Subgroups — C₇×#60649 #13237865

Order 1792.

3178 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#577	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#576	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3500	C₂	1	2	yes	maximal, complemented
5	C₇×#3156	C₂	1	3	yes	maximal, complemented
6	C₇×#3155	C₂	1	3	yes	maximal, complemented
7	C₇×#3154	C₂	1	6	yes	maximal, complemented
8	C₇×#3153	C₂	1	6	yes	maximal, complemented
9	C₇×#3147	C₂	1	3	yes	maximal, complemented
10	C₂×C₇×#495	C₂	1	3	yes	maximal, complemented
11	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₇×#578	—	2	8	no	—
15	C₇×#577	—	2	16	no	—
16	C₇×#576	—	2	8	no	—
17	C₂×C₇×#137	C₂²	1	6	yes	complemented
18	C₂×C₇×#137	C₂²	1	6	yes	complemented
19	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	—
20	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
21	C₂×C₇×#135	C₂²	1	3	yes	—
22	C₂×C₇×#135	C₂²	1	3	yes	complemented
23	C₂×C₇×#135	C₂²	1	6	yes	complemented
24	C₂²×C₇×Q₁₆	C₂²	1	3	yes	complemented
25	C₂²×C₇×SD₁₆	C₂²	1	6	yes	complemented
26	C₂²×C₇×D₈	C₂²	1	3	yes	—
27	C₂×C₇×#131	C₂²	1	1	yes	characteristic
28	C₂×C₇×#131	C₂²	1	1	yes	characteristic
29	C₇×#548	—	2	4	no	—
30	C₇×#546	—	2	12	no	—
31	C₇×#536	C₂²	1	6	yes	complemented
32	C₇×#535	C₂²	1	6	yes	complemented
33	C₇×#532	C₂²	1	6	yes	complemented
34	C₂×C₇×#124	C₂²	1	6	yes	complemented
35	C₄×C₇×#41	C₂²	1	1	yes	characteristic
36	C₇×#497	C₂²	1	12	yes	complemented
37	C₇×#496	C₂²	1	12	yes	complemented
38	C₇×#495	C₂²	1	6	yes	complemented
39	C₇×#495	C₂²	1	3	yes	complemented
40	C₇×#495	C₂²	1	6	yes	complemented
41	C₇×#495	C₂²	1	6	yes	complemented
42	C₇×#495	C₂²	1	1	yes	characteristic, complemented
43	C₇×#495	C₂²	1	6	yes	complemented
44	C₇×#494	C₂²	1	3	yes	complemented
45	C₇×#493	C₂²	1	3	yes	complemented
46	C₇×#492	C₂²	1	6	yes	complemented
47	C₇×#405	C₂²	1	3	yes	complemented
48	C₇×#405	C₂²	1	3	yes	complemented
49	C₄×C₇×M₁₆	C₂²	1	3	yes	complemented
50	C₂×C₄×C₇×C₈	C₂²	1	3	yes	complemented
51	#60649	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₇×2₋¹⁺⁴	—	2	8	no	—
53	C₇×2₋¹⁺⁴	—	2	8	no	—
54	C₇×H(5,𝔽₂)	—	2	8	no	—
55	C₇×H(5,𝔽₂)	—	2	8	no	—
56	C₂×C₇×#41	—	2	2	no	—
57	C₂×C₇×#41	—	2	6	no	—
58	C₂×C₇×#41	C₂³	1	6	yes	—
59	C₂×C₇×#41	C₂³	1	6	yes	—
60	C₂×C₇×#41	C₂³	1	1	yes	characteristic
61	C₂²×C₇×Q₈	—	2	2	no	—
62	C₂²×C₇×Q₈	C₂³	1	6	yes	—
63	C₂²×C₇×D₄	—	2	6	no	—
64	C₂²×C₇×D₄	C₂³	1	6	yes	—
65	C₇×#137	—	2	24	no	—
66	C₇×#137	—	2	24	no	—
67	C₇×AGL(1,ℤ/8)	—	2	24	no	—
68	C₇×AGL(1,ℤ/8)	—	2	24	no	—
69	C₇×#135	—	2	12	no	—
70	C₇×#135	—	2	24	no	—
