Subgroups — C₂×C₇×#3151 #13237852

Order 1792.

3950 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
2	C₂×C₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3151	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₇×#496	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×#493	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#492	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×#41	C₂²	1	2	yes	—
10	C₂³×C₇×D₄	—	2	2	no	—
11	C₂³×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₇×AGL(1,ℤ/8)	—	2	32	no	—
13	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	32	yes	complemented
14	C₂²×C₇×SD₁₆	C₂²	1	8	yes	complemented
15	C₂²×C₇×D₈	C₂²	1	8	yes	complemented
16	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
17	C₇×#532	—	2	4	no	—
18	C₇×#532	C₂²	1	8	yes	complemented
19	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₇×#127	—	2	2	no	—
21	C₂×C₇×#127	—	2	4	no	—
22	C₂×C₇×#127	C₂²	1	2	yes	—
23	C₂×C₇×#127	C₂²	1	2	yes	—
24	C₂×C₇×#127	C₂²	1	8	yes	complemented
25	C₂×C₇×#127	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₇×#124	C₂²	1	2	yes	complemented
27	C₂×C₇×#121	—	2	2	no	—
28	C₂×C₄×C₇×D₄	—	2	2	no	—
29	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
30	C₇×#496	C₂²	1	32	yes	complemented
31	C₇×#493	C₂²	1	16	yes	complemented
32	C₇×#492	C₂²	1	16	yes	complemented
33	C₄×C₇×M₁₆	C₂²	1	8	yes	complemented
34	C₂×C₇×#97	C₂²	1	2	yes	complemented
35	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
36	C₂×#3151	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂⁵×C₇	—	2	2	no	—
38	C₂×C₇×#41	—	2	4	no	—
39	C₂×C₇×#41	—	2	16	no	—
40	C₂×C₇×#41	C₂³	1	8	yes	—
41	C₂²×C₇×Q₈	C₂³	1	2	yes	—
42	C₂²×C₇×D₄	—	2	2	no	—
43	C₂²×C₇×D₄	—	2	16	no	—
44	C₂²×C₇×D₄	—	2	4	no	—
45	C₂²×C₇×D₄	—	2	4	no	—
46	C₂²×C₇×D₄	—	2	16	no	—
47	C₂²×C₇×D₄	—	2	4	no	—
48	C₂²×C₇×D₄	—	2	2	no	—
49	C₂²×C₇×D₄	—	2	2	no	—
50	C₂²×C₇×D₄	—	4	16	no	—
51	C₂²×C₇×D₄	C₂³	1	4	yes	—
52	C₂²×C₇×D₄	C₂³	1	8	yes	—
53	C₂²×C₇×D₄	C₂³	1	2	yes	—
54	C₂³×C₄×C₇	—	2	2	no	—
55	C₂³×C₄×C₇	D₄	1	2	yes	—
56	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
57	C₇×AGL(1,ℤ/8)	—	2	128	no	—
58	C₂×C₇×SD₁₆	—	2	32	no	—
59	C₂×C₇×SD₁₆	C₂³	1	32	yes	complemented
60	C₂×C₇×D₈	—	2	32	no	—
61	C₂×C₇×D₈	C₂³	1	32	yes	complemented
62	C₂×C₇×M₁₆	D₄	1	16	yes	complemented
63	C₂×C₇×M₁₆	C₂³	1	8	yes	—
64	C₂²×C₇×C₈	C₂³	1	4	yes	—
65	C₇×C₄⋊Q₈	—	2	2	no	—
66	C₇×C₄⋊Q₈	C₂³	1	4	yes	complemented
67	C₇×#127	—	2	4	no	—
68	C₇×#127	—	2	4	no	—
69	C₇×#127	—	2	16	no	—
70	C₇×#127	—	2	2	no	—
71	C₇×#127	C₂³	1	8	yes	—
72	C₇×#127	C₂³	1	8	yes	—
