Subgroups — C₇×#60818 #13238034

Order 1792.

2138 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3498	C₂	1	1	yes	maximal, characteristic
3	C₇×#3222	C₂	1	8	yes	maximal, complemented
4	C₇×#3210	C₂	1	4	yes	maximal, complemented
5	C₇×#3208	C₂	1	4	yes	maximal, complemented
6	C₇×#3152	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3043	C₂	1	4	yes	maximal, complemented
11	C₇×#3009	C₂	1	4	yes	maximal, complemented
12	C₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₂×C₇×#137	C₂²	1	4	yes	complemented
15	C₂²×C₇×SD₁₆	C₂²	1	4	yes	complemented
16	C₇×#557	—	2	2	no	—
17	C₇×#556	—	2	2	no	—
18	C₇×#552	—	2	4	no	—
19	C₇×#550	—	2	2	no	—
20	C₇×D₄²	—	2	4	no	—
21	C₇×D₄²	—	2	2	no	—
22	C₇×#541	C₂²	1	4	yes	complemented
23	C₇×#536	C₂²	1	2	yes	—
24	C₇×#535	C₂²	1	4	yes	—
25	C₇×#535	C₂²	1	2	yes	—
26	C₇×#533	C₂²	1	4	yes	complemented
27	C₇×#533	C₂²	1	1	yes	characteristic, complemented
28	C₇×#532	C₂²	1	2	yes	—
29	C₇×#532	C₂²	1	1	yes	characteristic
30	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₇×#127	C₂²	1	2	yes	—
32	C₂×C₇×#127	C₂²	1	1	yes	characteristic
33	C₇×#519	C₂²	1	2	yes	complemented
34	C₄×C₇×#41	C₂²	1	1	yes	characteristic
35	C₇×#497	C₂²	1	4	yes	complemented
36	C₇×#494	C₂²	1	2	yes	complemented
37	C₇×#492	C₂²	1	2	yes	complemented
38	C₇×#492	C₂²	1	4	yes	complemented
39	C₇×#492	C₂²	1	4	yes	complemented
40	C₇×#490	C₂²	1	4	yes	complemented
41	C₇×#488	C₂²	1	4	yes	complemented
42	C₇×#487	C₂²	1	2	yes	complemented
43	C₇×#486	C₂²	1	2	yes	complemented
44	C₇×#486	C₂²	1	4	yes	complemented
45	C₇×#464	C₂²	1	16	yes	complemented
46	C₇×#460	C₂²	1	16	yes	complemented
47	C₇×#452	C₂²	1	16	yes	complemented
48	C₇×#450	C₂²	1	16	yes	complemented
49	C₇×#433	C₂²	1	4	yes	complemented
50	C₇×#431	C₂²	1	4	yes	complemented
51	C₇×#424	C₂²	1	2	yes	complemented
52	C₇×#418	C₂²	1	4	yes	complemented
53	C₇×#416	C₂²	1	4	yes	complemented
54	C₇×#408	C₂²	1	2	yes	—
55	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
56	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
57	#60818	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₇×H(5,𝔽₂)	—	4	8	no	—
59	C₂×C₇×#41	—	2	2	no	—
60	C₂×C₇×#41	C₂³	1	2	yes	—
61	C₂×C₇×#41	C₂³	1	4	yes	—
62	C₂×C₇×#41	C₂³	1	1	yes	characteristic
63	C₂²×C₇×Q₈	C₂³	1	2	yes	—
64	C₂²×C₇×D₄	—	2	4	no	—
65	C₂²×C₇×D₄	—	2	2	no	—
66	C₂²×C₇×D₄	—	2	2	no	—
67	C₂²×C₇×D₄	—	2	4	no	—
68	C₂²×C₇×D₄	C₂³	1	2	yes	—
69	C₂²×C₇×D₄	C₂³	1	4	yes	—
70	C₇×#137	—	4	8	no	—
71	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
72	C₂×C₇×SD₁₆	—	4	8	no	—
73	C₂×C₇×SD₁₆	C₂³	1	16	yes	—
74	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
75	C₂×C₇×M₁₆	C₂³	1	2	yes	—
76	C₂²×C₇×C₈	C₂³	1	2	yes	—
77	C₇×C₄⋊Q₈	—	2	2	no	—
78	C₇×C₄⋊Q₈	—	2	2	no	—
79	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₇×#127	—	2	4	no	—
84	C₇×#127	—	2	2	no	—
85	C₇×#127	C₂³	1	1	yes	characteristic
86	C₇×#127	C₂³	1	2	yes	—
87	C₇×#127	C₂³	1	2	yes	—
88	C₇×#127	C₂³	1	4	yes	—
89	C₇×#126	—	2	8	no	—
90	C₇×#125	—	2	8	no	—