71	C₇×#135	—	2	12	no	—
72	C₂×C₇×Q₁₆	—	2	12	no	—
73	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₇×Q₁₆	C₂³	1	3	yes	—
75	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
76	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
77	C₂×C₇×SD₁₆	—	2	24	no	—
78	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
79	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
80	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
81	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
82	C₂×C₇×SD₁₆	C₂³	1	12	yes	—
83	C₂×C₇×D₈	—	2	12	no	—
84	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
85	C₂×C₇×D₈	C₂³	1	3	yes	—
86	C₂×C₇×D₈	C₂³	1	6	yes	—
87	C₂×C₇×D₈	C₂³	1	6	yes	—
88	C₇×#131	D₄	1	8	yes	complemented
89	C₇×#131	D₄	1	8	yes	complemented
90	C₂×C₇×M₁₆	C₂³	1	3	yes	—
91	C₂×C₇×M₁₆	C₂³	1	3	yes	—
92	C₂²×C₇×C₈	C₂³	1	3	yes	—
93	C₂²×C₇×C₈	C₂³	1	3	yes	—
94	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
95	C₇×#127	C₂³	1	6	yes	—
96	C₇×#124	—	2	12	no	—
97	C₇×#124	C₂³	1	6	yes	—
98	C₇×#124	C₂³	1	2	yes	—
99	C₇×#124	C₂³	1	12	yes	—
100	C₇×#123	—	2	12	no	—
101	C₇×C₂²⋊Q₈	—	2	12	no	—
102	C₇×#121	—	2	12	no	—
103	C₇×#121	—	2	12	no	—
104	C₇×#120	—	2	12	no	—
105	C₄×C₇×Q₈	D₄	1	2	yes	—
106	C₄×C₇×D₄	—	2	12	no	—
107	C₄×C₇×D₄	D₄	1	6	yes	—
108	C₇×#117	C₂³	1	3	yes	—
109	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
110	C₂×C₄²×C₇	C₂³	1	3	yes	—
111	C₇×#97	C₂³	1	6	yes	—
112	C₄×C₇×C₈	C₂³	1	3	yes	—
113	C₄×C₇×C₈	C₂³	1	3	yes	—
114	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
115	C₄×C₇×C₈	C₂³	1	3	yes	—
116	C₂×#578	C₂×C₇	1	1	yes	characteristic, complemented
117	C₂×#577	C₂×C₇	1	2	yes	complemented
118	C₂×#576	C₂×C₇	1	1	yes	characteristic, complemented
119	#3500	C₂×C₇	1	2	yes	complemented
120	#3156	C₂×C₇	1	3	yes	complemented
121	#3155	C₂×C₇	1	3	yes	complemented
122	#3154	C₂×C₇	1	6	yes	complemented
123	#3153	C₂×C₇	1	6	yes	complemented
124	#3147	C₂×C₇	1	3	yes	complemented
125	C₂×#495	C₂×C₇	1	3	yes	complemented
126	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
127	C₂⁴×C₇	—	2	6	no	—
128	C₇×#41	—	2	12	no	—
129	C₇×#41	—	2	12	no	—
130	C₇×#41	—	2	12	no	—
131	C₇×#41	—	2	12	no	—
132	C₇×#41	—	4	8	no	—
133	C₇×#41	—	4	24	no	—
134	C₇×#41	C₂×D₄	1	4	yes	—
135	C₇×#41	C₂×D₄	1	4	yes	—
136	C₂×C₇×Q₈	—	2	6	no	—
137	C₂×C₇×Q₈	—	2	12	no	—
138	C₂×C₇×Q₈	—	2	12	no	—
139	C₂×C₇×Q₈	—	4	8	no	—
140	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₇×Q₈	C₂⁴	1	6	yes	—
142	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
143	C₂×C₇×D₄	—	2	6	no	—
144	C₂×C₇×D₄	—	2	6	no	—