73	C₇×#127	C₂³	1	4	yes	complemented
74	C₇×#124	—	2	4	no	—
75	C₇×#124	C₂³	1	8	yes	complemented
76	C₇×#121	—	2	16	no	—
77	C₄×C₇×D₄	—	2	16	no	—
78	C₂×C₇×C₄⋊C₄	—	2	2	no	—
79	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
80	C₂×C₄²×C₇	D₄	1	2	yes	—
81	C₂×C₄²×C₇	D₄	1	4	yes	—
82	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₇	C₂³	1	4	yes	—
85	C₇×#97	C₂³	1	8	yes	complemented
86	C₄×C₇×C₈	C₂³	1	8	yes	complemented
87	C₂²×AGL(1,ℤ/8)	C₂×C₇	1	4	yes	complemented
88	C₂×#532	C₂×C₇	1	1	yes	characteristic, complemented
89	C₂²×#127	C₂×C₇	1	1	yes	characteristic, complemented
90	#3151	C₂×C₇	1	16	yes	direct product factor
91	C₂×#496	C₂×C₇	1	4	yes	complemented
92	C₂×#493	C₂×C₇	1	2	yes	complemented
93	C₂×#492	C₂×C₇	1	2	yes	complemented
94	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
95	C₂⁴×C₇	—	2	8	no	—
96	C₂⁴×C₇	—	2	2	no	—
97	C₂⁴×C₇	—	2	4	no	—
98	C₂⁴×C₇	—	4	16	no	—
99	C₂⁴×C₇	—	4	8	no	—
100	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
101	C₇×#41	—	2	32	no	—
102	C₇×#41	—	4	16	no	—
103	C₂×C₇×Q₈	—	2	2	no	—
104	C₂×C₇×Q₈	—	2	8	no	—
105	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
106	C₂×C₇×D₄	—	2	8	no	—
107	C₂×C₇×D₄	—	2	8	no	—
108	C₂×C₇×D₄	—	2	16	no	—
109	C₂×C₇×D₄	—	2	32	no	—
110	C₂×C₇×D₄	—	2	8	no	—
111	C₂×C₇×D₄	—	2	2	no	—
112	C₂×C₇×D₄	—	2	8	no	—
113	C₂×C₇×D₄	—	2	4	no	—
114	C₂×C₇×D₄	—	2	16	no	—
115	C₂×C₇×D₄	—	2	16	no	—
116	C₂×C₇×D₄	—	4	16	no	—
117	C₂×C₇×D₄	—	4	64	no	—
118	C₂×C₇×D₄	—	4	8	no	—
119	C₂×C₇×D₄	—	4	8	no	—
120	C₂×C₇×D₄	—	4	16	no	—
121	C₂×C₇×D₄	—	4	8	no	—
122	C₂×C₇×D₄	—	4	16	no	—
123	C₂×C₇×D₄	C₂⁴	1	4	yes	—
124	C₂×C₇×D₄	C₂⁴	1	8	yes	—
125	C₂²×C₄×C₇	—	2	8	no	—
126	C₂²×C₄×C₇	—	2	2	no	—
127	C₂²×C₄×C₇	—	2	8	no	—
128	C₂²×C₄×C₇	—	4	8	no	—
129	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
130	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
134	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
137	C₇×SD₁₆	—	2	64	no	—
138	C₇×D₈	—	2	64	no	—
139	C₇×M₁₆	C₂×D₄	1	32	yes	complemented
140	C₂×C₇×C₈	C₂×D₄	1	16	yes	complemented
141	C₂×C₇×C₈	C₂⁴	1	8	yes	—
142	C₇×C₄⋊C₄	—	2	8	no	—
143	C₇×C₂²⋊C₄	—	2	16	no	—
144	C₄²×C₇	C₂×D₄	1	8	yes	—
145	C₄²×C₇	C₂×D₄	1	2	yes	—
146	C₄²×C₇	C₂×D₄	1	2	yes	—
147	C₄²×C₇	C₂⁴	1	2	yes	—
148	C₄²×C₇	C₂⁴	1	2	yes	—
149	C₂²×#41	C₂²×C₇	1	2	yes	—
150	C₂³×D₄	—	2	2	no	—
151	C₂³×D₄	C₂²×C₇	1	2	yes	—
152	C₂×AGL(1,ℤ/8)	—	2	32	no	—
153	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	32	yes	complemented
154	C₂²×SD₁₆	C₂²×C₇	1	8	yes	complemented
155	C₂²×D₈	C₂²×C₇	1	8	yes	complemented
156	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
157	#532	—	2	4	no	—
