₄×C₇×D₄	—	4	2	no	—
58	C₄×C₇×D₄	—	4	4	no	—
59	C₂×C₇×C₄⋊C₄	—	2	1	no	—
60	C₂×C₇×C₄⋊C₄	—	2	2	no	—
61	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
66	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
69	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
70	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₄²×C₇	—	2	1	no	—
72	C₂×C₄²×C₇	—	2	2	no	—
73	C₂×C₄²×C₇	—	2	1	no	—
74	C₂×C₄²×C₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
76	C₇×#106	—	2	2	no	—
77	C₇×#106	—	2	2	no	—
78	C₇×C₄⋊C₈	—	2	2	no	—
79	C₇×#102	—	2	4	no	—
80	C₇×#102	—	4	2	no	—
81	C₇×#102	C₂×C₄	1	8	yes	complemented
82	C₇×#102	C₂×C₄	1	8	yes	complemented
83	C₇×#97	—	4	2	no	—
84	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
85	C₇×#95	—	2	1	no	—
86	C₇×#95	—	2	1	no	—
87	C₇×#95	—	2	2	no	—
88	C₇×#95	—	2	2	no	—
89	C₇×#95	D₄	1	2	yes	—
90	C₂×#486	C₂×C₇	1	1	yes	characteristic
91	#2366	C₂×C₇	1	1	yes	characteristic
92	#2365	C₂×C₇	1	1	yes	characteristic, complemented
93	#1988	C₂×C₇	1	1	yes	characteristic, complemented
94	#1982	C₂×C₇	1	1	yes	characteristic
95	#1936	C₂×C₇	1	2	yes	complemented
96	#1926	C₂×C₇	1	2	yes	complemented
97	#1864	C₂×C₇	1	2	yes	complemented
98	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
99	#1770	C₂×C₇	1	1	yes	characteristic, complemented
100	#1762	C₂×C₇	1	2	yes	complemented
101	C₂⁴×C₇	—	2	2	no	—
102	C₂×C₇×Q₈	—	2	4	no	—
103	C₂×C₇×Q₈	—	4	2	no	—
104	C₂×C₇×Q₈	—	4	2	no	—
105	C₂×C₇×D₄	—	2	4	no	—
106	C₂×C₇×D₄	—	2	4	no	—
107	C₂×C₇×D₄	—	2	2	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	—	4	2	no	—
113	C₂×C₇×D₄	—	4	2	no	—
114	C₂×C₇×D₄	—	8	1	no	—
115	C₂×C₇×D₄	—	8	1	no	—
116	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
117	C₂²×C₄×C₇	—	2	1	no	—
118	C₂²×C₄×C₇	—	2	2	no	—
119	C₂²×C₄×C₇	—	2	1	no	—
120	C₂²×C₄×C₇	—	2	1	no	—
121	C₂²×C₄×C₇	—	2	1	no	—
122	C₂²×C₄×C₇	—	4	4	no	—
123	C₂²×C₄×C₇	—	4	4	no	—
124	C₂²×C₄×C₇	—	8	2	no	—
125	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
130	C₂×C₇×C₈	—	2	1	no	—
131	C₂×C₇×C₈	—	2	1	no	—
132	C₂×C₇×C₈	—	2	1	no	—
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	2	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	—	2	2	no	—
137	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
138	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
139	C₇×C₄⋊C₄	—	2	4	no	—
140	C₇×C₄⋊C₄	—	2	2	no	—
141	C₇×C₄⋊C₄	—	2	2	no	—
142	C₇×C₄⋊C₄	—	2	2	no	—
143	C₇×C₄⋊C₄	—	2	1	no	—
144	C₇×C₄⋊C₄	—	2	1	no	—
145	C₇×C₄⋊C₄	—	2	1	no	—
146	C₇×C₄⋊C₄	—	2	1	no	—
147	C₇×C₄⋊C₄	—	2	1	no	—
148	C₇×C₄⋊C₄	—	2	1	no	—
149	C₇×C₄⋊C₄	—	4	4	no	—
150	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
151	C₇×C₄⋊C₄	#41	1	2	yes	—
152	C₇×C₄⋊C₄	#41	1	2	yes	—
153	C₇×C₂²⋊C₄	—	4	4	no	—
154	C₇×C₂²⋊C₄	—	4	4	no	—
155	C₇×C₂²⋊C₄	—	8	2	no	—
156	C₄²×C₇	—	2	2	no	—
157	C₄²×C₇	—	2	2	no	—
158	C₄²×C₇	—	4	4	no	—
159	C₄²×C₇	—	4	2	no	—
160	C₄²×C₇	—	4	2	no	—
161	C₄²×C₇	C₂²×C₄	1	2	yes	—