91	C₇×#125	—	2	4	no	—
92	C₇×#125	C₂³	1	2	yes	—
93	C₇×#124	—	2	2	no	—
94	C₇×#124	C₂³	1	2	yes	—
95	C₇×#124	C₂³	1	4	yes	—
96	C₇×#123	—	2	4	no	—
97	C₇×C₂²⋊Q₈	—	2	2	no	—
98	C₇×C₂²⋊Q₈	—	2	4	no	—
99	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
100	C₇×#121	—	2	2	no	—
101	C₇×#121	—	2	4	no	—
102	C₇×#121	—	2	4	no	—
103	C₇×#121	—	2	8	no	—
104	C₇×#120	—	2	4	no	—
105	C₇×#120	—	2	8	no	—
106	C₄×C₇×Q₈	—	2	4	no	—
107	C₄×C₇×Q₈	—	2	4	no	—
108	C₄×C₇×Q₈	—	2	2	no	—
109	C₄×C₇×Q₈	D₄	1	2	yes	—
110	C₄×C₇×Q₈	C₂³	1	8	yes	—
111	C₄×C₇×D₄	—	2	4	no	—
112	C₄×C₇×D₄	—	2	2	no	—
113	C₄×C₇×D₄	D₄	1	4	yes	—
114	C₄×C₇×D₄	D₄	1	2	yes	—
115	C₇×#117	—	2	4	no	—
116	C₇×#117	—	2	4	no	—
117	C₇×#117	C₂³	1	2	yes	—
118	C₇×#117	C₂³	1	1	yes	characteristic
119	C₇×#117	C₂³	1	2	yes	—
120	C₂×C₇×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₇	C₂³	1	2	yes	—
123	C₇×C₄⋊C₈	C₂³	1	8	yes	—
124	C₇×#103	C₂³	1	8	yes	—
125	C₇×#102	C₂³	1	8	yes	—
126	C₇×#102	C₂³	1	16	yes	—
127	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₇×#97	C₂³	1	2	yes	—
129	C₄×C₇×C₈	C₂³	1	2	yes	—
130	C₄×C₇×C₈	C₂³	1	2	yes	—
131	C₄×C₇×C₈	C₂³	1	2	yes	—
132	#3509	C₂×C₇	1	1	yes	characteristic, complemented
133	#3498	C₂×C₇	1	1	yes	characteristic
134	#3222	C₂×C₇	1	8	yes	complemented
135	#3210	C₂×C₇	1	4	yes	complemented
136	#3208	C₂×C₇	1	4	yes	complemented
137	#3152	C₂×C₇	1	1	yes	characteristic, complemented
138	C₂×#492	C₂×C₇	1	1	yes	characteristic, complemented
139	#3138	C₂×C₇	1	1	yes	characteristic, complemented
140	#3131	C₂×C₇	1	1	yes	characteristic, complemented
141	#3043	C₂×C₇	1	4	yes	complemented
142	#3009	C₂×C₇	1	4	yes	complemented
143	#2922	C₂×C₇	1	1	yes	characteristic, complemented
144	C₂⁴×C₇	—	2	4	no	—
145	C₂⁴×C₇	—	2	2	no	—
146	C₇×#41	—	2	4	no	—
147	C₇×#41	—	2	8	no	—
148	C₇×#41	—	4	8	no	—
149	C₇×#41	—	4	8	no	—
150	C₂×C₇×Q₈	—	2	2	no	—
151	C₂×C₇×Q₈	—	2	2	no	—
152	C₂×C₇×Q₈	—	2	8	no	—
153	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
154	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
155	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
156	C₂×C₇×D₄	—	2	4	no	—
157	C₂×C₇×D₄	—	2	2	no	—
158	C₂×C₇×D₄	—	2	4	no	—
159	C₂×C₇×D₄	—	2	2	no	—
160	C₂×C₇×D₄	—	2	4	no	—
161	C₂×C₇×D₄	—	2	2	no	—
162	C₂×C₇×D₄	—	2	8	no	—
163	C₂×C₇×D₄	—	2	4	no	—
164	C₂×C₇×D₄	—	2	8	no	—
165	C₂×C₇×D₄	—	4	8	no	—
166	C₂×C₇×D₄	—	4	16	no	—
167	C₂×C₇×D₄	—	4	8	no	—
168	C₂×C₇×D₄	—	8	4	no	—
169	C₂×C₇×D₄	—	8	8	no	—
170	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
171	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
172	C₂×C₇×D₄	C₂⁴	1	4	yes	—
173	C₂×C₇×D₄	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₇	—	2	2	no	—
175	C₂²×C₄×C₇	—	2	4	no	—
176	C₂²×C₄×C₇	—	2	2	no	—
177	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
181	C₇×Q₁₆	—	4	8	no	—
182	C₇×SD₁₆	—	4	16	no	—
183	C₇×SD₁₆	—	4	8	no	—
184	C₇×M₁₆	—	2	4	no	—
185	C₂×C₇×C₈	—	2	4	no	—
186	C₂×C₇×C₈	C₂⁴	1	4	yes	—
187	C₂×C₇×C₈	C₂⁴	1	4	yes	—
188	C₇×C₄⋊C₄	—	2	4	no	—