158	#532	C₂²×C₇	1	8	yes	complemented
159	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
160	C₂×#127	—	2	2	no	—
161	C₂×#127	—	2	4	no	—
162	C₂×#127	C₂²×C₇	1	2	yes	—
163	C₂×#127	C₂²×C₇	1	2	yes	—
164	C₂×#127	C₂²×C₇	1	8	yes	complemented
165	C₂×#127	C₂²×C₇	1	1	yes	characteristic, complemented
166	C₂×#124	C₂²×C₇	1	2	yes	complemented
167	C₂×#121	—	2	2	no	—
168	C₂×C₄×D₄	—	2	2	no	—
169	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
170	#496	C₂²×C₇	1	32	yes	complemented
171	#493	C₂²×C₇	1	16	yes	complemented
172	#492	C₂²×C₇	1	16	yes	complemented
173	C₄×M₁₆	C₂²×C₇	1	8	yes	complemented
174	C₂×#97	C₂²×C₇	1	2	yes	complemented
175	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
176	C₂³×C₇	—	2	4	no	—
177	C₂³×C₇	—	2	4	no	—
178	C₂³×C₇	—	2	8	no	—
179	C₂³×C₇	—	4	32	no	—
180	C₂³×C₇	—	4	4	no	—
181	C₂³×C₇	—	4	16	no	—
182	C₂³×C₇	—	4	8	no	—
183	C₂³×C₇	—	4	16	no	—
184	C₂³×C₇	—	4	8	no	—
185	C₂³×C₇	—	8	32	no	—
186	C₂³×C₇	#127	1	4	yes	—
187	C₂³×C₇	C₂²×D₄	1	2	yes	—
188	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
189	C₇×Q₈	—	2	8	no	—
190	C₇×Q₈	—	4	4	no	—
191	C₇×D₄	—	2	8	no	—
192	C₇×D₄	—	2	16	no	—
193	C₇×D₄	—	4	32	no	—
194	C₇×D₄	—	4	16	no	—
195	C₇×D₄	—	4	16	no	—
196	C₇×D₄	—	4	8	no	—
197	C₇×D₄	—	4	4	no	—
198	C₇×D₄	—	4	16	no	—
199	C₇×D₄	—	4	32	no	—
200	C₂×C₄×C₇	—	2	4	no	—
201	C₂×C₄×C₇	—	2	16	no	—
202	C₂×C₄×C₇	—	2	4	no	—
203	C₂×C₄×C₇	—	4	4	no	—
204	C₂×C₄×C₇	—	4	16	no	—
205	C₂×C₄×C₇	#127	1	4	yes	—
206	C₂×C₄×C₇	#127	1	8	yes	—
207	C₂×C₄×C₇	AGL(1,ℤ/8)	1	8	yes	complemented
208	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
209	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
213	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
214	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
215	C₇×C₈	C₂²×D₄	1	16	yes	complemented
216	C₂⁵	—	2	2	no	—
217	C₂×#41	—	2	4	no	—
218	C₂×#41	—	2	16	no	—
219	C₂×#41	C₂³×C₇	1	8	yes	—
220	C₂²×Q₈	C₂³×C₇	1	2	yes	—
221	C₂²×D₄	—	2	2	no	—
222	C₂²×D₄	—	2	16	no	—
223	C₂²×D₄	—	2	2	no	—
224	C₂²×D₄	—	2	2	no	—
225	C₂²×D₄	—	2	4	no	—
226	C₂²×D₄	—	2	4	no	—
227	C₂²×D₄	—	2	4	no	—
228	C₂²×D₄	—	2	16	no	—
229	C₂²×D₄	—	4	16	no	—
230	C₂²×D₄	C₂³×C₇	1	2	yes	—
231	C₂²×D₄	C₂³×C₇	1	4	yes	—
232	C₂²×D₄	C₂³×C₇	1	8	yes	—
233	C₂³×C₄	—	2	2	no	—
234	C₂³×C₄	C₇×D₄	1	2	yes	—
235	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
236	AGL(1,ℤ/8)	—	2	128	no	—
237	C₂×SD₁₆	—	2	32	no	—
238	C₂×SD₁₆	C₂³×C₇	1	32	yes	complemented
239	C₂×D₈	—	2	32	no	—
240	C₂×D₈	C₂³×C₇	1	32	yes	complemented
241	C₂×M₁₆	C₇×D₄	1	16	yes	complemented
242	C₂×M₁₆	C₂³×C₇	1	8	yes	—
