Subgroups — C₇×#22146 #13199362

Order 1792.

896 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#461	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2293	C₂	1	1	yes	maximal, characteristic
4	C₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1968	C₂	1	2	yes	maximal
6	C₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1933	C₂	1	2	yes	maximal, complemented
8	C₇×#1880	C₂	1	2	yes	maximal
9	C₇×#1865	C₂	1	2	yes	maximal, complemented
10	C₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×#117	—	2	1	no	—
17	C₇×#461	C₄	1	8	yes	complemented
18	C₂×C₇×#106	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#103	C₂²	1	2	yes	—
22	C₂×C₇×#103	C₂²	1	2	yes	—
23	C₂×C₇×#103	C₂²	1	2	yes	—
24	C₂×C₇×#102	C₂²	1	2	yes	—
25	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
26	C₇×#391	C₂²	1	2	yes	—
27	C₇×#390	C₂²	1	1	yes	characteristic
28	C₇×#390	C₂²	1	1	yes	characteristic
29	C₇×#389	—	2	1	no	—
30	C₇×#389	—	2	1	no	—
31	C₇×#389	C₂²	1	1	yes	characteristic
32	C₇×#389	C₂²	1	1	yes	characteristic
33	C₇×#388	—	2	1	no	—
34	C₇×#388	—	2	1	no	—
35	C₇×#384	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₄⋊C₄	—	2	1	no	—
37	C₄×C₇×C₄⋊C₄	—	2	1	no	—
38	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
40	C₄³×C₇	C₂²	1	1	yes	characteristic
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#336	C₂²	1	2	yes	—
44	#22146	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	C₂³	1	2	yes	—
46	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₇	—	2	1	no	—
48	C₂×C₇×SD₁₆	—	4	2	no	—
49	C₂×C₇×SD₁₆	—	4	2	no	—
50	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
51	C₂²×C₇×C₈	C₂³	1	2	yes	—
52	C₂²×C₇×C₈	C₂³	1	2	yes	—
53	C₇×C₄⋊Q₈	—	2	2	no	—
54	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
55	C₄×C₇×Q₈	—	4	2	no	—
56	C₄×C₇×Q₈	—	4	2	no	—
57	C₄×C₇×D₄	—	2	2	no	—
58	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
59	C₇×#117	—	4	4	no	—
60	C₂×C₇×C₄⋊C₄	—	2	1	no	—
61	C₂×C₇×C₄⋊C₄	—	2	1	no	—
62	C₂×C₇×C₄⋊C₄	—	2	1	no	—
63	C₂×C₇×C₄⋊C₄	—	2	1	no	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₇×C₄⋊C₄	—	2	1	no	—
66	C₂×C₇×C₄⋊C₄	—	2	2	no	—
67	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
72	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
73	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
74	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
75	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
76	C₂×C₄²×C₇	—	2	1	no	—
77	C₂×C₄²×C₇	—	2	2	no	—
78	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
85	C₇×#106	—	4	2	no	—
86	C₇×C₄⋊C₈	—	2	2	no	—
87	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
88	C₇×#103	—	4	2	no	—
89	C₇×#103	—	4	2	no	—
90	C₇×#103	C₂×C₄	1	8	yes	—
91	C₇×#102	—	4	2	no	—
92	C₄×C₇×C₈	—	4	2	no	—
93	C₇×#95	—	2	1	no	—
94	C₇×#95	—	2	1	no	—
95	C₇×#95	—	2	2	no	—
96	C₇×#95	—	2	2	no	—
97	C₇×#95	—	2	1	no	—
98	C₂×#461	C₂×C₇	1	1	yes	characteristic
99	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
100	#2293	C₂×C₇	1	1	yes	characteristic
101	#1981	C₂×C₇	1	1	yes	characteristic, complemented
102	#1968	C₂×C₇	1	2	yes	—
103	#1948	C₂×C₇	1	1	yes	characteristic, complemented
104	#1933	C₂×C₇	1	2	yes	complemented
105	#1880	C₂×C₇	1	2	yes	—
106	#1865	C₂×C₇	1	2	yes	complemented
107	#1802	C₂×C₇	1	1	yes	characteristic
108	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
109	C₂⁴×C₇	—	2	1	no	—
110	C₂×C₇×Q₈	—	2	2	no	—
111	C₂×C₇×Q₈	—	2	2	no	—
