Subgroups — C₇×#19837 #13197053

Order 1792.

628 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3373	C₂	1	1	yes	maximal, characteristic
2	C₇×#3365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3359	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1720	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1700	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1681	C₂	1	1	yes	maximal, characteristic
9	C₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1653	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1651	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1640	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1627	C₂	1	1	yes	maximal, characteristic
14	C₇×#1584	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×Q₁₆	—	2	1	no	—
17	C₇×#559	C₂²	1	1	yes	characteristic
18	C₇×#557	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
20	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
21	C₇×#498	C₂²	1	1	yes	characteristic
22	C₇×#494	C₂²	1	1	yes	characteristic
23	C₇×#489	—	2	1	no	—
24	C₇×#488	—	2	1	no	—
25	C₇×#487	—	2	1	no	—
26	C₇×#487	C₂²	1	1	yes	characteristic
27	C₇×#486	C₂²	1	1	yes	characteristic
28	C₇×#486	C₂²	1	1	yes	characteristic
29	C₇×#486	C₂²	1	1	yes	characteristic
30	C₇×#478	—	2	1	no	—
31	C₇×#476	—	2	1	no	—
32	C₇×#475	—	2	1	no	—
33	C₇×#475	C₂²	1	1	yes	characteristic
34	C₇×#475	C₂²	1	1	yes	characteristic
35	C₇×#475	C₂²	1	1	yes	characteristic
36	C₇×#467	—	2	1	no	—
37	C₇×#464	—	2	1	no	—
38	C₇×#463	—	2	1	no	—
39	C₇×#462	C₂²	1	1	yes	characteristic
40	C₇×#462	C₂²	1	1	yes	characteristic
41	C₇×#461	—	2	1	no	—
42	C₇×#461	C₂²	1	1	yes	characteristic
43	C₇×#460	C₂²	1	1	yes	characteristic
44	C₇×#459	C₂²	1	1	yes	characteristic
45	C₇×#451	—	2	1	no	—
46	C₇×#451	—	2	1	no	—
47	C₇×#446	C₂²	1	1	yes	characteristic
48	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
50	C₇×#442	—	2	1	no	—
51	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
52	C₄×C₇×SD₁₆	—	2	1	no	—
53	C₄×C₇×SD₁₆	—	2	1	no	—
54	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
55	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
56	C₇×#335	C₂²	1	1	yes	characteristic
57	C₇×#335	C₂²	1	1	yes	characteristic
58	C₇×#332	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#331	C₂²	1	1	yes	characteristic
62	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×#322	C₂²	1	1	yes	characteristic
67	#19837	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₇×Q₈	—	2	1	no	—
69	C₂²×C₇×Q₈	—	2	1	no	—
70	C₂×C₇×Q₁₆	—	2	1	no	—
71	C₂×C₇×Q₁₆	—	2	1	no	—
72	C₂×C₇×Q₁₆	—	2	1	no	—
73	C₂×C₇×Q₁₆	—	2	1	no	—
74	C₂×C₇×Q₁₆	—	4	4	no	—
75	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
