Subgroups — C₇×#19812 #13197028

Order 1792.

616 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3386	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×SD₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#3363	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1692	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1683	C₂	1	1	yes	maximal, characteristic
8	C₇×#1674	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1649	C₂	1	1	yes	maximal, characteristic
11	C₇×#1648	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1645	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1637	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
16	C₂²×C₇×Q₁₆	—	2	1	no	—
17	C₇×Q₈²	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×#501	—	2	1	no	—
22	C₇×#500	C₂²	1	1	yes	characteristic
23	C₇×#500	C₂²	1	1	yes	characteristic
24	C₇×#498	—	2	1	no	—
25	C₇×#498	C₂²	1	1	yes	characteristic
26	C₇×#494	C₂²	1	1	yes	characteristic
27	C₇×#487	—	2	1	no	—
28	C₇×#487	C₂²	1	1	yes	characteristic
29	C₇×#487	C₂²	1	1	yes	characteristic
30	C₇×#478	—	2	1	no	—
31	C₇×#477	—	2	1	no	—
32	C₇×#477	—	2	1	no	—
33	C₇×#477	C₂²	1	1	yes	characteristic
34	C₇×#476	—	2	1	no	—
35	C₇×#476	C₂²	1	1	yes	characteristic
36	C₇×#475	—	2	1	no	—
37	C₇×#475	C₂²	1	1	yes	characteristic
38	C₇×#475	C₂²	1	1	yes	characteristic
39	C₇×#474	C₂²	1	1	yes	characteristic
40	C₇×#467	—	2	1	no	—
41	C₇×#462	—	2	1	no	—
42	C₇×#462	C₂²	1	1	yes	characteristic
43	C₇×#462	C₂²	1	1	yes	characteristic
44	C₇×#462	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×#440	—	2	1	no	—
51	C₄×C₇×Q₁₆	—	2	1	no	—
52	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
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×#333	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#332	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	#19812	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	—	2	1	no	—
75	C₂×C₇×Q₁₆	—	4	4	no	—
76	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
77	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
78	C₂²×C₇×C₈	—	2	1	no	—
79	C₇×C₄⋊Q₈	—	2	1	no	—
80	C₇×C₄⋊Q₈	—	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	—	4	1	no	—
85	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₇×#125	—	2	1	no	—
90	C₇×#125	—	4	1	no	—
91	C₇×C₂²⋊Q₈	—	2	1	no	—
92	C₇×C₂²⋊Q₈	—	2	1	no	—
93	C₇×C₂²⋊Q₈	—	4	1	no	—
94	C₇×C₂²⋊Q₈	—	4	1	no	—
95	C₄×C₇×Q₈	—	2	1	no	—
96	C₄×C₇×Q₈	—	2	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	C₂³	1	1	yes	characteristic
105	C₇×#107	—	2	1	no	—
106	C₇×#107	—	2	1	no	—
107	C₇×#107	—	2	1	no	—
108	C₇×#107	—	2	1	no	—
109	C₇×#107	—	4	1	no	—
110	C₇×#107	D₄	1	1	yes	characteristic
111	C₇×#106	—	2	1	no	—
112	C₇×#106	—	2	1	no	—
113	C₇×#106	—	2	1	no	—
114	C₇×#106	—	2	1	no	—
115	C₇×#106	—	2	1	no	—
116	C₇×#106	D₄	1	1	yes	characteristic
117	C₇×C₄⋊C₈	—	2	1	no	—
118	C₇×C₄⋊C₈	—	2	1	no	—
119	C₇×C₄⋊C₈	—	2	1	no	—
120	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
129	C₇×#103	—	2	1	no	—
130	C₇×#103	—	2	1	no	—
131	C₇×#103	—	2	1	no	—
132	C₇×#103	—	4	1	no	—
133	C₇×#103	D₄	1	1	yes	characteristic
134	C₇×#103	D₄	1	1	yes	characteristic
135	C₇×#103	D₄	1	1	yes	characteristic
136	C₇×#102	—	2	1	no	—
137	C₇×#102	—	2	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	#3386	C₂×C₇	1	1	yes	characteristic
147	Q₈×SD₁₆	C₂×C₇	1	1	yes	characteristic
148	#3363	C₂×C₇	1	1	yes	characteristic, complemented
149	D₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
150	#1723	C₂×C₇	1	1	yes	characteristic, complemented
151	#1692	C₂×C₇	1	1	yes	characteristic, complemented
152	#1683	C₂×C₇	1	1	yes	characteristic
