Subgroups — C₇×#34141 #13211357

Order 1792.

3298 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	C₂×C₇×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×#41	C₂	1	2	yes	maximal, complemented
3	C₇×#2367	C₂	1	4	yes	maximal, complemented
4	C₇×#2366	C₂	1	2	yes	maximal, complemented
5	C₇×#2365	C₂	1	2	yes	maximal, complemented
6	C₇×#2364	C₂	1	4	yes	maximal, complemented
7	C₇×#2361	C₂	1	4	yes	maximal, complemented
8	C₇×#2297	C₂	1	2	yes	maximal, complemented
9	C₇×#2293	C₂	1	2	yes	maximal, complemented
10	C₇×#2290	C₂	1	4	yes	maximal, complemented
11	C₄²×C₇×D₄	C₂	1	2	yes	maximal, complemented
12	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×C₇×#41	C₂²	1	2	yes	—
15	C₇×#532	C₄	1	16	yes	complemented
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#127	C₂²	1	1	yes	characteristic
18	C₂×C₇×#124	C₂²	1	2	yes	—
19	C₂×C₇×#121	C₂²	1	2	yes	—
20	C₂×C₇×#121	C₂²	1	2	yes	—
21	C₄×C₇×#41	—	2	16	no	—
22	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
23	C₂×C₄×C₇×D₄	C₂²	1	8	yes	—
24	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
25	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
26	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₇×#117	C₂²	1	4	yes	—
29	C₂×C₇×#117	C₂²	1	1	yes	characteristic
30	C₂×C₇×#117	C₂²	1	1	yes	characteristic
31	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂²×C₄²×C₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₇	C₂²	1	4	yes	—
34	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
35	C₇×#391	C₂²	1	4	yes	—
36	C₇×#390	C₂²	1	4	yes	—
37	C₇×#390	C₂²	1	4	yes	—
38	C₇×#389	C₂²	1	4	yes	—
39	C₇×#388	C₂²	1	8	yes	—
40	C₇×#388	C₂²	1	8	yes	—
41	C₇×#387	C₂²	1	8	yes	—
42	C₇×#386	C₂²	1	8	yes	—
43	C₇×#386	C₂²	1	8	yes	—
44	C₇×#385	C₂²	1	8	yes	—
45	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
46	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
47	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
49	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
50	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
51	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
52	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
53	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
54	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
55	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
56	C₄×C₇×C₂²⋊C₄	C₂²	1	8	yes	—
57	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
58	C₇×#376	C₂²	1	2	yes	—
59	C₂×C₇×#95	C₂²	1	2	yes	—
60	C₄³×C₇	C₂²	1	2	yes	—
61	#34141	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₇×#41	—	2	8	no	—
63	C₂×C₇×#41	—	2	8	no	—
64	C₂×C₇×#41	C₂×C₄	1	16	yes	complemented
65	C₂²×C₇×Q₈	C₂³	1	2	yes	—
66	C₂²×C₇×D₄	C₂³	1	4	yes	—
67	C₂²×C₇×D₄	C₂³	1	2	yes	—
68	C₂²×C₇×D₄	C₂³	1	2	yes	—
69	C₂²×C₇×D₄	C₂³	1	2	yes	—
70	C₂³×C₄×C₇	C₂³	1	4	yes	—
71	C₂³×C₄×C₇	C₂³	1	8	yes	—
72	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₇	C₂³	1	4	yes	—
74	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
76	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
77	C₇×#127	C₂×C₄	1	8	yes	complemented
78	C₇×#124	C₂×C₄	1	16	yes	complemented
79	C₇×#121	C₂×C₄	1	16	yes	complemented