112	C₂×C₇×Q₈	—	2	2	no	—
113	C₂×C₇×Q₈	—	2	2	no	—
114	C₂×C₇×Q₈	—	2	2	no	—
115	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
116	C₂×C₇×D₄	—	2	1	no	—
117	C₂×C₇×D₄	—	2	1	no	—
118	C₂×C₇×D₄	—	2	1	no	—
119	C₂×C₇×D₄	—	2	1	no	—
120	C₂×C₇×D₄	—	2	1	no	—
121	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₇	—	2	2	no	—
123	C₂²×C₄×C₇	—	2	1	no	—
124	C₂²×C₄×C₇	—	2	4	no	—
125	C₂²×C₄×C₇	—	2	1	no	—
126	C₂²×C₄×C₇	—	2	1	no	—
127	C₂²×C₄×C₇	—	2	2	no	—
128	C₂²×C₄×C₇	—	4	2	no	—
129	C₂²×C₄×C₇	—	4	2	no	—
130	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
137	C₇×SD₁₆	—	4	8	no	—
138	C₂×C₇×C₈	—	2	2	no	—
139	C₂×C₇×C₈	—	2	2	no	—
140	C₂×C₇×C₈	—	2	2	no	—
141	C₂×C₇×C₈	—	2	2	no	—
142	C₂×C₇×C₈	—	2	2	no	—
143	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
144	C₇×C₄⋊C₄	—	2	4	no	—
145	C₇×C₄⋊C₄	—	2	4	no	—
146	C₇×C₄⋊C₄	—	2	1	no	—
147	C₇×C₄⋊C₄	—	2	2	no	—
148	C₇×C₄⋊C₄	—	2	1	no	—
149	C₇×C₄⋊C₄	—	2	1	no	—
150	C₇×C₄⋊C₄	—	2	1	no	—
151	C₇×C₄⋊C₄	—	2	2	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	4	2	no	—
154	C₇×C₄⋊C₄	—	4	4	no	—
155	C₇×C₄⋊C₄	—	4	2	no	—
156	C₇×C₄⋊C₄	—	4	2	no	—
157	C₇×C₄⋊C₄	—	4	2	no	—
158	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
159	C₇×C₂²⋊C₄	—	2	4	no	—
160	C₇×C₂²⋊C₄	—	4	2	no	—
161	C₇×C₂²⋊C₄	—	4	2	no	—
162	C₇×C₂²⋊C₄	—	4	2	no	—
163	C₇×C₂²⋊C₄	—	4	2	no	—
164	C₄²×C₇	—	2	2	no	—
165	C₄²×C₇	—	2	2	no	—
166	C₄²×C₇	—	2	1	no	—
167	C₄²×C₇	—	2	1	no	—
168	C₄²×C₇	—	2	1	no	—
169	C₄²×C₇	—	2	1	no	—
170	C₄²×C₇	—	4	1	no	—
171	C₄²×C₇	—	4	4	no	—
172	C₄²×C₇	—	4	2	no	—
173	C₄²×C₇	SD₁₆	1	4	yes	complemented
174	C₄²×C₇	C₂²×C₄	1	2	yes	—
175	C₄²×C₇	#41	1	2	yes	—
176	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
177	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
179	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
180	C₂×#117	—	2	1	no	—
181	#461	C₄×C₇	1	8	yes	complemented
182	C₂×#106	C₂²×C₇	1	2	yes	—
183	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
185	C₂×#103	C₂²×C₇	1	2	yes	—
186	C₂×#103	C₂²×C₇	1	2	yes	—
187	C₂×#103	C₂²×C₇	1	2	yes	—
188	C₂×#102	C₂²×C₇	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
190	#391	C₂²×C₇	1	2	yes	—
191	#390	C₂²×C₇	1	1	yes	characteristic
192	#390	C₂²×C₇	1	1	yes	characteristic
193	#389	—	2	1	no	—
194	#389	—	2	1	no	—
195	#389	C₂²×C₇	1	1	yes	characteristic
196	#389	C₂²×C₇	1	1	yes	characteristic
197	#388	—	2	1	no	—
198	#388	—	2	1	no	—
199	#384	C₂²×C₇	1	1	yes	characteristic
200	C₄×C₄⋊C₄	—	2	1	no	—
201	C₄×C₄⋊C₄	—	2	1	no	—
202	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
203	C₄×C₂²⋊C₄	—	2	1	no	—
204	C₄³	C₂²×C₇	1	1	yes	characteristic
205	#340	C₂²×C₇	1	2	yes	—
206	#340	C₂²×C₇	1	2	yes	—
207	#336	C₂²×C₇	1	2	yes	—
208	C₂³×C₇	—	2	2	no	—
209	C₂³×C₇	—	4	1	no	—
210	C₂³×C₇	—	4	1	no	—
211	C₂³×C₇	—	4	1	no	—
212	C₂³×C₇	—	4	1	no	—
213	C₂³×C₇	—	4	1	no	—
214	C₂³×C₇	—	4	1	no	—
215	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
216	C₇×Q₈	—	2	4	no	—
217	C₇×Q₈	—	2	4	no	—
218	C₇×Q₈	—	4	2	no	—
219	C₇×Q₈	—	4	2	no	—
220	C₇×D₄	—	2	2	no	—
221	C₇×D₄	—	2	2	no	—
222	C₇×D₄	—	4	1	no	—
223	C₇×D₄	—	4	1	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	1	no	—
226	C₂×C₄×C₇	—	2	1	no	—
227	C₂×C₄×C₇	—	2	1	no	—
228	C₂×C₄×C₇	—	2	1	no	—
229	C₂×C₄×C₇	—	2	1	no	—
230	C₂×C₄×C₇	—	2	1	no	—
231	C₂×C₄×C₇	—	2	1	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	4	no	—
234	C₂×C₄×C₇	—	2	1	no	—
235	C₂×C₄×C₇	—	2	1	no	—
236	C₂×C₄×C₇	—	2	1	no	—
237	C₂×C₄×C₇	—	4	2	no	—