243	C₂²×C₈	C₂³×C₇	1	4	yes	—
244	C₄⋊Q₈	—	2	2	no	—
245	C₄⋊Q₈	C₂³×C₇	1	4	yes	complemented
246	#127	—	2	4	no	—
247	#127	—	2	4	no	—
248	#127	—	2	16	no	—
249	#127	—	2	2	no	—
250	#127	C₂³×C₇	1	8	yes	—
251	#127	C₂³×C₇	1	8	yes	—
252	#127	C₂³×C₇	1	4	yes	complemented
253	#124	—	2	4	no	—
254	#124	C₂³×C₇	1	8	yes	complemented
255	#121	—	2	16	no	—
256	C₄×D₄	—	2	16	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₂²⋊C₄	—	2	4	no	—
259	C₂×C₄²	C₇×D₄	1	2	yes	—
260	C₂×C₄²	C₇×D₄	1	4	yes	—
261	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₇	1	4	yes	—
264	#97	C₂³×C₇	1	8	yes	complemented
265	C₄×C₈	C₂³×C₇	1	8	yes	complemented
266	C₂²×C₇	—	2	8	no	—
267	C₂²×C₇	—	2	4	no	—
268	C₂²×C₇	—	4	4	no	—
269	C₂²×C₇	—	4	8	no	—
270	C₂²×C₇	—	4	16	no	—
271	C₂²×C₇	—	4	8	no	—
272	C₂²×C₇	—	8	32	no	—
273	C₂²×C₇	—	8	8	no	—
274	C₂²×C₇	—	8	16	no	—
275	C₂²×C₇	C₂×#127	1	4	yes	—
276	C₂²×C₇	C₂×#127	1	2	yes	—
277	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
278	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₇	—	2	4	no	—
280	C₄×C₇	—	4	4	no	—
281	C₄×C₇	C₂×#127	1	4	yes	—
282	C₄×C₇	C₂×#127	1	2	yes	—
283	C₄×C₇	C₂×AGL(1,ℤ/8)	1	8	yes	complemented
284	C₄×C₇	C₂³×D₄	1	2	yes	—
285	C₂⁴	—	2	2	no	—
286	C₂⁴	—	2	8	no	—
287	C₂⁴	—	2	4	no	—
288	C₂⁴	—	4	16	no	—
289	C₂⁴	—	4	8	no	—
290	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
291	#41	—	2	32	no	—
292	#41	—	4	16	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	8	no	—
295	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
296	C₂×D₄	—	2	8	no	—
297	C₂×D₄	—	2	8	no	—
298	C₂×D₄	—	2	16	no	—
299	C₂×D₄	—	2	32	no	—
300	C₂×D₄	—	2	8	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	8	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	16	no	—
305	C₂×D₄	—	2	16	no	—
306	C₂×D₄	—	4	16	no	—
307	C₂×D₄	—	4	64	no	—
308	C₂×D₄	—	4	8	no	—
309	C₂×D₄	—	4	8	no	—
310	C₂×D₄	—	4	16	no	—
311	C₂×D₄	—	4	16	no	—
312	C₂×D₄	—	4	8	no	—
313	C₂×D₄	C₂⁴×C₇	1	4	yes	—
314	C₂×D₄	C₂⁴×C₇	1	8	yes	—
315	C₂²×C₄	—	2	8	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	4	8	no	—
319	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
320	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
321	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
322	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
323	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
324	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
325	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
326	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
327	SD₁₆	—	2	64	no	—
328	D₈	—	2	64	no	—