189	C₇×C₄⋊C₄	—	2	4	no	—
190	C₇×C₄⋊C₄	—	2	2	no	—
191	C₇×C₄⋊C₄	—	2	8	no	—
192	C₇×C₄⋊C₄	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	8	no	—
195	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
196	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
197	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
198	C₇×C₂²⋊C₄	—	2	2	no	—
199	C₇×C₂²⋊C₄	—	2	8	no	—
200	C₇×C₂²⋊C₄	—	2	8	no	—
201	C₇×C₂²⋊C₄	—	2	4	no	—
202	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
203	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
204	C₄²×C₇	—	2	8	no	—
205	C₄²×C₇	—	2	2	no	—
206	C₄²×C₇	C₂×D₄	1	4	yes	—
207	C₄²×C₇	C₂×D₄	1	2	yes	—
208	C₄²×C₇	C₂⁴	1	1	yes	characteristic
209	C₄²×C₇	C₂⁴	1	1	yes	characteristic
210	C₄²×C₇	C₂⁴	1	2	yes	—
211	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
212	C₂×#137	C₂²×C₇	1	4	yes	complemented
213	C₂²×SD₁₆	C₂²×C₇	1	4	yes	complemented
214	#557	—	2	2	no	—
215	#556	—	2	2	no	—
216	#552	—	2	4	no	—
217	#550	—	2	2	no	—
218	D₄²	—	2	4	no	—
219	D₄²	—	2	2	no	—
220	#541	C₂²×C₇	1	4	yes	complemented
221	#536	C₂²×C₇	1	2	yes	—
222	#535	C₂²×C₇	1	4	yes	—
223	#535	C₂²×C₇	1	2	yes	—
224	#533	C₂²×C₇	1	4	yes	complemented
225	#533	C₂²×C₇	1	1	yes	characteristic, complemented
226	#532	C₂²×C₇	1	1	yes	characteristic
227	#532	C₂²×C₇	1	2	yes	—
228	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
229	C₂×#127	C₂²×C₇	1	1	yes	characteristic
230	C₂×#127	C₂²×C₇	1	2	yes	—
231	#519	C₂²×C₇	1	2	yes	complemented
232	C₄×#41	C₂²×C₇	1	1	yes	characteristic
233	#497	C₂²×C₇	1	4	yes	complemented
234	#494	C₂²×C₇	1	2	yes	complemented
235	#492	C₂²×C₇	1	4	yes	complemented
236	#492	C₂²×C₇	1	4	yes	complemented
237	#492	C₂²×C₇	1	2	yes	complemented
238	#490	C₂²×C₇	1	4	yes	complemented
239	#488	C₂²×C₇	1	4	yes	complemented
240	#487	C₂²×C₇	1	2	yes	complemented
241	#486	C₂²×C₇	1	2	yes	complemented
242	#486	C₂²×C₇	1	4	yes	complemented
243	#464	C₂²×C₇	1	16	yes	complemented
244	#460	C₂²×C₇	1	16	yes	complemented
245	#452	C₂²×C₇	1	16	yes	complemented
246	#450	C₂²×C₇	1	16	yes	complemented
247	#433	C₂²×C₇	1	4	yes	complemented
248	#431	C₂²×C₇	1	4	yes	complemented
249	#424	C₂²×C₇	1	2	yes	complemented
250	#418	C₂²×C₇	1	4	yes	complemented
251	#416	C₂²×C₇	1	4	yes	complemented
252	#408	C₂²×C₇	1	2	yes	—
253	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
254	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
255	C₂³×C₇	—	2	4	no	—
256	C₂³×C₇	—	2	2	no	—
257	C₂³×C₇	—	4	8	no	—
258	C₂³×C₇	—	4	4	no	—
259	C₂³×C₇	—	8	4	no	—
260	C₂³×C₇	—	8	8	no	—
261	C₂³×C₇	C₂²×D₄	1	2	yes	—
262	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
263	C₇×Q₈	—	2	2	no	—
264	C₇×Q₈	—	2	8	no	—
265	C₇×Q₈	—	2	4	no	—
266	C₇×Q₈	—	4	4	no	—
267	C₇×Q₈	—	4	4	no	—
268	C₇×D₄	—	2	4	no	—
269	C₇×D₄	—	2	2	no	—
270	C₇×D₄	—	2	8	no	—
271	C₇×D₄	—	2	4	no	—
272	C₇×D₄	—	4	8	no	—
273	C₇×D₄	—	4	8	no	—
274	C₇×D₄	—	4	4	no	—
275	C₇×D₄	—	4	4	no	—
276	C₇×D₄	—	4	8	no	—
277	C₇×D₄	—	4	4	no	—
278	C₇×D₄	—	8	8	no	—