76	C₂×C₇×SD₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
79	C₂×C₇×D₈	—	2	1	no	—
80	C₂²×C₇×C₈	—	2	1	no	—
81	C₇×C₄⋊Q₈	—	2	1	no	—
82	C₇×C₄⋊Q₈	—	2	1	no	—
83	C₇×C₄⋊Q₈	—	2	1	no	—
84	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₇×#127	C₂³	1	1	yes	characteristic
88	C₇×#125	—	2	1	no	—
89	C₇×#125	—	4	1	no	—
90	C₇×#124	—	2	1	no	—
91	C₇×#124	—	2	1	no	—
92	C₇×#123	—	4	1	no	—
93	C₇×C₂²⋊Q₈	—	2	1	no	—
94	C₇×C₂²⋊Q₈	—	2	1	no	—
95	C₇×C₂²⋊Q₈	—	4	1	no	—
96	C₇×C₂²⋊Q₈	—	4	1	no	—
97	C₄×C₇×Q₈	—	2	1	no	—
98	C₄×C₇×Q₈	—	2	1	no	—
99	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
100	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
101	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
102	C₄×C₇×D₄	—	2	1	no	—
103	C₄×C₇×D₄	—	2	1	no	—
104	C₄×C₇×D₄	—	2	1	no	—
105	C₄×C₇×D₄	—	2	1	no	—
106	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
107	C₇×#107	D₄	1	1	yes	characteristic
108	C₇×#106	—	2	1	no	—
109	C₇×#106	—	2	1	no	—
110	C₇×#106	—	2	1	no	—
111	C₇×#106	—	4	1	no	—
112	C₇×#106	D₄	1	1	yes	characteristic
113	C₇×C₄⋊C₈	—	2	1	no	—
114	C₇×C₄⋊C₈	—	2	1	no	—
115	C₇×C₄⋊C₈	—	2	1	no	—
116	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₇×#103	—	2	1	no	—
121	C₇×#103	—	2	1	no	—
122	C₇×#103	—	2	1	no	—
123	C₇×#103	—	2	1	no	—
124	C₇×#103	—	2	1	no	—
125	C₇×#103	—	2	1	no	—
126	C₇×#103	—	2	1	no	—
127	C₇×#103	—	2	1	no	—
128	C₇×#103	D₄	1	1	yes	characteristic
129	C₇×#103	D₄	1	1	yes	characteristic
130	C₇×#103	D₄	1	1	yes	characteristic
131	C₇×#102	—	2	1	no	—
132	C₇×#102	—	2	1	no	—
133	C₇×#102	—	2	1	no	—
134	C₇×#102	—	2	1	no	—
135	C₇×#102	—	2	1	no	—
136	C₇×#102	—	2	1	no	—
137	C₇×#102	—	4	1	no	—
138	C₇×#102	D₄	1	1	yes	characteristic
139	C₇×C₂²⋊C₈	—	2	1	no	—
140	C₇×#97	—	2	1	no	—
141	C₄×C₇×C₈	—	2	1	no	—
142	C₄×C₇×C₈	—	2	1	no	—
143	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
146	#3373	C₂×C₇	1	1	yes	characteristic
147	#3365	C₂×C₇	1	1	yes	characteristic, complemented
148	#3359	C₂×C₇	1	1	yes	characteristic, complemented
149	D₄×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
150	#1720	C₂×C₇	1	1	yes	characteristic, complemented
151	#1700	C₂×C₇	1	1	yes	characteristic, complemented
152	#1682	C₂×C₇	1	1	yes	characteristic, complemented
153	#1681	C₂×C₇	1	1	yes	characteristic
154	#1670	C₂×C₇	1	1	yes	characteristic, complemented
155	#1653	C₂×C₇	1	1	yes	characteristic, complemented
156	#1651	C₂×C₇	1	1	yes	characteristic, complemented
157	#1640	C₂×C₇	1	1	yes	characteristic, complemented
158	#1627	C₂×C₇	1	1	yes	characteristic
159	#1584	C₂×C₇	1	1	yes	characteristic, complemented
160	SD₁₆⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
161	C₂×C₇×Q₈	—	2	1	no	—
162	C₂×C₇×Q₈	—	2	1	no	—
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	4	1	no	—