145	C₂×C₇×D₄	—	2	12	no	—
146	C₂×C₇×D₄	—	2	12	no	—
147	C₂×C₇×D₄	—	2	6	no	—
148	C₂×C₇×D₄	—	2	12	no	—
149	C₂×C₇×D₄	—	4	24	no	—
150	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
151	C₂×C₇×D₄	C₂⁴	1	6	yes	—
152	C₂×C₇×D₄	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₇	—	2	6	no	—
154	C₂²×C₄×C₇	—	2	6	no	—
155	C₂²×C₄×C₇	—	2	6	no	—
156	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
157	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
158	C₇×Q₁₆	—	2	6	no	—
159	C₇×Q₁₆	—	2	12	no	—
160	C₇×Q₁₆	—	2	2	no	—
161	C₇×Q₁₆	—	2	12	no	—
162	C₇×SD₁₆	—	2	12	no	—
163	C₇×SD₁₆	—	2	24	no	—
164	C₇×SD₁₆	—	2	4	no	—
165	C₇×SD₁₆	—	2	12	no	—
166	C₇×SD₁₆	—	2	12	no	—
167	C₇×D₈	—	2	6	no	—
168	C₇×D₈	—	2	12	no	—
169	C₇×D₈	—	2	2	no	—
170	C₇×D₈	—	2	12	no	—
171	C₇×M₁₆	C₂×D₄	1	12	yes	—
172	C₇×M₁₆	C₂×D₄	1	12	yes	—
173	C₂×C₇×C₈	C₂×D₄	1	12	yes	—
174	C₂×C₇×C₈	C₂×D₄	1	12	yes	—
175	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
176	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
177	C₂×C₇×C₈	C₂⁴	1	3	yes	—
178	C₂×C₇×C₈	C₂⁴	1	3	yes	—
179	C₇×C₄⋊C₄	—	2	12	no	—
180	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
181	C₇×C₂²⋊C₄	—	2	12	no	—
182	C₇×C₂²⋊C₄	—	2	24	no	—
183	C₇×C₂²⋊C₄	—	2	4	no	—
184	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
185	C₄²×C₇	C₂×D₄	1	6	yes	—
186	C₄²×C₇	C₂⁴	1	1	yes	characteristic
187	C₄²×C₇	C₂⁴	1	3	yes	—
188	C₂×2₋¹⁺⁴	C₂²×C₇	1	2	yes	—
189	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
190	#578	—	2	8	no	—
191	#577	—	2	16	no	—
192	#576	—	2	8	no	—
193	C₂×#137	C₂²×C₇	1	6	yes	complemented
194	C₂×#137	C₂²×C₇	1	6	yes	complemented
195	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	6	yes	—
196	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	6	yes	complemented
197	C₂×#135	C₂²×C₇	1	3	yes	—
198	C₂×#135	C₂²×C₇	1	6	yes	complemented
199	C₂×#135	C₂²×C₇	1	3	yes	complemented
200	C₂²×Q₁₆	C₂²×C₇	1	3	yes	complemented
201	C₂²×SD₁₆	C₂²×C₇	1	6	yes	complemented
202	C₂²×D₈	C₂²×C₇	1	3	yes	—
203	C₂×#131	C₂²×C₇	1	1	yes	characteristic
204	C₂×#131	C₂²×C₇	1	1	yes	characteristic
205	#548	—	2	4	no	—
206	#546	—	2	12	no	—
207	#536	C₂²×C₇	1	6	yes	complemented
208	#535	C₂²×C₇	1	6	yes	complemented
209	#532	C₂²×C₇	1	6	yes	complemented
210	C₂×#124	C₂²×C₇	1	6	yes	complemented
211	C₄×#41	C₂²×C₇	1	1	yes	characteristic
212	#497	C₂²×C₇	1	12	yes	complemented
213	#496	C₂²×C₇	1	12	yes	complemented
214	#495	C₂²×C₇	1	3	yes	complemented
215	#495	C₂²×C₇	1	6	yes	complemented
216	#495	C₂²×C₇	1	6	yes	complemented
217	#495	C₂²×C₇	1	6	yes	complemented