162	C₄²×C₇	#41	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
164	C₂×#127	C₂²×C₇	1	1	yes	characteristic
165	C₂×C₄×Q₈	—	2	1	no	—
166	C₂×C₄×D₄	—	2	1	no	—
167	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
168	#486	C₄×C₇	1	8	yes	complemented
169	C₂×#106	C₂²×C₇	1	1	yes	characteristic
170	C₂×#106	C₂²×C₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
172	C₂×#102	C₂²×C₇	1	2	yes	—
173	C₂×#102	C₂²×C₇	1	2	yes	—
174	C₂×#102	C₂²×C₇	1	2	yes	—
175	C₂×#102	C₂²×C₇	1	2	yes	—
176	C₂×#97	C₂²×C₇	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
178	#391	—	2	1	no	—
179	#390	—	2	1	no	—
180	#390	C₂²×C₇	1	2	yes	—
181	#389	C₂²×C₇	1	2	yes	—
182	#387	—	2	2	no	—
183	#386	—	2	2	no	—
184	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
185	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
186	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
187	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
188	#340	C₂²×C₇	1	2	yes	—
189	#340	C₂²×C₇	1	2	yes	—
190	#340	C₂²×C₇	1	1	yes	characteristic
191	#340	C₂²×C₇	1	1	yes	characteristic
192	#336	C₂²×C₇	1	2	yes	—
193	C₂³×C₇	—	2	4	no	—
194	C₂³×C₇	—	4	2	no	—
195	C₂³×C₇	—	4	2	no	—
196	C₂³×C₇	—	4	2	no	—
197	C₂³×C₇	—	4	2	no	—
198	C₂³×C₇	—	4	2	no	—
199	C₂³×C₇	—	4	2	no	—
200	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
201	C₇×Q₈	—	4	8	no	—
202	C₇×D₄	—	2	4	no	—
203	C₇×D₄	—	2	4	no	—
204	C₇×D₄	—	4	2	no	—
205	C₇×D₄	—	4	2	no	—
206	C₇×D₄	—	4	8	no	—
207	C₇×D₄	—	8	4	no	—
208	C₂×C₄×C₇	—	2	1	no	—
209	C₂×C₄×C₇	—	2	1	no	—
210	C₂×C₄×C₇	—	2	4	no	—
211	C₂×C₄×C₇	—	2	2	no	—
212	C₂×C₄×C₇	—	2	2	no	—
213	C₂×C₄×C₇	—	2	2	no	—
214	C₂×C₄×C₇	—	2	2	no	—
215	C₂×C₄×C₇	—	2	2	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	2	2	no	—
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	4	1	no	—
220	C₂×C₄×C₇	—	4	2	no	—
221	C₂×C₄×C₇	—	4	1	no	—
222	C₂×C₄×C₇	—	4	2	no	—
223	C₂×C₄×C₇	—	4	1	no	—
224	C₂×C₄×C₇	—	4	1	no	—
225	C₂×C₄×C₇	—	4	1	no	—
226	C₂×C₄×C₇	—	4	1	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	1	no	—
229	C₂×C₄×C₇	—	4	1	no	—
230	C₂×C₄×C₇	—	4	1	no	—
231	C₂×C₄×C₇	—	8	8	no	—
232	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
233	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
234	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
235	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
236	C₂×C₄×C₇	C₂³×C₄	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	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
242	C₇×C₈	—	2	2	no	—
243	C₇×C₈	—	2	2	no	—
244	C₇×C₈	—	4	2	no	—
245	C₂²×Q₈	—	2	1	no	—
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	C₂³×C₇	1	2	yes	—
249	C₂³×C₄	—	2	2	no	—
250	C₂²×C₈	C₂³×C₇	1	2	yes	—
251	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
253	C₄⋊Q₈	—	2	2	no	—
254	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
255	#127	—	2	2	no	—
256	#127	C₂×C₄×C₇	1	4	yes	—
257	C₄×Q₈	—	4	4	no	—
258	C₄×D₄	—	2	4	no	—