153	#1674	C₂×C₇	1	1	yes	characteristic, complemented
154	#1670	C₂×C₇	1	1	yes	characteristic, complemented
155	#1649	C₂×C₇	1	1	yes	characteristic
156	#1648	C₂×C₇	1	1	yes	characteristic, complemented
157	#1645	C₂×C₇	1	1	yes	characteristic, complemented
158	#1637	C₂×C₇	1	1	yes	characteristic
159	#1584	C₂×C₇	1	1	yes	characteristic, complemented
160	SD₁₆⋊C₈	C₂×C₇	1	1	yes	characteristic
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	—	2	1	no	—
165	C₂×C₇×Q₈	—	4	1	no	—
166	C₂×C₇×Q₈	—	4	1	no	—
167	C₂×C₇×Q₈	—	4	1	no	—
168	C₂×C₇×Q₈	—	4	2	no	—
169	C₂×C₇×Q₈	—	4	2	no	—
170	C₂×C₇×Q₈	—	8	1	no	—
171	C₂×C₇×Q₈	—	8	1	no	—
172	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
174	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₇	—	2	1	no	—
177	C₂²×C₄×C₇	—	4	1	no	—
178	C₂²×C₄×C₇	—	4	1	no	—
179	C₇×Q₁₆	—	2	2	no	—
180	C₇×Q₁₆	—	4	2	no	—
181	C₇×Q₁₆	—	4	2	no	—
182	C₇×Q₁₆	—	4	2	no	—
183	C₇×Q₁₆	—	4	2	no	—
184	C₇×Q₁₆	—	8	1	no	—
185	C₇×SD₁₆	Q₁₆	1	4	yes	complemented
186	C₂×C₇×C₈	—	2	1	no	—
187	C₂×C₇×C₈	—	2	1	no	—
188	C₂×C₇×C₈	—	2	1	no	—
189	C₂×C₇×C₈	—	2	1	no	—
190	C₂×C₇×C₈	—	2	1	no	—
191	C₂×C₇×C₈	—	4	2	no	—
192	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
193	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
196	C₇×C₄⋊C₄	—	2	1	no	—
197	C₇×C₄⋊C₄	—	2	1	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
208	C₇×C₄⋊C₄	—	4	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	C₄²×C₇	—	2	1	no	—
222	C₄²×C₇	—	2	1	no	—
223	C₄²×C₇	—	2	1	no	—
224	C₄²×C₇	—	4	1	no	—
225	C₄²×C₇	—	4	1	no	—
226	C₄²×C₇	C₂⁴	1	1	yes	characteristic
227	C₂²×Q₁₆	—	2	1	no	—
228	Q₈²	C₂²×C₇	1	1	yes	characteristic
229	#557	C₂²×C₇	1	1	yes	characteristic
230	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
231	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
232	#501	—	2	1	no	—
233	#500	C₂²×C₇	1	1	yes	characteristic
234	#500	C₂²×C₇	1	1	yes	characteristic
235	#498	—	2	1	no	—
236	#498	C₂²×C₇	1	1	yes	characteristic
237	#494	C₂²×C₇	1	1	yes	characteristic
238	#487	—	2	1	no	—
239	#487	C₂²×C₇	1	1	yes	characteristic
240	#487	C₂²×C₇	1	1	yes	characteristic
241	#478	—	2	1	no	—
242	#477	—	2	1	no	—
243	#477	—	2	1	no	—
244	#477	C₂²×C₇	1	1	yes	characteristic
245	#476	—	2	1	no	—
246	#476	C₂²×C₇	1	1	yes	characteristic
247	#475	—	2	1	no	—
248	#475	C₂²×C₇	1	1	yes	characteristic
249	#475	C₂²×C₇	1	1	yes	characteristic
250	#474	C₂²×C₇	1	1	yes	characteristic
251	#467	—	2	1	no	—
252	#462	—	2	1	no	—
253	#462	C₂²×C₇	1	1	yes	characteristic
254	#462	C₂²×C₇	1	1	yes	characteristic
255	#462	C₂²×C₇	1	1	yes	characteristic
256	#451	—	2	1	no	—
257	#451	—	2	1	no	—
258	#446	C₂²×C₇	1	1	yes	characteristic
259	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
260	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
261	#440	—	2	1	no	—
262	C₄×Q₁₆	—	2	1	no	—
263	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
264	C₄×SD₁₆	—	2	1	no	—
265	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
266	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
267	#335	C₂²×C₇	1	1	yes	characteristic
268	#335	C₂²×C₇	1	1	yes	characteristic
269	#333	C₂²×C₇	1	1	yes	characteristic
270	#332	C₂²×C₇	1	1	yes	characteristic
271	#332	C₂²×C₇	1	1	yes	characteristic
272	#332	C₂²×C₇	1	1	yes	characteristic
273	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
274	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
275	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
276	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
277	#322	C₂²×C₇	1	1	yes	characteristic
278	C₂³×C₇	—	2	1	no	—
279	C₇×Q₈	—	2	1	no	—