218	#495	C₂²×C₇	1	1	yes	characteristic, complemented
219	#495	C₂²×C₇	1	6	yes	complemented
220	#494	C₂²×C₇	1	3	yes	complemented
221	#493	C₂²×C₇	1	3	yes	complemented
222	#492	C₂²×C₇	1	6	yes	complemented
223	#405	C₂²×C₇	1	3	yes	complemented
224	#405	C₂²×C₇	1	3	yes	complemented
225	C₄×M₁₆	C₂²×C₇	1	3	yes	complemented
226	C₂×C₄×C₈	C₂²×C₇	1	3	yes	complemented
227	C₂³×C₇	—	2	6	no	—
228	C₂³×C₇	—	2	2	no	—
229	C₂³×C₇	—	4	12	no	—
230	C₂³×C₇	—	8	12	no	—
231	C₂³×C₇	C₂²×D₄	1	3	yes	—
232	C₇×Q₈	—	2	6	no	—
233	C₇×Q₈	—	2	2	no	—
234	C₇×Q₈	—	2	6	no	—
235	C₇×Q₈	—	2	2	no	—
236	C₇×Q₈	—	4	12	no	—
237	C₇×Q₈	#127	1	4	yes	—
238	C₇×D₄	—	2	6	no	—
239	C₇×D₄	—	2	2	no	—
240	C₇×D₄	—	2	6	no	—
241	C₇×D₄	—	2	2	no	—
242	C₇×D₄	—	4	12	no	—
243	C₇×D₄	—	4	12	no	—
244	C₇×D₄	—	4	12	no	—
245	C₇×D₄	—	8	12	no	—
246	C₇×D₄	#127	1	12	yes	—
247	C₂×C₄×C₇	—	2	6	no	—
248	C₂×C₄×C₇	—	2	2	no	—
249	C₂×C₄×C₇	—	4	6	no	—
250	C₂×C₄×C₇	—	4	12	no	—
251	C₂×C₄×C₇	—	4	12	no	—
252	C₂×C₄×C₇	—	4	12	no	—
253	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
255	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
256	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
257	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
258	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
259	C₇×C₈	C₂²×D₄	1	2	yes	—
260	C₇×C₈	C₂²×D₄	1	2	yes	—
261	C₇×C₈	C₂²×D₄	1	6	yes	—
262	C₇×C₈	C₂²×D₄	1	6	yes	—
263	2₋¹⁺⁴	—	2	8	no	—
264	2₋¹⁺⁴	—	2	8	no	—
265	H(5,𝔽₂)	—	2	8	no	—
266	H(5,𝔽₂)	—	2	8	no	—
267	C₂×#41	—	2	2	no	—
268	C₂×#41	—	2	6	no	—
269	C₂×#41	C₂³×C₇	1	6	yes	—
270	C₂×#41	C₂³×C₇	1	6	yes	—
271	C₂×#41	C₂³×C₇	1	1	yes	characteristic
272	C₂²×Q₈	—	2	2	no	—
273	C₂²×Q₈	C₂³×C₇	1	6	yes	—
274	C₂²×D₄	—	2	6	no	—
275	C₂²×D₄	C₂³×C₇	1	6	yes	—
276	#137	—	2	24	no	—
277	#137	—	2	24	no	—
278	AGL(1,ℤ/8)	—	2	24	no	—
279	AGL(1,ℤ/8)	—	2	24	no	—
280	#135	—	2	12	no	—
281	#135	—	2	24	no	—
282	#135	—	2	12	no	—
283	C₂×Q₁₆	—	2	12	no	—
284	C₂×Q₁₆	C₂³×C₇	1	3	yes	—
285	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
286	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
287	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
288	C₂×SD₁₆	—	2	24	no	—
289	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
290	C₂×SD₁₆	C₂³×C₇	1	12	yes	—
291	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
292	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
293	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
294	C₂×D₈	—	2	12	no	—
295	C₂×D₈	C₂³×C₇	1	3	yes	—
296	C₂×D₈	C₂³×C₇	1	6	yes	—
297	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