279	C₇×D₄	—	8	4	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	8	no	—
282	C₂×C₄×C₇	—	2	4	no	—
283	C₂×C₄×C₇	—	2	4	no	—
284	C₂×C₄×C₇	—	2	4	no	—
285	C₂×C₄×C₇	—	2	2	no	—
286	C₂×C₄×C₇	—	4	8	no	—
287	C₂×C₄×C₇	—	4	4	no	—
288	C₂×C₄×C₇	—	4	4	no	—
289	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
290	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
295	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
296	C₇×C₈	—	2	4	no	—
297	C₇×C₈	—	2	4	no	—
298	H(5,𝔽₂)	—	4	8	no	—
299	C₂×#41	—	2	2	no	—
300	C₂×#41	C₂³×C₇	1	2	yes	—
301	C₂×#41	C₂³×C₇	1	4	yes	—
302	C₂×#41	C₂³×C₇	1	1	yes	characteristic
303	C₂²×Q₈	C₂³×C₇	1	2	yes	—
304	C₂²×D₄	—	2	4	no	—
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	—	2	4	no	—
308	C₂²×D₄	C₂³×C₇	1	2	yes	—
309	C₂²×D₄	C₂³×C₇	1	4	yes	—
310	#137	—	4	8	no	—
311	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
312	C₂×SD₁₆	—	4	8	no	—
313	C₂×SD₁₆	C₂³×C₇	1	16	yes	—
314	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
315	C₂×M₁₆	C₂³×C₇	1	2	yes	—
316	C₂²×C₈	C₂³×C₇	1	2	yes	—
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
320	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
322	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
323	#127	—	2	4	no	—
324	#127	—	2	2	no	—
325	#127	C₂³×C₇	1	2	yes	—
326	#127	C₂³×C₇	1	1	yes	characteristic
327	#127	C₂³×C₇	1	4	yes	—
328	#127	C₂³×C₇	1	2	yes	—
329	#126	—	2	8	no	—
330	#125	—	2	8	no	—
331	#125	—	2	4	no	—
332	#125	C₂³×C₇	1	2	yes	—
333	#124	—	2	2	no	—
334	#124	C₂³×C₇	1	4	yes	—
335	#124	C₂³×C₇	1	2	yes	—
336	#123	—	2	4	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₂²⋊Q₈	—	2	4	no	—
339	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
340	#121	—	2	2	no	—
341	#121	—	2	4	no	—
342	#121	—	2	4	no	—
343	#121	—	2	8	no	—
344	#120	—	2	4	no	—
345	#120	—	2	8	no	—
346	C₄×Q₈	—	2	4	no	—
347	C₄×Q₈	—	2	4	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	C₇×D₄	1	2	yes	—
350	C₄×Q₈	C₂³×C₇	1	8	yes	—
351	C₄×D₄	—	2	4	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	C₇×D₄	1	4	yes	—
354	C₄×D₄	C₇×D₄	1	2	yes	—
355	#117	—	2	4	no	—
356	#117	—	2	4	no	—
357	#117	C₂³×C₇	1	2	yes	—
358	#117	C₂³×C₇	1	1	yes	characteristic
359	#117	C₂³×C₇	1	2	yes	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₇	1	2	yes	—
363	C₄⋊C₈	C₂³×C₇	1	8	yes	—
364	#103	C₂³×C₇	1	8	yes	—
365	#102	C₂³×C₇	1	8	yes	—
366	#102	C₂³×C₇	1	16	yes	—
367	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
368	#97	C₂³×C₇	1	2	yes	—
369	C₄×C₈	C₂³×C₇	1	2	yes	—
370	C₄×C₈	C₂³×C₇	1	2	yes	—
371	C₄×C₈	C₂³×C₇	1	2	yes	—
372	C₂²×C₇	—	2	2	no	—
373	C₂²×C₇	—	2	2	no	—
374	C₂²×C₇	—	2	4	no	—
375	C₂²×C₇	—	4	4	no	—
376	C₂²×C₇	—	4	2	no	—
377	C₂²×C₇	—	4	8	no	—
378	C₂²×C₇	—	4	4	no	—
379	C₂²×C₇	—	16	4	no	—
380	C₂²×C₇	—	16	8	no	—
381	C₂²×C₇	#535	1	2	yes	—
382	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
383	C₄×C₇	—	2	2	no	—
384	C₄×C₇	—	2	4	no	—
385	C₄×C₇	—	2	2	no	—
386	C₄×C₇	—	4	2	no	—
387	C₄×C₇	—	4	4	no	—
388	C₄×C₇	—	4	4	no	—
389	C₄×C₇	#535	1	2	yes	—