80	C₇×#121	C₂×C₄	1	16	yes	complemented
81	C₄×C₇×Q₈	—	2	16	no	—
82	C₄×C₇×D₄	—	2	16	no	—
83	C₄×C₇×D₄	—	2	16	no	—
84	C₄×C₇×D₄	—	2	32	no	—
85	C₄×C₇×D₄	C₂×C₄	1	16	yes	complemented
86	C₄×C₇×D₄	C₂×C₄	1	16	yes	complemented
87	C₇×#117	—	2	4	no	—
88	C₇×#117	—	2	16	no	—
89	C₇×#117	D₄	1	8	yes	complemented
90	C₂×C₇×C₄⋊C₄	—	2	8	no	—
91	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
92	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
93	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
94	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
95	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
96	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
97	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
98	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
99	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
100	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
101	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
102	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
103	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
104	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
105	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
106	C₂×C₄²×C₇	—	2	8	no	—
107	C₂×C₄²×C₇	—	2	16	no	—
108	C₂×C₄²×C₇	C₂×C₄	1	8	yes	complemented
109	C₂×C₄²×C₇	C₂³	1	4	yes	—
110	C₂×C₄²×C₇	C₂³	1	8	yes	—
111	C₂×C₄²×C₇	C₂³	1	4	yes	—
112	C₂×C₄²×C₇	C₂³	1	2	yes	—
113	C₂×C₄²×C₇	C₂³	1	2	yes	—
114	C₂×C₄²×C₇	C₂³	1	2	yes	—
115	C₂×C₄²×C₇	C₂³	1	4	yes	—
116	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₇	C₂³	1	2	yes	—
121	C₇×#95	C₂³	1	2	yes	—
122	C₇×#95	C₂³	1	4	yes	—
123	C₇×#95	C₂³	1	2	yes	—
124	C₇×#95	C₂³	1	4	yes	—
125	C₇×#95	C₂³	1	4	yes	—
126	C₇×#95	C₂³	1	8	yes	—
127	C₇×#95	C₂³	1	8	yes	—
128	C₂×#532	C₂×C₇	1	1	yes	characteristic
129	C₂×C₄×#41	C₂×C₇	1	2	yes	complemented
130	#2367	C₂×C₇	1	4	yes	complemented
131	#2366	C₂×C₇	1	2	yes	complemented
132	#2365	C₂×C₇	1	2	yes	complemented
133	#2364	C₂×C₇	1	4	yes	complemented
134	#2361	C₂×C₇	1	4	yes	complemented
135	#2297	C₂×C₇	1	2	yes	complemented
136	#2293	C₂×C₇	1	2	yes	complemented
137	#2290	C₂×C₇	1	4	yes	complemented
138	C₄²×D₄	C₂×C₇	1	2	yes	complemented
139	C₄×#117	C₂×C₇	1	1	yes	characteristic, complemented
140	C₂×C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
141	C₂⁴×C₇	C₂⁴	1	4	yes	—
142	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
143	C₇×#41	—	2	64	no	—
144	C₂×C₇×Q₈	—	2	4	no	—
145	C₂×C₇×Q₈	—	2	4	no	—
146	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
147	C₂×C₇×D₄	—	2	8	no	—
148	C₂×C₇×D₄	—	2	8	no	—
149	C₂×C₇×D₄	—	2	4	no	—
150	C₂×C₇×D₄	—	2	8	no	—
151	C₂×C₇×D₄	—	2	4	no	—
152	C₂×C₇×D₄	—	4	4	no	—
153	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
154	C₂×C₇×D₄	C₂²×C₄	1	16	yes	—
155	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
156	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
157	C₂²×C₄×C₇	—	2	8	no	—
158	C₂²×C₄×C₇	—	2	4	no	—
159	C₂²×C₄×C₇	—	2	8	no	—
160	C₂²×C₄×C₇	—	2	8	no	—
161	C₂²×C₄×C₇	—	2	16	no	—
162	C₂²×C₄×C₇	—	2	8	no	—
163	C₂²×C₄×C₇	—	2	16	no	—
164	C₂²×C₄×C₇	—	4	2	no	—
165	C₂²×C₄×C₇	—	4	8	no	—
166	C₂²×C₄×C₇	—	4	4	no	—