329	M₁₆	C₂×C₇×D₄	1	32	yes	complemented
330	C₂×C₈	C₂×C₇×D₄	1	16	yes	complemented
331	C₂×C₈	C₂⁴×C₇	1	8	yes	—
332	C₄⋊C₄	—	2	8	no	—
333	C₂²⋊C₄	—	2	16	no	—
334	C₄²	C₂×C₇×D₄	1	8	yes	—
335	C₄²	C₂×C₇×D₄	1	2	yes	—
336	C₄²	C₂×C₇×D₄	1	2	yes	—
337	C₄²	C₂⁴×C₇	1	2	yes	—
338	C₄²	C₂⁴×C₇	1	2	yes	—
339	C₂×C₇	—	2	4	no	—
340	C₂×C₇	—	8	4	no	—
341	C₂×C₇	—	8	8	no	—
342	C₂×C₇	#3151	1	4	yes	direct product factor
343	C₂×C₇	C₂²×#127	1	1	yes	characteristic
344	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
345	C₂³	—	2	4	no	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	8	no	—
348	C₂³	—	4	32	no	—
349	C₂³	—	4	4	no	—
350	C₂³	—	4	16	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	4	16	no	—
353	C₂³	—	4	8	no	—
354	C₂³	—	8	32	no	—
355	C₂³	C₇×#127	1	4	yes	—
356	C₂³	C₂²×C₇×D₄	1	2	yes	—
357	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
358	Q₈	—	2	8	no	—
359	Q₈	—	4	4	no	—
360	D₄	—	2	8	no	—
361	D₄	—	2	16	no	—
362	D₄	—	4	32	no	—
363	D₄	—	4	16	no	—
364	D₄	—	4	16	no	—
365	D₄	—	4	8	no	—
366	D₄	—	4	4	no	—
367	D₄	—	4	16	no	—
368	D₄	—	4	32	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	16	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	16	no	—
374	C₂×C₄	C₇×#127	1	4	yes	—
375	C₂×C₄	C₇×#127	1	8	yes	—
376	C₂×C₄	C₇×AGL(1,ℤ/8)	1	8	yes	complemented
377	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
378	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
379	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
380	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
381	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
382	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
383	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
384	C₈	C₂²×C₇×D₄	1	16	yes	complemented
385	C₇	C₂×#3151	1	1	yes	characteristic, direct product factor, Sylow
386	C₂²	—	2	4	no	—
387	C₂²	—	2	8	no	—
388	C₂²	—	4	16	no	—
389	C₂²	—	4	8	no	—
390	C₂²	—	4	4	no	—
391	C₂²	—	4	8	no	—
392	C₂²	—	8	32	no	—
393	C₂²	—	8	16	no	—
394	C₂²	—	8	8	no	—
395	C₂²	C₂×C₇×#127	1	4	yes	—
396	C₂²	C₂×C₇×#127	1	2	yes	—
397	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	4	yes	—
398	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
399	C₄	—	2	4	no	—
400	C₄	—	4	4	no	—
401	C₄	C₂×C₇×#127	1	4	yes	—
402	C₄	C₂×C₇×#127	1	2	yes	—
403	C₄	C₂×C₇×AGL(1,ℤ/8)	1	8	yes	complemented
404	C₄	C₂³×C₇×D₄	1	2	yes	—
405	C₂	—	2	4	no	—
406	C₂	—	8	8	no	—
407	C₂	—	8	4	no	—
408	C₂	C₇×#3151	1	4	yes	direct product factor
409	C₂	C₂²×C₇×#127	1	1	yes	characteristic
410	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₇×#532	C₂	semidirect