390	C₄×C₇	#577	1	4	yes	—
391	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	2	no	—
394	#41	—	2	4	no	—
395	#41	—	2	8	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	8	no	—
401	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
402	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
403	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	4	8	no	—
414	C₂×D₄	—	4	16	no	—
415	C₂×D₄	—	4	8	no	—
416	C₂×D₄	—	8	4	no	—
417	C₂×D₄	—	8	8	no	—
418	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
419	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
420	C₂×D₄	C₂⁴×C₇	1	4	yes	—
421	C₂×D₄	C₂⁴×C₇	1	2	yes	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
426	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
428	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
429	Q₁₆	—	4	8	no	—
430	SD₁₆	—	4	16	no	—
431	SD₁₆	—	4	8	no	—
432	M₁₆	—	2	4	no	—
433	C₂×C₈	—	2	4	no	—
434	C₂×C₈	C₂⁴×C₇	1	4	yes	—
435	C₂×C₈	C₂⁴×C₇	1	4	yes	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
444	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
451	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
452	C₄²	—	2	8	no	—
453	C₄²	—	2	2	no	—
454	C₄²	C₂×C₇×D₄	1	4	yes	—
455	C₄²	C₂×C₇×D₄	1	2	yes	—
456	C₄²	C₂⁴×C₇	1	1	yes	characteristic
457	C₄²	C₂⁴×C₇	1	1	yes	characteristic
458	C₄²	C₂⁴×C₇	1	2	yes	—
459	C₂×C₇	—	2	2	no	—
460	C₂×C₇	—	4	2	no	—
461	C₂×C₇	—	8	4	no	—
462	C₂×C₇	—	8	2	no	—
463	C₂×C₇	C₂×#535	1	1	yes	characteristic
464	C₂×C₇	C₂×#577	1	2	yes	—
465	C₂³	—	2	4	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	4	8	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	8	4	no	—
470	C₂³	—	8	8	no	—
471	C₂³	C₂²×C₇×D₄	1	2	yes	—
472	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
473	Q₈	—	2	2	no	—
474	Q₈	—	2	8	no	—
475	Q₈	—	2	4	no	—
476	Q₈	—	4	4	no	—
477	Q₈	—	4	4	no	—
478	D₄	—	2	4	no	—
479	D₄	—	2	8	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	2	no	—
482	D₄	—	4	8	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	4	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	4	no	—
488	D₄	—	8	8	no	—
489	D₄	—	8	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
500	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
501	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
502	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
503	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
504	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
505	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
506	C₈	—	2	4	no	—
507	C₈	—	2	4	no	—
508	C₇	#60818	1	1	yes	characteristic, direct product factor, Sylow
509	C₂²	—	2	2	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	8	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	16	4	no	—
517	C₂²	—	16	8	no	—
518	C₂²	C₇×#535	1	2	yes	—
519	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
520	C₄	—	2	4	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	4	4	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	4	no	—