167	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₇	C₂²×C₄	1	16	yes	—
169	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
172	C₂²×C₄×C₇	#41	1	4	yes	—
173	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
184	C₇×C₄⋊C₄	—	2	16	no	—
185	C₇×C₄⋊C₄	—	2	8	no	—
186	C₇×C₄⋊C₄	—	2	4	no	—
187	C₇×C₄⋊C₄	—	2	4	no	—
188	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
189	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
190	C₇×C₄⋊C₄	C₂×D₄	1	8	yes	—
191	C₇×C₄⋊C₄	#41	1	8	yes	complemented
192	C₇×C₂²⋊C₄	—	2	4	no	—
193	C₇×C₂²⋊C₄	—	2	16	no	—
194	C₇×C₂²⋊C₄	—	2	16	no	—
195	C₇×C₂²⋊C₄	—	4	8	no	—
196	C₇×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
197	C₇×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
198	C₇×C₂²⋊C₄	C₂×D₄	1	8	yes	—
199	C₄²×C₇	—	2	4	no	—
200	C₄²×C₇	—	2	4	no	—
201	C₄²×C₇	—	2	16	no	—
202	C₄²×C₇	—	2	8	no	—
203	C₄²×C₇	—	2	4	no	—
204	C₄²×C₇	—	2	8	no	—
205	C₄²×C₇	—	4	4	no	—
206	C₄²×C₇	C₂²×C₄	1	4	yes	—
207	C₄²×C₇	C₂²×C₄	1	4	yes	—
208	C₄²×C₇	C₂²×C₄	1	4	yes	—
209	C₄²×C₇	C₂²×C₄	1	4	yes	—
210	C₄²×C₇	C₂×D₄	1	8	yes	—
211	C₂²×#41	C₂²×C₇	1	2	yes	—
212	#532	C₄×C₇	1	16	yes	complemented
213	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
214	C₂×#127	C₂²×C₇	1	1	yes	characteristic
215	C₂×#124	C₂²×C₇	1	2	yes	—
216	C₂×#121	C₂²×C₇	1	2	yes	—
217	C₂×#121	C₂²×C₇	1	2	yes	—
218	C₄×#41	—	2	16	no	—
219	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
220	C₂×C₄×D₄	C₂²×C₇	1	8	yes	—
221	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
222	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
223	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
224	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
225	C₂×#117	C₂²×C₇	1	4	yes	—
226	C₂×#117	C₂²×C₇	1	1	yes	characteristic
227	C₂×#117	C₂²×C₇	1	1	yes	characteristic
228	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
229	C₂²×C₄²	C₂²×C₇	1	2	yes	—
230	C₂²×C₄²	C₂²×C₇	1	4	yes	—
231	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
232	#391	C₂²×C₇	1	4	yes	—
233	#390	C₂²×C₇	1	4	yes	—
234	#390	C₂²×C₇	1	4	yes	—
235	#389	C₂²×C₇	1	4	yes	—
236	#388	C₂²×C₇	1	8	yes	—
237	#388	C₂²×C₇	1	8	yes	—
238	#387	C₂²×C₇	1	8	yes	—
239	#386	C₂²×C₇	1	8	yes	—
240	#386	C₂²×C₇	1	8	yes	—
241	#385	C₂²×C₇	1	8	yes	—
242	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
243	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
244	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
246	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
250	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
251	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
252	C₄×C₂²⋊C₄	C₂²×C₇	1	8	yes	—
253	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
254	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
255	#376	C₂²×C₇	1	2	yes	—
256	C₂×#95	C₂²×C₇	1	2	yes	—
257	C₄³	C₂²×C₇	1	2	yes	—
258	C₂³×C₇	—	2	2	no	—
259	C₂³×C₇	—	2	2	no	—
260	C₂³×C₇	—	2	4	no	—
261	C₂³×C₇	—	2	8	no	—
262	C₂³×C₇	—	2	4	no	—
263	C₂³×C₇	—	2	8	no	—
264	C₂³×C₇	C₂³×C₄	1	8	yes	—
265	C₂³×C₇	C₂³×C₄	1	2	yes	—
266	C₂³×C₇	C₂²×D₄	1	2	yes	—
267	C₂³×C₇	C₂×#41	1	2	yes	—
268	C₂³×C₇	C₂⁵	1	1	yes	characteristic
269	C₇×Q₈	—	2	16	no	—
270	C₇×D₄	—	2	32	no	—
271	C₇×D₄	—	2	16	no	—