238	C₂×C₄×C₇	—	4	2	no	—
239	C₂×C₄×C₇	—	4	1	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	8	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	—	4	1	no	—
245	C₂×C₄×C₇	—	4	1	no	—
246	C₂×C₄×C₇	—	4	1	no	—
247	C₂×C₄×C₇	—	4	2	no	—
248	C₂×C₄×C₇	—	4	1	no	—
249	C₂×C₄×C₇	—	4	1	no	—
250	C₂×C₄×C₇	—	4	2	no	—
251	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
254	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
255	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
256	C₂×C₄×C₇	#135	1	2	yes	—
257	C₂×C₄×C₇	#137	1	2	yes	—
258	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
259	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
263	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
264	C₇×C₈	—	2	4	no	—
265	C₇×C₈	—	4	2	no	—
266	C₂²×Q₈	C₂³×C₇	1	2	yes	—
267	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
268	C₂³×C₄	—	2	1	no	—
269	C₂×SD₁₆	—	4	2	no	—
270	C₂×SD₁₆	—	4	2	no	—
271	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	—
272	C₂²×C₈	C₂³×C₇	1	2	yes	—
273	C₂²×C₈	C₂³×C₇	1	2	yes	—
274	C₄⋊Q₈	—	2	2	no	—
275	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
276	C₄×Q₈	—	4	2	no	—
277	C₄×Q₈	—	4	2	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
280	#117	—	4	4	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₄²	—	2	1	no	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
300	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
301	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
302	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
305	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
306	#106	—	4	2	no	—
307	C₄⋊C₈	—	2	2	no	—
308	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
309	#103	—	4	2	no	—
310	#103	—	4	2	no	—
311	#103	C₂×C₄×C₇	1	8	yes	—
312	#102	—	4	2	no	—
313	C₄×C₈	—	4	2	no	—
314	#95	—	2	1	no	—
315	#95	—	2	1	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	1	no	—
319	C₂²×C₇	—	4	2	no	—
320	C₂²×C₇	—	4	2	no	—
321	C₂²×C₇	—	4	2	no	—
322	C₂²×C₇	—	8	1	no	—
323	C₂²×C₇	—	8	1	no	—
324	C₂²×C₇	—	8	1	no	—
325	C₂²×C₇	—	8	1	no	—
326	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
327	C₂²×C₇	#535	1	1	yes	characteristic
328	C₂²×C₇	#537	1	1	yes	characteristic
329	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
330	C₂²×C₇	C₂×#135	1	1	yes	characteristic
331	C₂²×C₇	C₂×#137	1	1	yes	characteristic
332	C₂²×C₇	#575	1	1	yes	characteristic
333	C₄×C₇	—	2	2	no	—
334	C₄×C₇	—	2	2	no	—
335	C₄×C₇	—	2	2	no	—
336	C₄×C₇	—	2	2	no	—
337	C₄×C₇	—	4	4	no	—
338	C₄×C₇	—	4	2	no	—
339	C₄×C₇	—	4	1	no	—
340	C₄×C₇	—	8	1	no	—
341	C₄×C₇	—	8	2	no	—
342	C₄×C₇	—	8	1	no	—
343	C₄×C₇	C₄×SD₁₆	1	4	yes	—
344	C₄×C₇	#518	1	1	yes	characteristic
345	C₄×C₇	#520	1	1	yes	characteristic
346	C₄×C₇	#540	1	1	yes	characteristic
347	C₄×C₇	#542	1	1	yes	characteristic
348	C₂⁴	—	2	1	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
370	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
371	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
372	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
373	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
374	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
375	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
376	SD₁₆	—	4	8	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	4	4	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
398	C₂²⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	—	4	2	no	—