280	C₇×Q₈	—	2	1	no	—
281	C₇×Q₈	—	2	1	no	—
282	C₇×Q₈	—	2	1	no	—
283	C₇×Q₈	—	4	1	no	—
284	C₇×Q₈	—	4	1	no	—
285	C₇×Q₈	—	4	1	no	—
286	C₇×Q₈	—	4	1	no	—
287	C₇×Q₈	—	4	1	no	—
288	C₇×Q₈	—	4	1	no	—
289	C₇×Q₈	—	8	2	no	—
290	C₇×Q₈	—	8	1	no	—
291	C₇×Q₈	—	8	2	no	—
292	C₇×Q₈	—	16	1	no	—
293	C₇×Q₈	C₂×Q₁₆	1	2	yes	complemented
294	C₇×Q₈	AGL(1,ℤ/8)	1	2	yes	—
295	C₇×D₄	—	2	1	no	—
296	C₇×D₄	C₂×Q₁₆	1	2	yes	—
297	C₂×C₄×C₇	—	2	1	no	—
298	C₂×C₄×C₇	—	2	1	no	—
299	C₂×C₄×C₇	—	2	1	no	—
300	C₂×C₄×C₇	—	2	1	no	—
301	C₂×C₄×C₇	—	2	1	no	—
302	C₂×C₄×C₇	—	2	1	no	—
303	C₂×C₄×C₇	—	2	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	1	no	—
307	C₂×C₄×C₇	—	4	1	no	—
308	C₂×C₄×C₇	—	4	1	no	—
309	C₂×C₄×C₇	—	4	1	no	—
310	C₂×C₄×C₇	—	8	2	no	—
311	C₂×C₄×C₇	—	8	2	no	—
312	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
315	C₇×C₈	—	2	1	no	—
316	C₇×C₈	—	2	1	no	—
317	C₇×C₈	—	2	1	no	—
318	C₇×C₈	—	4	1	no	—
319	C₇×C₈	—	4	1	no	—
320	C₇×C₈	—	4	1	no	—
321	C₇×C₈	—	4	1	no	—
322	C₇×C₈	—	4	1	no	—
323	C₇×C₈	C₂×Q₁₆	1	2	yes	complemented
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×Q₈	—	2	1	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×Q₁₆	—	2	1	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	—	4	4	no	—
332	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
333	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
334	C₂²×C₈	—	2	1	no	—
335	C₄⋊Q₈	—	2	1	no	—
336	C₄⋊Q₈	—	2	1	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	4	1	no	—
341	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
343	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
344	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
345	#125	—	2	1	no	—
346	#125	—	4	1	no	—
347	C₂²⋊Q₈	—	2	1	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	4	1	no	—
350	C₂²⋊Q₈	—	4	1	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
356	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
357	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
361	#107	—	2	1	no	—
362	#107	—	2	1	no	—
363	#107	—	2	1	no	—
364	#107	—	2	1	no	—
365	#107	—	4	1	no	—
366	#107	C₇×D₄	1	1	yes	characteristic
367	#106	—	2	1	no	—
368	#106	—	2	1	no	—
369	#106	—	2	1	no	—
370	#106	—	2	1	no	—
371	#106	—	2	1	no	—
372	#106	C₇×D₄	1	1	yes	characteristic
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
377	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
378	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
380	#103	—	2	1	no	—
381	#103	—	2	1	no	—
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	—	4	1	no	—
389	#103	C₇×D₄	1	1	yes	characteristic
390	#103	C₇×D₄	1	1	yes	characteristic
391	#103	C₇×D₄	1	1	yes	characteristic
392	#102	—	2	1	no	—
393	#102	—	2	1	no	—
394	#102	C₇×D₄	1	1	yes	characteristic
395	C₂²⋊C₈	—	2	1	no	—
396	#97	—	2	1	no	—
397	C₄×C₈	—	2	1	no	—
398	C₄×C₈	—	2	1	no	—
399	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
402	C₂²×C₇	—	2	2	no	—
403	C₂²×C₇	—	4	1	no	—
404	C₂²×C₇	—	4	1	no	—
405	C₂²×C₇	D₄²	1	1	yes	characteristic
406	C₄×C₇	—	2	2	no	—
407	C₄×C₇	—	2	2	no	—
408	C₄×C₇	—	2	1	no	—
409	C₄×C₇	—	4	1	no	—
410	C₄×C₇	—	4	1	no	—
411	C₄×C₇	—	4	1	no	—
412	C₄×C₇	—	4	1	no	—
413	C₄×C₇	—	4	1	no	—
414	C₄×C₇	—	8	1	no	—
415	C₄×C₇	—	8	1	no	—
416	C₄×C₇	—	8	1	no	—
417	C₄×C₇	—	8	1	no	—
418	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
419	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
420	C₄×C₇	#577	1	1	yes	characteristic
421	C₄×C₇	#578	1	1	yes	characteristic
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	4	1	no	—
427	C₂×Q₈	—	4	1	no	—
428	C₂×Q₈	—	4	1	no	—
429	C₂×Q₈	—	4	2	no	—
430	C₂×Q₈	—	4	2	no	—