526	C₄	C₇×#535	1	2	yes	—
527	C₄	C₇×#577	1	4	yes	—
528	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
529	C₂	—	2	2	no	—
530	C₂	—	4	2	no	—
531	C₂	—	8	4	no	—
532	C₂	—	8	2	no	—
533	C₂	C₂×C₇×#535	1	1	yes	characteristic
534	C₂	C₂×C₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3509	C₂	semidirect
3. C₇×#3222	C₂	semidirect
4. C₇×#3210	C₂	semidirect
5. C₇×#3208	C₂	semidirect
6. C₇×#3152	C₂	semidirect
7. C₂×C₇×#492	C₂	semidirect
8. C₇×#3138	C₂	semidirect
9. C₇×#3131	C₂	semidirect
10. C₇×#3043	C₂	semidirect
11. C₇×#3009	C₂	semidirect
12. C₇×#2922	C₂	semidirect
14. C₂×C₇×#137	C₂²	semidirect
15. C₂²×C₇×SD₁₆	C₂²	semidirect
22. C₇×#541	C₂²	semidirect
26. C₇×#533	C₂²	semidirect
27. C₇×#533	C₂²	semidirect
30. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
33. C₇×#519	C₂²	semidirect
35. C₇×#497	C₂²	semidirect
36. C₇×#494	C₂²	semidirect
37. C₇×#492	C₂²	semidirect
38. C₇×#492	C₂²	semidirect
39. C₇×#492	C₂²	semidirect
40. C₇×#490	C₂²	semidirect
41. C₇×#488	C₂²	semidirect
42. C₇×#487	C₂²	semidirect
43. C₇×#486	C₂²	semidirect
44. C₇×#486	C₂²	semidirect
45. C₇×#464	C₂²	semidirect
46. C₇×#460	C₂²	semidirect
47. C₇×#452	C₂²	semidirect
48. C₇×#450	C₂²	semidirect
49. C₇×#433	C₂²	semidirect
50. C₇×#431	C₂²	semidirect
51. C₇×#424	C₂²	semidirect
52. C₇×#418	C₂²	semidirect
53. C₇×#416	C₂²	semidirect
55. C₄×C₇×M₁₆	C₂²	semidirect
56. C₂×C₄×C₇×C₈	C₂²	semidirect
57. #60818	C₇	direct (N × H)
132. #3509	C₂×C₇	semidirect
134. #3222	C₂×C₇	semidirect
135. #3210	C₂×C₇	semidirect
136. #3208	C₂×C₇	semidirect
137. #3152	C₂×C₇	semidirect
138. C₂×#492	C₂×C₇	semidirect
139. #3138	C₂×C₇	semidirect
140. #3131	C₂×C₇	semidirect
141. #3043	C₂×C₇	semidirect
142. #3009	C₂×C₇	semidirect
143. #2922	C₂×C₇	semidirect
212. C₂×#137	C₂²×C₇	semidirect
213. C₂²×SD₁₆	C₂²×C₇	semidirect
220. #541	C₂²×C₇	semidirect
224. #533	C₂²×C₇	semidirect
225. #533	C₂²×C₇	semidirect
228. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
231. #519	C₂²×C₇	semidirect
233. #497	C₂²×C₇	semidirect
234. #494	C₂²×C₇	semidirect
235. #492	C₂²×C₇	semidirect
236. #492	C₂²×C₇	semidirect
237. #492	C₂²×C₇	semidirect
238. #490	C₂²×C₇	semidirect
239. #488	C₂²×C₇	semidirect
240. #487	C₂²×C₇	semidirect
241. #486	C₂²×C₇	semidirect
242. #486	C₂²×C₇	semidirect
243. #464	C₂²×C₇	semidirect
244. #460	C₂²×C₇	semidirect
245. #452	C₂²×C₇	semidirect
246. #450	C₂²×C₇	semidirect
247. #433	C₂²×C₇	semidirect
248. #431	C₂²×C₇	semidirect
249. #424	C₂²×C₇	semidirect
250. #418	C₂²×C₇	semidirect
251. #416	C₂²×C₇	semidirect
253. C₄×M₁₆	C₂²×C₇	semidirect
254. C₂×C₄×C₈	C₂²×C₇	semidirect
508. C₇	#60818	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×#3509	2
C₇×#3498	2
C₇×#3222	2
C₇×#3210	2
C₇×#3208	2
C₇×#3152	2
C₂×C₇×#492	2
C₇×#3138	2
C₇×#3131	2
C₇×#3043	2
C₇×#3009	2
C₇×#2922	2
#60818	7

Subgroups — C7×#60818 #13238034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60818 #13238034