165	C₂×C₇×Q₈	—	4	1	no	—
166	C₂×C₇×Q₈	—	4	2	no	—
167	C₂×C₇×Q₈	—	4	2	no	—
168	C₂×C₇×Q₈	—	8	1	no	—
169	C₂×C₇×Q₈	—	8	1	no	—
170	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
171	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₇×D₄	—	2	1	no	—
174	C₂×C₇×D₄	—	2	1	no	—
175	C₂×C₇×D₄	—	2	1	no	—
176	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₇	—	2	1	no	—
178	C₂²×C₄×C₇	—	4	1	no	—
179	C₂²×C₄×C₇	—	4	1	no	—
180	C₂²×C₄×C₇	—	4	1	no	—
181	C₇×Q₁₆	—	2	2	no	—
182	C₇×Q₁₆	—	4	2	no	—
183	C₇×Q₁₆	—	4	2	no	—
184	C₇×Q₁₆	—	4	2	no	—
185	C₇×Q₁₆	—	8	1	no	—
186	C₇×SD₁₆	—	2	2	no	—
187	C₇×SD₁₆	—	4	2	no	—
188	C₇×SD₁₆	—	4	2	no	—
189	C₇×D₈	—	4	2	no	—
190	C₂×C₇×C₈	—	2	1	no	—
191	C₂×C₇×C₈	—	2	1	no	—
192	C₂×C₇×C₈	—	2	1	no	—
193	C₂×C₇×C₈	—	2	1	no	—
194	C₂×C₇×C₈	—	2	1	no	—
195	C₂×C₇×C₈	—	4	2	no	—
196	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
198	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
200	C₇×C₄⋊C₄	—	2	1	no	—
201	C₇×C₄⋊C₄	—	2	1	no	—
202	C₇×C₄⋊C₄	—	2	1	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	—	2	1	no	—
205	C₇×C₄⋊C₄	—	2	1	no	—
206	C₇×C₄⋊C₄	—	2	1	no	—
207	C₇×C₄⋊C₄	—	2	1	no	—
208	C₇×C₄⋊C₄	—	2	1	no	—
209	C₇×C₄⋊C₄	—	4	1	no	—
210	C₇×C₄⋊C₄	—	4	1	no	—
211	C₇×C₄⋊C₄	—	4	1	no	—
212	C₇×C₄⋊C₄	—	4	1	no	—
213	C₇×C₄⋊C₄	—	4	1	no	—
214	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
215	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
216	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
217	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
218	C₇×C₂²⋊C₄	—	2	1	no	—
219	C₇×C₂²⋊C₄	—	4	1	no	—
220	C₇×C₂²⋊C₄	—	4	1	no	—
221	C₇×C₂²⋊C₄	—	4	1	no	—
222	C₇×C₂²⋊C₄	—	4	1	no	—
223	C₇×C₂²⋊C₄	—	4	1	no	—
224	C₄²×C₇	—	2	1	no	—
225	C₄²×C₇	—	2	1	no	—
226	C₄²×C₇	—	2	1	no	—
227	C₄²×C₇	—	4	1	no	—
228	C₄²×C₇	C₂⁴	1	1	yes	characteristic
229	C₂²×Q₁₆	—	2	1	no	—
230	#559	C₂²×C₇	1	1	yes	characteristic
231	#557	C₂²×C₇	1	1	yes	characteristic
232	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
233	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
234	#498	C₂²×C₇	1	1	yes	characteristic
235	#494	C₂²×C₇	1	1	yes	characteristic
236	#489	—	2	1	no	—
237	#488	—	2	1	no	—
238	#487	—	2	1	no	—
239	#487	C₂²×C₇	1	1	yes	characteristic
240	#486	C₂²×C₇	1	1	yes	characteristic
241	#486	C₂²×C₇	1	1	yes	characteristic
242	#486	C₂²×C₇	1	1	yes	characteristic
243	#478	—	2	1	no	—
244	#476	—	2	1	no	—
245	#475	—	2	1	no	—
246	#475	C₂²×C₇	1	1	yes	characteristic
247	#475	C₂²×C₇	1	1	yes	characteristic
248	#475	C₂²×C₇	1	1	yes	characteristic
249	#467	—	2	1	no	—
250	#464	—	2	1	no	—
251	#463	—	2	1	no	—
252	#462	C₂²×C₇	1	1	yes	characteristic
253	#462	C₂²×C₇	1	1	yes	characteristic
254	#461	—	2	1	no	—
255	#461	C₂²×C₇	1	1	yes	characteristic