298	C₂×D₈	C₂³×C₇	1	6	yes	—
299	#131	C₇×D₄	1	8	yes	complemented
300	#131	C₇×D₄	1	8	yes	complemented
301	C₂×M₁₆	C₂³×C₇	1	3	yes	—
302	C₂×M₁₆	C₂³×C₇	1	3	yes	—
303	C₂²×C₈	C₂³×C₇	1	3	yes	—
304	C₂²×C₈	C₂³×C₇	1	3	yes	—
305	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
306	#127	C₂³×C₇	1	6	yes	—
307	#124	—	2	12	no	—
308	#124	C₂³×C₇	1	6	yes	—
309	#124	C₂³×C₇	1	12	yes	—
310	#124	C₂³×C₇	1	2	yes	—
311	#123	—	2	12	no	—
312	C₂²⋊Q₈	—	2	12	no	—
313	#121	—	2	12	no	—
314	#121	—	2	12	no	—
315	#120	—	2	12	no	—
316	C₄×Q₈	C₇×D₄	1	2	yes	—
317	C₄×D₄	—	2	12	no	—
318	C₄×D₄	C₇×D₄	1	6	yes	—
319	#117	C₂³×C₇	1	3	yes	—
320	C₂×C₂²⋊C₄	—	2	12	no	—
321	C₂×C₄²	C₂³×C₇	1	3	yes	—
322	#97	C₂³×C₇	1	6	yes	—
323	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₇	1	3	yes	—
325	C₄×C₈	C₂³×C₇	1	3	yes	—
326	C₄×C₈	C₂³×C₇	1	3	yes	—
327	C₂²×C₇	—	2	6	no	—
328	C₂²×C₇	—	4	4	no	—
329	C₂²×C₇	—	4	12	no	—
330	C₂²×C₇	—	4	2	no	—
331	C₂²×C₇	—	4	6	no	—
332	C₂²×C₇	—	16	12	no	—
333	C₂²×C₇	C₂×#127	1	6	yes	—
334	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
335	C₄×C₇	—	2	2	no	—
336	C₄×C₇	—	2	6	no	—
337	C₄×C₇	—	4	2	no	—
338	C₄×C₇	—	4	6	no	—
339	C₄×C₇	C₂×#127	1	6	yes	—
340	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
341	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
342	C₂⁴	—	2	6	no	—
343	#41	—	2	12	no	—
344	#41	—	2	12	no	—
345	#41	—	2	12	no	—
346	#41	—	2	12	no	—
347	#41	—	4	8	no	—
348	#41	—	4	24	no	—
349	#41	C₂×C₇×D₄	1	4	yes	—
350	#41	C₂×C₇×D₄	1	4	yes	—
351	C₂×Q₈	—	2	6	no	—
352	C₂×Q₈	—	2	12	no	—
353	C₂×Q₈	—	2	12	no	—
354	C₂×Q₈	—	4	8	no	—
355	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
356	C₂×Q₈	C₂⁴×C₇	1	6	yes	—
357	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
358	C₂×D₄	—	2	6	no	—
359	C₂×D₄	—	2	6	no	—
360	C₂×D₄	—	2	12	no	—
361	C₂×D₄	—	2	12	no	—
362	C₂×D₄	—	2	6	no	—
363	C₂×D₄	—	2	12	no	—
364	C₂×D₄	—	4	24	no	—
365	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
366	C₂×D₄	C₂⁴×C₇	1	6	yes	—
367	C₂×D₄	C₂⁴×C₇	1	2	yes	—
368	C₂²×C₄	—	2	6	no	—
369	C₂²×C₄	—	2	6	no	—
370	C₂²×C₄	—	2	6	no	—
371	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
372	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
373	Q₁₆	—	2	6	no	—
374	Q₁₆	—	2	12	no	—
375	Q₁₆	—	2	2	no	—
376	Q₁₆	—	2	12	no	—
377	SD₁₆	—	2	12	no	—
378	SD₁₆	—	2	24	no	—
379	SD₁₆	—	2	4	no	—
380	SD₁₆	—	2	12	no	—
381	SD₁₆	—	2	12	no	—
382	D₈	—	2	6	no	—
383	D₈	—	2	12	no	—
384	D₈	—	2	2	no	—
385	D₈	—	2	12	no	—
386	M₁₆	C₂×C₇×D₄	1	12	yes	—
387	M₁₆	C₂×C₇×D₄	1	12	yes	—
388	C₂×C₈	C₂×C₇×D₄	1	12	yes	—