400	C₂²⋊C₄	—	4	2	no	—
401	C₂²⋊C₄	—	4	2	no	—
402	C₂²⋊C₄	—	4	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	4	1	no	—
410	C₄²	—	4	4	no	—
411	C₄²	—	4	2	no	—
412	C₄²	C₇×SD₁₆	1	4	yes	complemented
413	C₄²	C₂²×C₄×C₇	1	2	yes	—
414	C₄²	C₇×#41	1	2	yes	—
415	C₂×C₇	—	8	2	no	—
416	C₂×C₇	#2324	1	1	yes	characteristic
417	C₂×C₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
418	C₂×C₇	C₄×#137	1	1	yes	characteristic
419	C₂×C₇	#3206	1	1	yes	characteristic
420	C₂×C₇	#3212	1	1	yes	characteristic
421	C₂×C₇	#3246	1	1	yes	characteristic
422	C₂×C₇	#3248	1	1	yes	characteristic
423	C₂³	—	2	2	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
431	Q₈	—	2	4	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	4	1	no	—
438	D₄	—	4	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
467	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
468	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
469	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
470	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
471	C₂×C₄	C₇×#135	1	2	yes	—
472	C₂×C₄	C₇×#137	1	2	yes	—
473	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
474	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
475	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
476	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
477	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
479	C₈	—	2	4	no	—
480	C₈	—	4	2	no	—
481	C₇	#22146	1	1	yes	characteristic, direct product factor, Sylow
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	8	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
490	C₂²	C₇×#535	1	1	yes	characteristic
491	C₂²	C₇×#537	1	1	yes	characteristic
492	C₂²	C₂²×C₇×SD₁₆	1	1	yes	characteristic
493	C₂²	C₂×C₇×#135	1	1	yes	characteristic
494	C₂²	C₂×C₇×#137	1	1	yes	characteristic
495	C₂²	C₇×#575	1	1	yes	characteristic
496	C₄	—	2	2	no	—
497	C₄	—	2	2	no	—
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	4	no	—
503	C₄	—	8	1	no	—
504	C₄	—	8	2	no	—
505	C₄	—	8	1	no	—
506	C₄	C₄×C₇×SD₁₆	1	4	yes	—
507	C₄	C₇×#518	1	1	yes	characteristic
508	C₄	C₇×#520	1	1	yes	characteristic
509	C₄	C₇×#540	1	1	yes	characteristic
510	C₄	C₇×#542	1	1	yes	characteristic
511	C₂	—	8	2	no	—
512	C₂	C₇×#2324	1	1	yes	characteristic
513	C₂	C₂×C₄×C₇×SD₁₆	1	1	yes	characteristic
514	C₂	C₄×C₇×#137	1	1	yes	characteristic
515	C₂	C₇×#3206	1	1	yes	characteristic
516	C₂	C₇×#3212	1	1	yes	characteristic
517	C₂	C₇×#3246	1	1	yes	characteristic
518	C₂	C₇×#3248	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	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×C₄⋊Q₈	C₂	semidirect
4. C₇×#1981	C₂	semidirect
6. C₇×#1948	C₂	semidirect
7. C₇×#1933	C₂	semidirect
9. C₇×#1865	C₂	semidirect
11. C₄×C₇×C₄⋊C₈	C₂	semidirect
17. C₇×#461	C₄	semidirect
44. #22146	C₇	direct (N × H)
99. C₄×C₄⋊Q₈	C₂×C₇	semidirect
101. #1981	C₂×C₇	semidirect
103. #1948	C₂×C₇	semidirect
104. #1933	C₂×C₇	semidirect
106. #1865	C₂×C₇	semidirect
108. C₄×C₄⋊C₈	C₂×C₇	semidirect
173. C₄²×C₇	SD₁₆	semidirect
181. #461	C₄×C₇	semidirect
412. C₄²	C₇×SD₁₆	semidirect
481. C₇	#22146	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×#461	2
C₄×C₇×C₄⋊Q₈	2
C₇×#2293	2
C₇×#1981	2
C₇×#1968	2
C₇×#1948	2
C₇×#1933	2
C₇×#1880	2
C₇×#1865	2
C₇×#1802	2
C₄×C₇×C₄⋊C₈	2
#22146	7

Subgroups — C7×#22146 #13199362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22146 #13199362