256	#460	C₂²×C₇	1	1	yes	characteristic
257	#459	C₂²×C₇	1	1	yes	characteristic
258	#451	—	2	1	no	—
259	#451	—	2	1	no	—
260	#446	C₂²×C₇	1	1	yes	characteristic
261	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
262	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
263	#442	—	2	1	no	—
264	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
265	C₄×SD₁₆	—	2	1	no	—
266	C₄×SD₁₆	—	2	1	no	—
267	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
268	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
269	#335	C₂²×C₇	1	1	yes	characteristic
270	#335	C₂²×C₇	1	1	yes	characteristic
271	#332	C₂²×C₇	1	1	yes	characteristic
272	#332	C₂²×C₇	1	1	yes	characteristic
273	#332	C₂²×C₇	1	1	yes	characteristic
274	#331	C₂²×C₇	1	1	yes	characteristic
275	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
276	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
277	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
278	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
279	#322	C₂²×C₇	1	1	yes	characteristic
280	C₂³×C₇	—	2	1	no	—
281	C₂³×C₇	—	4	1	no	—
282	C₇×Q₈	—	2	1	no	—
283	C₇×Q₈	—	2	1	no	—
284	C₇×Q₈	—	2	1	no	—
285	C₇×Q₈	—	2	1	no	—
286	C₇×Q₈	—	2	1	no	—
287	C₇×Q₈	—	4	1	no	—
288	C₇×Q₈	—	4	1	no	—
289	C₇×Q₈	—	4	1	no	—
290	C₇×Q₈	—	4	1	no	—
291	C₇×Q₈	—	4	1	no	—
292	C₇×Q₈	—	4	1	no	—
293	C₇×Q₈	—	8	2	no	—
294	C₇×Q₈	—	8	2	no	—
295	C₇×Q₈	AGL(1,ℤ/8)	1	2	yes	—
296	C₇×D₄	—	2	1	no	—
297	C₇×D₄	—	4	1	no	—
298	C₇×D₄	—	4	1	no	—
299	C₇×D₄	—	4	1	no	—
300	C₇×D₄	—	4	1	no	—
301	C₇×D₄	—	8	1	no	—
302	C₇×D₄	#135	1	2	yes	—
303	C₂×C₄×C₇	—	2	1	no	—
304	C₂×C₄×C₇	—	2	1	no	—
305	C₂×C₄×C₇	—	2	1	no	—
306	C₂×C₄×C₇	—	2	1	no	—
307	C₂×C₄×C₇	—	2	1	no	—
308	C₂×C₄×C₇	—	2	1	no	—
309	C₂×C₄×C₇	—	2	1	no	—
310	C₂×C₄×C₇	—	4	1	no	—
311	C₂×C₄×C₇	—	4	1	no	—
312	C₂×C₄×C₇	—	4	1	no	—
313	C₂×C₄×C₇	—	4	1	no	—
314	C₂×C₄×C₇	—	4	1	no	—
315	C₂×C₄×C₇	—	8	2	no	—
316	C₂×C₄×C₇	—	8	2	no	—
317	C₂×C₄×C₇	—	16	1	no	—
318	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
321	C₇×C₈	—	2	1	no	—
322	C₇×C₈	—	2	1	no	—
323	C₇×C₈	—	2	1	no	—
324	C₇×C₈	—	2	1	no	—
325	C₇×C₈	—	4	1	no	—
326	C₇×C₈	—	4	1	no	—
327	C₇×C₈	—	4	1	no	—
328	C₇×C₈	—	4	1	no	—
329	C₇×C₈	—	4	1	no	—
330	C₂²×Q₈	—	2	1	no	—
331	C₂²×Q₈	—	2	1	no	—
332	C₂×Q₁₆	—	2	1	no	—
333	C₂×Q₁₆	—	2	1	no	—
334	C₂×Q₁₆	—	2	1	no	—
335	C₂×Q₁₆	—	2	1	no	—
336	C₂×Q₁₆	—	4	4	no	—
337	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
338	C₂×SD₁₆	—	2	1	no	—
339	C₂×SD₁₆	—	2	1	no	—
340	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
341	C₂×D₈	—	2	1	no	—
342	C₂²×C₈	—	2	1	no	—
343	C₄⋊Q₈	—	2	1	no	—
344	C₄⋊Q₈	—	2	1	no	—
345	C₄⋊Q₈	—	2	1	no	—
346	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
349	#127	C₂³×C₇	1	1	yes	characteristic