389	C₂×C₈	C₂×C₇×D₄	1	12	yes	—
390	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
391	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
392	C₂×C₈	C₂⁴×C₇	1	3	yes	—
393	C₂×C₈	C₂⁴×C₇	1	3	yes	—
394	C₄⋊C₄	—	2	12	no	—
395	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
396	C₂²⋊C₄	—	2	12	no	—
397	C₂²⋊C₄	—	2	24	no	—
398	C₂²⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
400	C₄²	C₂×C₇×D₄	1	6	yes	—
401	C₄²	C₂⁴×C₇	1	1	yes	characteristic
402	C₄²	C₂⁴×C₇	1	3	yes	—
403	C₂×C₇	—	2	6	no	—
404	C₂×C₇	—	8	2	no	—
405	C₂×C₇	—	8	6	no	—
406	C₂×C₇	C₂²×#127	1	1	yes	characteristic
407	C₂×C₇	#3584	1	2	yes	—
408	C₂³	—	2	6	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	4	12	no	—
411	C₂³	—	8	12	no	—
412	C₂³	C₂²×C₇×D₄	1	3	yes	—
413	Q₈	—	2	6	no	—
414	Q₈	—	2	2	no	—
415	Q₈	—	2	6	no	—
416	Q₈	—	2	2	no	—
417	Q₈	—	4	12	no	—
418	Q₈	C₇×#127	1	4	yes	—
419	D₄	—	2	6	no	—
420	D₄	—	2	2	no	—
421	D₄	—	2	6	no	—
422	D₄	—	2	2	no	—
423	D₄	—	4	12	no	—
424	D₄	—	4	12	no	—
425	D₄	—	4	12	no	—
426	D₄	—	8	12	no	—
427	D₄	C₇×#127	1	12	yes	—
428	C₂×C₄	—	2	6	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	6	no	—
431	C₂×C₄	—	4	12	no	—
432	C₂×C₄	—	4	12	no	—
433	C₂×C₄	—	4	12	no	—
434	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
435	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
436	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
437	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
438	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
439	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
440	C₈	C₂²×C₇×D₄	1	2	yes	—
441	C₈	C₂²×C₇×D₄	1	2	yes	—
442	C₈	C₂²×C₇×D₄	1	6	yes	—
443	C₈	C₂²×C₇×D₄	1	6	yes	—
444	C₇	#60649	1	1	yes	characteristic, direct product factor, Sylow
445	C₂²	—	2	6	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	12	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	6	no	—
450	C₂²	—	16	12	no	—
451	C₂²	C₂×C₇×#127	1	6	yes	—
452	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	2	6	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	6	no	—
457	C₄	C₂×C₇×#127	1	6	yes	—
458	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
459	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
460	C₂	—	2	6	no	—
461	C₂	—	8	6	no	—
462	C₂	—	8	2	no	—
463	C₂	C₂²×C₇×#127	1	1	yes	characteristic
464	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
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. C₂×C₇×#578	C₂	semidirect
2. C₂×C₇×#577	C₂	semidirect
3. C₂×C₇×#576	C₂	semidirect
4. C₇×#3500	C₂	semidirect
5. C₇×#3156	C₂	semidirect
