Subgroups — C₂×C₇×#3155 #13237857

Order 1792.

3534 conjugacy classes of subgroups.

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

Subgroups — C2×C7×#3155 #13237857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3155 #13237857