259	C₄×D₄	—	4	2	no	—
260	C₄×D₄	—	4	4	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₄²	—	2	1	no	—
274	C₂×C₄²	—	2	2	no	—
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	C₂³×C₇	1	2	yes	—
277	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
278	#106	—	2	2	no	—
279	#106	—	2	2	no	—
280	C₄⋊C₈	—	2	2	no	—
281	#102	—	2	4	no	—
282	#102	—	4	2	no	—
283	#102	C₂×C₄×C₇	1	8	yes	complemented
284	#102	C₂×C₄×C₇	1	8	yes	complemented
285	#97	—	4	2	no	—
286	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
287	#95	—	2	1	no	—
288	#95	—	2	1	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	C₇×D₄	1	2	yes	—
292	C₂²×C₇	—	4	4	no	—
293	C₂²×C₇	—	4	4	no	—
294	C₂²×C₇	—	4	4	no	—
295	C₂²×C₇	—	8	2	no	—
296	C₂²×C₇	—	8	2	no	—
297	C₂²×C₇	—	8	2	no	—
298	C₂²×C₇	—	8	2	no	—
299	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
300	C₂²×C₇	#546	1	1	yes	characteristic
301	C₂²×C₇	#548	1	1	yes	characteristic
302	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂²×C₇	#577	1	1	yes	characteristic
305	C₂²×C₇	#577	1	1	yes	characteristic
306	C₄×C₇	—	2	2	no	—
307	C₄×C₇	—	4	2	no	—
308	C₄×C₇	—	4	2	no	—
309	C₄×C₇	—	4	4	no	—
310	C₄×C₇	—	4	2	no	—
311	C₄×C₇	—	4	2	no	—
312	C₄×C₇	—	8	1	no	—
313	C₄×C₇	—	8	1	no	—
314	C₄×C₇	#442	1	4	yes	—
315	C₄×C₇	C₄×#41	1	1	yes	characteristic
316	C₄×C₇	#520	1	1	yes	characteristic
317	C₄×C₇	#556	1	1	yes	characteristic
318	C₄×C₇	#559	1	1	yes	characteristic
319	C₂⁴	—	2	2	no	—
320	C₂×Q₈	—	2	4	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	4	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	4	2	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	8	1	no	—
333	C₂×D₄	—	8	1	no	—
334	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	4	4	no	—
341	C₂²×C₄	—	4	4	no	—
342	C₂²×C₄	—	8	2	no	—
343	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
344	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
345	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
346	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
347	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
356	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
357	C₄⋊C₄	—	2	4	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
369	C₄⋊C₄	C₇×#41	1	2	yes	—
370	C₄⋊C₄	C₇×#41	1	2	yes	—
371	C₂²⋊C₄	—	4	4	no	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₂²⋊C₄	—	8	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	4	4	no	—
377	C₄²	—	4	2	no	—
378	C₄²	—	4	2	no	—
379	C₄²	C₂²×C₄×C₇	1	2	yes	—
380	C₄²	C₇×#41	1	2	yes	—
381	C₂×C₇	—	8	4	no	—
382	C₂×C₇	#2367	1	1	yes	characteristic
383	C₂×C₇	C₂×#442	1	1	yes	characteristic
384	C₂×C₇	#2949	1	1	yes	characteristic
385	C₂×C₇	#3324	1	1	yes	characteristic
386	C₂×C₇	#3332	1	1	yes	characteristic
387	C₂×C₇	#3368	1	1	yes	characteristic
388	C₂×C₇	#3373	1	1	yes	characteristic
389	C₂³	—	2	4	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
397	Q₈	—	4	8	no	—
398	D₄	—	2	4	no	—