6. C₇×#3155	C₂	semidirect
7. C₇×#3154	C₂	semidirect
8. C₇×#3153	C₂	semidirect
9. C₇×#3147	C₂	semidirect
10. C₂×C₇×#495	C₂	semidirect
11. C₄×C₇×#131	C₂	semidirect
17. C₂×C₇×#137	C₂²	semidirect
18. C₂×C₇×#137	C₂²	semidirect
20. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
22. C₂×C₇×#135	C₂²	semidirect
23. C₂×C₇×#135	C₂²	semidirect
24. C₂²×C₇×Q₁₆	C₂²	semidirect
25. C₂²×C₇×SD₁₆	C₂²	semidirect
31. C₇×#536	C₂²	semidirect
32. C₇×#535	C₂²	semidirect
33. C₇×#532	C₂²	semidirect
34. C₂×C₇×#124	C₂²	semidirect
36. C₇×#497	C₂²	semidirect
37. C₇×#496	C₂²	semidirect
38. C₇×#495	C₂²	semidirect
39. C₇×#495	C₂²	semidirect
40. C₇×#495	C₂²	semidirect
41. C₇×#495	C₂²	semidirect
42. C₇×#495	C₂²	semidirect
43. C₇×#495	C₂²	semidirect
44. C₇×#494	C₂²	semidirect
45. C₇×#493	C₂²	semidirect
46. C₇×#492	C₂²	semidirect
47. C₇×#405	C₂²	semidirect
48. C₇×#405	C₂²	semidirect
49. C₄×C₇×M₁₆	C₂²	semidirect
50. C₂×C₄×C₇×C₈	C₂²	semidirect
51. #60649	C₇	direct (N × H)
88. C₇×#131	D₄	semidirect
89. C₇×#131	D₄	semidirect
116. C₂×#578	C₂×C₇	semidirect
117. C₂×#577	C₂×C₇	semidirect
118. C₂×#576	C₂×C₇	semidirect
119. #3500	C₂×C₇	semidirect
120. #3156	C₂×C₇	semidirect
121. #3155	C₂×C₇	semidirect
122. #3154	C₂×C₇	semidirect
123. #3153	C₂×C₇	semidirect
124. #3147	C₂×C₇	semidirect
125. C₂×#495	C₂×C₇	semidirect
126. C₄×#131	C₂×C₇	semidirect
193. C₂×#137	C₂²×C₇	semidirect
194. C₂×#137	C₂²×C₇	semidirect
196. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
198. C₂×#135	C₂²×C₇	semidirect
199. C₂×#135	C₂²×C₇	semidirect
200. C₂²×Q₁₆	C₂²×C₇	semidirect
201. C₂²×SD₁₆	C₂²×C₇	semidirect
207. #536	C₂²×C₇	semidirect
208. #535	C₂²×C₇	semidirect
209. #532	C₂²×C₇	semidirect
210. C₂×#124	C₂²×C₇	semidirect
212. #497	C₂²×C₇	semidirect
213. #496	C₂²×C₇	semidirect
214. #495	C₂²×C₇	semidirect
215. #495	C₂²×C₇	semidirect
216. #495	C₂²×C₇	semidirect
217. #495	C₂²×C₇	semidirect
218. #495	C₂²×C₇	semidirect
219. #495	C₂²×C₇	semidirect
220. #494	C₂²×C₇	semidirect
221. #493	C₂²×C₇	semidirect
222. #492	C₂²×C₇	semidirect
223. #405	C₂²×C₇	semidirect
224. #405	C₂²×C₇	semidirect
225. C₄×M₁₆	C₂²×C₇	semidirect
226. C₂×C₄×C₈	C₂²×C₇	semidirect
299. #131	C₇×D₄	semidirect
300. #131	C₇×D₄	semidirect
444. C₇	#60649	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×#578	2
C₂×C₇×#577	2
C₂×C₇×#576	2
C₇×#3500	2
C₇×#3156	2
C₇×#3155	2
C₇×#3154	2
C₇×#3153	2
C₇×#3147	2
C₂×C₇×#495	2
C₄×C₇×#131	2
#60649	7

Subgroups — C7×#60649 #13237865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60649 #13237865