350	#125	—	2	1	no	—
351	#125	—	4	1	no	—
352	#124	—	2	1	no	—
353	#124	—	2	1	no	—
354	#123	—	4	1	no	—
355	C₂²⋊Q₈	—	2	1	no	—
356	C₂²⋊Q₈	—	2	1	no	—
357	C₂²⋊Q₈	—	4	1	no	—
358	C₂²⋊Q₈	—	4	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
362	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
363	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
369	#107	C₇×D₄	1	1	yes	characteristic
370	#106	—	2	1	no	—
371	#106	—	2	1	no	—
372	#106	—	2	1	no	—
373	#106	—	4	1	no	—
374	#106	C₇×D₄	1	1	yes	characteristic
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
382	#103	—	2	1	no	—
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#103	—	2	1	no	—
386	#103	—	2	1	no	—
387	#103	—	2	1	no	—
388	#103	—	2	1	no	—
389	#103	—	2	1	no	—
390	#103	C₇×D₄	1	1	yes	characteristic
391	#103	C₇×D₄	1	1	yes	characteristic
392	#103	C₇×D₄	1	1	yes	characteristic
393	#102	—	2	1	no	—
394	#102	—	2	1	no	—
395	#102	—	2	1	no	—
396	#102	—	2	1	no	—
397	#102	—	2	1	no	—
398	#102	—	2	1	no	—
399	#102	—	4	1	no	—
400	#102	C₇×D₄	1	1	yes	characteristic
401	C₂²⋊C₈	—	2	1	no	—
402	#97	—	2	1	no	—
403	C₄×C₈	—	2	1	no	—
404	C₄×C₈	—	2	1	no	—
405	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
407	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
408	C₂²×C₇	—	2	2	no	—
409	C₂²×C₇	—	4	1	no	—
410	C₂²×C₇	—	4	1	no	—
411	C₂²×C₇	—	8	1	no	—
412	C₂²×C₇	—	8	1	no	—
413	C₂²×C₇	—	8	1	no	—
414	C₂²×C₇	D₄²	1	1	yes	characteristic
415	C₄×C₇	—	2	2	no	—
416	C₄×C₇	—	2	1	no	—
417	C₄×C₇	—	4	1	no	—
418	C₄×C₇	—	4	1	no	—
419	C₄×C₇	—	4	1	no	—
420	C₄×C₇	—	4	1	no	—
421	C₄×C₇	—	4	1	no	—
422	C₄×C₇	—	4	1	no	—
423	C₄×C₇	—	8	1	no	—
424	C₄×C₇	—	8	1	no	—
425	C₄×C₇	—	8	1	no	—
426	C₄×C₇	C₂×#135	1	1	yes	characteristic
427	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
428	C₄×C₇	#577	1	1	yes	characteristic
429	C₄×C₇	#578	1	1	yes	characteristic
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	4	1	no	—
434	C₂×Q₈	—	4	1	no	—
435	C₂×Q₈	—	4	2	no	—
436	C₂×Q₈	—	4	2	no	—
437	C₂×Q₈	—	8	1	no	—
438	C₂×Q₈	—	8	1	no	—
439	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
440	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
441	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	—	4	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	Q₁₆	—	2	2	no	—
451	Q₁₆	—	4	2	no	—
452	Q₁₆	—	4	2	no	—
453	Q₁₆	—	4	2	no	—
454	Q₁₆	—	8	1	no	—
455	SD₁₆	—	2	2	no	—
456	SD₁₆	—	4	2	no	—
457	SD₁₆	—	4	2	no	—
458	D₈	—	4	2	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	4	2	no	—
465	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
466	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
467	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