431	C₂×Q₈	—	8	1	no	—
432	C₂×Q₈	—	8	1	no	—
433	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
434	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
435	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
436	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	4	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	Q₁₆	—	2	2	no	—
441	Q₁₆	—	4	2	no	—
442	Q₁₆	—	4	2	no	—
443	Q₁₆	—	4	2	no	—
444	Q₁₆	—	4	2	no	—
445	Q₁₆	—	8	1	no	—
446	SD₁₆	C₇×Q₁₆	1	4	yes	complemented
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	4	2	no	—
453	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
454	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
455	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
456	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	4	1	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
476	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
477	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
478	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	4	1	no	—
486	C₄²	—	4	1	no	—
487	C₄²	C₂⁴×C₇	1	1	yes	characteristic
488	C₂×C₇	—	4	2	no	—
489	C₂×C₇	#3265	1	1	yes	characteristic
490	C₂×C₇	D₄×Q₁₆	1	1	yes	characteristic
491	C₂×C₇	#3289	1	1	yes	characteristic
492	C₂³	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	4	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	8	2	no	—
504	Q₈	—	8	1	no	—
505	Q₈	—	8	2	no	—
506	Q₈	—	16	1	no	—
507	Q₈	C₂×C₇×Q₁₆	1	2	yes	complemented
508	Q₈	C₇×AGL(1,ℤ/8)	1	2	yes	—
509	D₄	—	2	1	no	—
510	D₄	C₂×C₇×Q₁₆	1	2	yes	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	8	2	no	—
525	C₂×C₄	—	8	2	no	—
526	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
527	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
528	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	C₂×C₇×Q₁₆	1	2	yes	complemented
538	C₇	#19812	1	1	yes	characteristic, direct product factor, Sylow
539	C₂²	—	2	2	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	C₇×D₄²	1	1	yes	characteristic
543	C₄	—	2	2	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	8	1	no	—
552	C₄	—	8	1	no	—
553	C₄	—	8	1	no	—
554	C₄	—	8	1	no	—
555	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
556	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
557	C₄	C₇×#577	1	1	yes	characteristic
558	C₄	C₇×#578	1	1	yes	characteristic
559	C₂	—	4	2	no	—
560	C₂	C₇×#3265	1	1	yes	characteristic
561	C₂	C₇×D₄×Q₁₆	1	1	yes	characteristic
562	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×D₄	C₂×D₄
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
3. C₇×#3363	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1692	C₂	semidirect
8. C₇×#1674	C₂	semidirect
9. C₇×#1670	C₂	semidirect
11. C₇×#1648	C₂	semidirect
12. C₇×#1645	C₂	semidirect
14. C₇×#1584	C₂	semidirect
67. #19812	C₇	direct (N × H)
148. #3363	C₂×C₇	semidirect
150. #1723	C₂×C₇	semidirect
151. #1692	C₂×C₇	semidirect
153. #1674	C₂×C₇	semidirect
154. #1670	C₂×C₇	semidirect
156. #1648	C₂×C₇	semidirect
157. #1645	C₂×C₇	semidirect
159. #1584	C₂×C₇	semidirect
185. C₇×SD₁₆	Q₁₆	semidirect
293. C₇×Q₈	C₂×Q₁₆	semidirect
323. C₇×C₈	C₂×Q₁₆	semidirect
446. SD₁₆	C₇×Q₁₆	semidirect
507. Q₈	C₂×C₇×Q₁₆	semidirect
537. C₈	C₂×C₇×Q₁₆	semidirect
538. C₇	#19812	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×#3386	2
C₇×Q₈×SD₁₆	2
C₇×#3363	2
C₇×D₄×Q₁₆	2
C₇×#1723	2
C₇×#1692	2
C₇×#1683	2
C₇×#1674	2
C₇×#1670	2
C₇×#1649	2
C₇×#1648	2
C₇×#1645	2
C₇×#1637	2
C₇×#1584	2
C₇×SD₁₆⋊C₈	2
#19812	7

Subgroups — C7×#19812 #13197028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19812 #13197028