3. C₂²×C₇×#127	C₂	semidirect
4. C₇×#3151	C₂	direct (N × H)
5. C₂×C₇×#496	C₂	semidirect
6. C₂×C₇×#493	C₂	semidirect
7. C₂×C₇×#492	C₂	semidirect
8. C₂×C₄×C₇×M₁₆	C₂	semidirect
13. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
14. C₂²×C₇×SD₁₆	C₂²	semidirect
15. C₂²×C₇×D₈	C₂²	semidirect
18. C₇×#532	C₂²	semidirect
19. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
24. C₂×C₇×#127	C₂²	semidirect
25. C₂×C₇×#127	C₂²	semidirect
26. C₂×C₇×#124	C₂²	semidirect
30. C₇×#496	C₂²	semidirect
31. C₇×#493	C₂²	semidirect
32. C₇×#492	C₂²	semidirect
33. C₄×C₇×M₁₆	C₂²	semidirect
34. C₂×C₇×#97	C₂²	semidirect
35. C₂×C₄×C₇×C₈	C₂²	semidirect
36. C₂×#3151	C₇	direct (N × H)
59. C₂×C₇×SD₁₆	C₂³	semidirect
61. C₂×C₇×D₈	C₂³	semidirect
62. C₂×C₇×M₁₆	D₄	semidirect
66. C₇×C₄⋊Q₈	C₂³	semidirect
73. C₇×#127	C₂³	semidirect
75. C₇×#124	C₂³	semidirect
85. C₇×#97	C₂³	semidirect
86. C₄×C₇×C₈	C₂³	semidirect
87. C₂²×AGL(1,ℤ/8)	C₂×C₇	semidirect
88. C₂×#532	C₂×C₇	semidirect
89. C₂²×#127	C₂×C₇	semidirect
90. #3151	C₂×C₇	direct (N × H)
91. C₂×#496	C₂×C₇	semidirect
92. C₂×#493	C₂×C₇	semidirect
93. C₂×#492	C₂×C₇	semidirect
94. C₂×C₄×M₁₆	C₂×C₇	semidirect
139. C₇×M₁₆	C₂×D₄	semidirect
140. C₂×C₇×C₈	C₂×D₄	semidirect
153. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
154. C₂²×SD₁₆	C₂²×C₇	semidirect
155. C₂²×D₈	C₂²×C₇	semidirect
158. #532	C₂²×C₇	semidirect
159. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
164. C₂×#127	C₂²×C₇	semidirect
165. C₂×#127	C₂²×C₇	semidirect
166. C₂×#124	C₂²×C₇	semidirect
170. #496	C₂²×C₇	semidirect
171. #493	C₂²×C₇	semidirect
172. #492	C₂²×C₇	semidirect
173. C₄×M₁₆	C₂²×C₇	semidirect
174. C₂×#97	C₂²×C₇	semidirect
175. C₂×C₄×C₈	C₂²×C₇	semidirect
207. C₂×C₄×C₇	AGL(1,ℤ/8)	semidirect
215. C₇×C₈	C₂²×D₄	semidirect
238. C₂×SD₁₆	C₂³×C₇	semidirect
240. C₂×D₈	C₂³×C₇	semidirect
241. C₂×M₁₆	C₇×D₄	semidirect
245. C₄⋊Q₈	C₂³×C₇	semidirect
252. #127	C₂³×C₇	semidirect
254. #124	C₂³×C₇	semidirect
264. #97	C₂³×C₇	semidirect
265. C₄×C₈	C₂³×C₇	semidirect
283. C₄×C₇	C₂×AGL(1,ℤ/8)	semidirect
329. M₁₆	C₂×C₇×D₄	semidirect
330. C₂×C₈	C₂×C₇×D₄	semidirect
342. C₂×C₇	#3151	direct (N × H)
376. C₂×C₄	C₇×AGL(1,ℤ/8)	semidirect
384. C₈	C₂²×C₇×D₄	semidirect
385. C₇	C₂×#3151	direct (N × H)
403. C₄	C₂×C₇×AGL(1,ℤ/8)	semidirect
408. C₂	C₇×#3151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×AGL(1,ℤ/8)	2
C₂×C₇×#532	2
C₂²×C₇×#127	2
C₇×#3151	2
C₂×C₇×#496	2
C₂×C₇×#493	2
C₂×C₇×#492	2
C₂×C₄×C₇×M₁₆	2
C₂×#3151	7

Subgroups — C2×C7×#3151 #13237852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3151 #13237852