468	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
484	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
485	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
486	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	C₂⁴×C₇	1	1	yes	characteristic
498	C₂×C₇	—	4	2	no	—
499	C₂×C₇	—	16	1	no	—
500	C₂×C₇	#3265	1	1	yes	characteristic
501	C₂×C₇	#3285	1	1	yes	characteristic
502	C₂×C₇	#3289	1	1	yes	characteristic
503	C₂³	—	2	1	no	—
504	C₂³	—	4	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	Q₈	—	4	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	8	2	no	—
517	Q₈	—	8	2	no	—
518	Q₈	C₇×AGL(1,ℤ/8)	1	2	yes	—
519	D₄	—	2	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	8	1	no	—
525	D₄	C₇×#135	1	2	yes	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	—	16	1	no	—
541	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
542	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
543	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₇	#19837	1	1	yes	characteristic, direct product factor, Sylow
554	C₂²	—	2	2	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	1	no	—
560	C₂²	C₇×D₄²	1	1	yes	characteristic
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	8	1	no	—
570	C₄	—	8	1	no	—
571	C₄	—	8	1	no	—
572	C₄	C₂×C₇×#135	1	1	yes	characteristic
573	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
574	C₄	C₇×#577	1	1	yes	characteristic
575	C₄	C₇×#578	1	1	yes	characteristic
576	C₂	—	4	2	no	—
577	C₂	—	16	1	no	—
578	C₂	C₇×#3265	1	1	yes	characteristic
579	C₂	C₇×#3285	1	1	yes	characteristic
580	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#459	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3365	C₂	semidirect
3. C₇×#3359	C₂	semidirect
4. C₇×D₄×Q₁₆	C₂	semidirect
5. C₇×#1720	C₂	semidirect
6. C₇×#1700	C₂	semidirect
7. C₇×#1682	C₂	semidirect
9. C₇×#1670	C₂	semidirect
10. C₇×#1653	C₂	semidirect
11. C₇×#1651	C₂	semidirect
12. C₇×#1640	C₂	semidirect
14. C₇×#1584	C₂	semidirect
15. C₇×SD₁₆⋊C₈	C₂	semidirect
67. #19837	C₇	direct (N × H)
147. #3365	C₂×C₇	semidirect
148. #3359	C₂×C₇	semidirect
149. D₄×Q₁₆	C₂×C₇	semidirect
150. #1720	C₂×C₇	semidirect
151. #1700	C₂×C₇	semidirect
152. #1682	C₂×C₇	semidirect
154. #1670	C₂×C₇	semidirect
155. #1653	C₂×C₇	semidirect
156. #1651	C₂×C₇	semidirect
157. #1640	C₂×C₇	semidirect
159. #1584	C₂×C₇	semidirect
160. SD₁₆⋊C₈	C₂×C₇	semidirect
553. C₇	#19837	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3373	2
C₇×#3365	2
C₇×#3359	2
C₇×D₄×Q₁₆	2
C₇×#1720	2
C₇×#1700	2
C₇×#1682	2
C₇×#1681	2
C₇×#1670	2
C₇×#1653	2
C₇×#1651	2
C₇×#1640	2
C₇×#1627	2
C₇×#1584	2
C₇×SD₁₆⋊C₈	2
#19837	7

Subgroups — C7×#19837 #13197053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19837 #13197053