272	C₇×D₄	—	2	16	no	—
273	C₇×D₄	—	4	8	no	—
274	C₂×C₄×C₇	—	2	4	no	—
275	C₂×C₄×C₇	—	2	4	no	—
276	C₂×C₄×C₇	—	2	4	no	—
277	C₂×C₄×C₇	—	2	2	no	—
278	C₂×C₄×C₇	—	2	4	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	2	no	—
283	C₂×C₄×C₇	—	2	32	no	—
284	C₂×C₄×C₇	—	2	8	no	—
285	C₂×C₄×C₇	—	2	4	no	—
286	C₂×C₄×C₇	—	2	4	no	—
287	C₂×C₄×C₇	—	2	4	no	—
288	C₂×C₄×C₇	—	2	4	no	—
289	C₂×C₄×C₇	—	2	4	no	—
290	C₂×C₄×C₇	—	2	4	no	—
291	C₂×C₄×C₇	—	4	4	no	—
292	C₂×C₄×C₇	—	4	32	no	—
293	C₂×C₄×C₇	—	4	16	no	—
294	C₂×C₄×C₇	C₄×D₄	1	16	yes	complemented
295	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
296	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
297	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
298	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
299	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
300	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
301	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
302	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
303	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₇	C₂×#41	1	8	yes	—
306	C₂×C₄×C₇	C₂×#41	1	2	yes	—
307	C₂×C₄×C₇	C₂×#41	1	2	yes	—
308	C₂×C₄×C₇	C₂×#41	1	4	yes	—
309	C₂×#41	—	2	8	no	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	C₂×C₄×C₇	1	16	yes	complemented
312	C₂²×Q₈	C₂³×C₇	1	2	yes	—
313	C₂²×D₄	C₂³×C₇	1	2	yes	—
314	C₂²×D₄	C₂³×C₇	1	4	yes	—
315	C₂²×D₄	C₂³×C₇	1	2	yes	—
316	C₂²×D₄	C₂³×C₇	1	2	yes	—
317	C₂³×C₄	C₂³×C₇	1	4	yes	—
318	C₂³×C₄	C₂³×C₇	1	8	yes	—
319	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₇	1	4	yes	—
321	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
323	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	complemented
324	#127	C₂×C₄×C₇	1	8	yes	complemented
325	#124	C₂×C₄×C₇	1	16	yes	complemented
326	#121	C₂×C₄×C₇	1	16	yes	complemented
327	#121	C₂×C₄×C₇	1	16	yes	complemented
328	C₄×Q₈	—	2	16	no	—
329	C₄×D₄	—	2	16	no	—
330	C₄×D₄	—	2	16	no	—
331	C₄×D₄	—	2	32	no	—
332	C₄×D₄	C₂×C₄×C₇	1	16	yes	complemented
333	C₄×D₄	C₂×C₄×C₇	1	16	yes	complemented
334	#117	—	2	4	no	—
335	#117	—	2	16	no	—
336	#117	C₇×D₄	1	8	yes	complemented
337	C₂×C₄⋊C₄	—	2	8	no	—
338	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
341	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
353	C₂×C₄²	—	2	8	no	—
354	C₂×C₄²	—	2	16	no	—
355	C₂×C₄²	C₂×C₄×C₇	1	8	yes	complemented
356	C₂×C₄²	C₂³×C₇	1	2	yes	—
357	C₂×C₄²	C₂³×C₇	1	2	yes	—
358	C₂×C₄²	C₂³×C₇	1	8	yes	—
359	C₂×C₄²	C₂³×C₇	1	4	yes	—
360	C₂×C₄²	C₂³×C₇	1	4	yes	—
361	C₂×C₄²	C₂³×C₇	1	2	yes	—
362	C₂×C₄²	C₂³×C₇	1	4	yes	—
363	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₇	1	2	yes	—
368	#95	C₂³×C₇	1	2	yes	—
369	#95	C₂³×C₇	1	4	yes	—
370	#95	C₂³×C₇	1	2	yes	—
371	#95	C₂³×C₇	1	4	yes	—
372	#95	C₂³×C₇	1	4	yes	—
373	#95	C₂³×C₇	1	8	yes	—
374	#95	C₂³×C₇	1	8	yes	—
375	C₂²×C₇	—	2	2	no	—
376	C₂²×C₇	—	2	2	no	—
377	C₂²×C₇	—	2	4	no	—
378	C₂²×C₇	—	2	4	no	—
379	C₂²×C₇	—	2	8	no	—
380	C₂²×C₇	—	2	8	no	—
381	C₂²×C₇	—	2	8	no	—
382	C₂²×C₇	—	4	8	no	—
383	C₂²×C₇	—	4	8	no	—
384	C₂²×C₇	C₂×C₄×D₄	1	4	yes	—
385	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
386	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