399	D₄	—	2	4	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	8	no	—
403	D₄	—	8	4	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	8	8	no	—
428	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
429	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
430	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
431	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
432	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
433	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
434	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
435	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
436	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
437	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
438	C₈	—	2	2	no	—
439	C₈	—	2	2	no	—
440	C₈	—	4	2	no	—
441	C₇	#23220	1	1	yes	characteristic, direct product factor, Sylow
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
450	C₂²	C₇×#546	1	1	yes	characteristic
451	C₂²	C₇×#548	1	1	yes	characteristic
452	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
453	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
454	C₂²	C₇×#577	1	1	yes	characteristic
455	C₂²	C₇×#577	1	1	yes	characteristic
456	C₄	—	2	2	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	8	1	no	—
463	C₄	—	8	1	no	—
464	C₄	C₇×#442	1	4	yes	—
465	C₄	C₄×C₇×#41	1	1	yes	characteristic
466	C₄	C₇×#520	1	1	yes	characteristic
467	C₄	C₇×#556	1	1	yes	characteristic
468	C₄	C₇×#559	1	1	yes	characteristic
469	C₂	—	8	4	no	—
470	C₂	C₇×#2367	1	1	yes	characteristic
471	C₂	C₂×C₇×#442	1	1	yes	characteristic
472	C₂	C₇×#2949	1	1	yes	characteristic
473	C₂	C₇×#3324	1	1	yes	characteristic
474	C₂	C₇×#3332	1	1	yes	characteristic
475	C₂	C₇×#3368	1	1	yes	characteristic
476	C₂	C₇×#3373	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
3. C₇×#2365	C₂	semidirect
4. C₇×#1988	C₂	semidirect
6. C₇×#1936	C₂	semidirect
7. C₇×#1926	C₂	semidirect
8. C₇×#1864	C₂	semidirect
9. C₄×C₇×#106	C₂	semidirect
10. C₇×#1770	C₂	semidirect
11. C₇×#1762	C₂	semidirect
17. C₇×#486	C₄	semidirect
42. #23220	C₇	direct (N × H)
52. C₇×C₄⋊Q₈	C₂×C₄	semidirect
81. C₇×#102	C₂×C₄	semidirect
82. C₇×#102	C₂×C₄	semidirect
84. C₄×C₇×C₈	C₂×C₄	semidirect
92. #2365	C₂×C₇	semidirect
93. #1988	C₂×C₇	semidirect
95. #1936	C₂×C₇	semidirect
96. #1926	C₂×C₇	semidirect
97. #1864	C₂×C₇	semidirect
98. C₄×#106	C₂×C₇	semidirect
99. #1770	C₂×C₇	semidirect
100. #1762	C₂×C₇	semidirect
168. #486	C₄×C₇	semidirect
254. C₄⋊Q₈	C₂×C₄×C₇	semidirect
283. #102	C₂×C₄×C₇	semidirect
284. #102	C₂×C₄×C₇	semidirect
286. C₄×C₈	C₂×C₄×C₇	semidirect
441. C₇	#23220	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×#486	2
C₇×#2366	2
C₇×#2365	2
C₇×#1988	2
C₇×#1982	2
C₇×#1936	2
C₇×#1926	2
C₇×#1864	2
C₄×C₇×#106	2
C₇×#1770	2
C₇×#1762	2
#23220	7

Subgroups — C7×#23220 #13200436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23220 #13200436