387	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
388	C₂²×C₇	C₂²×#41	1	2	yes	—
389	C₂²×C₇	#585	1	2	yes	—
390	C₄×C₇	—	2	8	no	—
391	C₄×C₇	—	2	2	no	—
392	C₄×C₇	—	2	2	no	—
393	C₄×C₇	—	2	2	no	—
394	C₄×C₇	—	2	8	no	—
395	C₄×C₇	—	2	8	no	—
396	C₄×C₇	—	4	4	no	—
397	C₄×C₇	—	4	4	no	—
398	C₄×C₇	C₂×C₄×D₄	1	4	yes	—
399	C₄×C₇	C₂×C₄×D₄	1	4	yes	—
400	C₄×C₇	C₄×#41	1	4	yes	complemented
401	C₂⁴	C₂⁴×C₇	1	4	yes	—
402	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
403	#41	—	2	64	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
407	C₂×D₄	—	2	8	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	4	4	no	—
413	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
414	C₂×D₄	C₂²×C₄×C₇	1	16	yes	—
415	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
416	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
417	C₂²×C₄	—	2	8	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	8	no	—
420	C₂²×C₄	—	2	8	no	—
421	C₂²×C₄	—	2	16	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	16	no	—
424	C₂²×C₄	—	4	2	no	—
425	C₂²×C₄	—	4	8	no	—
426	C₂²×C₄	—	4	4	no	—
427	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
428	C₂²×C₄	C₂²×C₄×C₇	1	16	yes	—
429	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
430	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
431	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
432	C₂²×C₄	C₇×#41	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
444	C₄⋊C₄	—	2	16	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
449	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
450	C₄⋊C₄	C₂×C₇×D₄	1	8	yes	—
451	C₄⋊C₄	C₇×#41	1	8	yes	complemented
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	16	no	—
454	C₂²⋊C₄	—	2	16	no	—
455	C₂²⋊C₄	—	4	8	no	—
456	C₂²⋊C₄	C₂²×C₄×C₇	1	16	yes	—
457	C₂²⋊C₄	C₂²×C₄×C₇	1	16	yes	—
458	C₂²⋊C₄	C₂×C₇×D₄	1	8	yes	—
459	C₄²	—	2	4	no	—
460	C₄²	—	2	4	no	—
461	C₄²	—	2	16	no	—
462	C₄²	—	2	8	no	—
463	C₄²	—	2	4	no	—
464	C₄²	—	2	8	no	—
465	C₄²	—	4	4	no	—
466	C₄²	C₂²×C₄×C₇	1	4	yes	—
467	C₄²	C₂²×C₄×C₇	1	4	yes	—
468	C₄²	C₂²×C₄×C₇	1	4	yes	—
469	C₄²	C₂²×C₄×C₇	1	4	yes	—
470	C₄²	C₂×C₇×D₄	1	8	yes	—
471	C₂×C₇	—	2	4	no	—
472	C₂×C₇	—	4	8	no	—
473	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
474	C₂×C₇	C₂×C₄×#41	1	1	yes	characteristic
475	C₂×C₇	#3428	1	1	yes	characteristic
476	C₂×C₇	D₄×#41	1	2	yes	—
477	C₂×C₇	#3480	1	2	yes	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	8	no	—
484	C₂³	C₂³×C₄×C₇	1	8	yes	—
485	C₂³	C₂³×C₄×C₇	1	2	yes	—
486	C₂³	C₂²×C₇×D₄	1	2	yes	—
487	C₂³	C₂×C₇×#41	1	2	yes	—
488	C₂³	C₂⁵×C₇	1	1	yes	characteristic
489	Q₈	—	2	16	no	—
490	D₄	—	2	32	no	—
491	D₄	—	2	16	no	—
492	D₄	—	2	16	no	—
493	D₄	—	4	8	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	32	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	32	no	—
513	C₂×C₄	—	4	16	no	—
514	C₂×C₄	C₄×C₇×D₄	1	16	yes	complemented
515	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
517	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
518	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
519	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
520	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
522	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
523	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
524	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
525	C₂×C₄	C₂×C₇×#41	1	8	yes	—
526	C₂×C₄	C₂×C₇×#41	1	2	yes	—
527	C₂×C₄	C₂×C₇×#41	1	2	yes	—
528	C₂×C₄	C₂×C₇×#41	1	4	yes	—
529	C₇	#34141	1	1	yes	characteristic, direct product factor, Sylow
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	8	no	—
539	C₂²	C₂×C₄×C₇×D₄	1	4	yes	—
540	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
541	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
542	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
543	C₂²	C₂²×C₇×#41	1	2	yes	—
544	C₂²	C₇×#585	1	2	yes	—
545	C₄	—	2	8	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	8	no	—
550	C₄	—	2	8	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	C₂×C₄×C₇×D₄	1	4	yes	—
554	C₄	C₂×C₄×C₇×D₄	1	4	yes	—
555	C₄	C₄×C₇×#41	1	4	yes	complemented
556	C₂	—	2	4	no	—
557	C₂	—	4	8	no	—
558	C₂	C₂²×C₄×C₇×D₄	1	1	yes	characteristic
559	C₂	C₂×C₄×C₇×#41	1	1	yes	characteristic
560	C₂	C₇×#3428	1	1	yes	characteristic
561	C₂	C₇×D₄×#41	1	2	yes	—
562	C₂	C₇×#3480	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂³	C₂⁵×C₇
socle	C₂³×C₇	C₂⁵
prime radical	C₂×C₇×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₇×#41	C₂	semidirect
3. C₇×#2367	C₂	semidirect
4. C₇×#2366	C₂	semidirect
5. C₇×#2365	C₂	semidirect
6. C₇×#2364	C₂	semidirect
7. C₇×#2361	C₂	semidirect
8. C₇×#2297	C₂	semidirect
9. C₇×#2293	C₂	semidirect
10. C₇×#2290	C₂	semidirect
11. C₄²×C₇×D₄	C₂	semidirect
12. C₄×C₇×#117	C₂	semidirect
13. C₂×C₄×C₇×C₄⋊C₄	C₂	semidirect
15. C₇×#532	C₄	semidirect
61. #34141	C₇	direct (N × H)
64. C₂×C₇×#41	C₂×C₄	semidirect
76. C₇×C₄⋊Q₈	C₂×C₄	semidirect
77. C₇×#127	C₂×C₄	semidirect
78. C₇×#124	C₂×C₄	semidirect
79. C₇×#121	C₂×C₄	semidirect
80. C₇×#121	C₂×C₄	semidirect
85. C₄×C₇×D₄	C₂×C₄	semidirect
86. C₄×C₇×D₄	C₂×C₄	semidirect
89. C₇×#117	D₄	semidirect
108. C₂×C₄²×C₇	C₂×C₄	semidirect
129. C₂×C₄×#41	C₂×C₇	semidirect
130. #2367	C₂×C₇	semidirect
131. #2366	C₂×C₇	semidirect
132. #2365	C₂×C₇	semidirect
133. #2364	C₂×C₇	semidirect
134. #2361	C₂×C₇	semidirect
135. #2297	C₂×C₇	semidirect
136. #2293	C₂×C₇	semidirect
137. #2290	C₂×C₇	semidirect
138. C₄²×D₄	C₂×C₇	semidirect
139. C₄×#117	C₂×C₇	semidirect
140. C₂×C₄×C₄⋊C₄	C₂×C₇	semidirect
191. C₇×C₄⋊C₄	#41	semidirect
212. #532	C₄×C₇	semidirect
294. C₂×C₄×C₇	C₄×D₄	semidirect
311. C₂×#41	C₂×C₄×C₇	semidirect
323. C₄⋊Q₈	C₂×C₄×C₇	semidirect
324. #127	C₂×C₄×C₇	semidirect
325. #124	C₂×C₄×C₇	semidirect
326. #121	C₂×C₄×C₇	semidirect
327. #121	C₂×C₄×C₇	semidirect
332. C₄×D₄	C₂×C₄×C₇	semidirect
333. C₄×D₄	C₂×C₄×C₇	semidirect
336. #117	C₇×D₄	semidirect
355. C₂×C₄²	C₂×C₄×C₇	semidirect
400. C₄×C₇	C₄×#41	semidirect
451. C₄⋊C₄	C₇×#41	semidirect
514. C₂×C₄	C₄×C₇×D₄	semidirect
529. C₇	#34141	direct (N × H)
555. C₄	C₄×C₇×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#532	2
C₂×C₄×C₇×#41	2
C₇×#2367	2
C₇×#2366	2
C₇×#2365	2
C₇×#2364	2
C₇×#2361	2
C₇×#2297	2
C₇×#2293	2
C₇×#2290	2
C₄²×C₇×D₄	2
C₄×C₇×#117	2
C₂×C₄×C₇×C₄⋊C₄	2
#34141	7

Subgroups — C7×#34141 #13211357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34141 #13211357