Subgroups — C₂²×C₇×#460 #13237622

Order 1792.

3478 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂²×C₇×#127	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#460	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#102	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂³×C₇×D₄	—	2	1	no	—
9	C₂³×C₇×D₄	—	2	1	no	—
10	C₂³×C₇×D₄	C₂²	1	2	yes	—
11	C₂²×C₇×SD₁₆	—	2	16	no	—
12	C₂²×C₇×SD₁₆	C₂²	1	24	yes	complemented
13	C₂³×C₇×C₈	C₂²	1	2	yes	—
14	C₂×C₇×#127	—	2	6	no	—
15	C₂×C₇×#127	C₂²	1	12	yes	—
16	C₂×C₄×C₇×Q₈	—	2	6	no	—
17	C₂×C₄×C₇×Q₈	C₂²	1	12	yes	complemented
18	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
19	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₇	—	2	1	no	—
21	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#460	C₂²	1	64	yes	direct product factor
23	C₂×C₇×C₄⋊C₈	C₂²	1	12	yes	complemented
24	C₂×C₇×#102	C₂²	1	24	yes	complemented
25	C₂²×#460	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂⁵×C₇	—	2	2	no	—
27	C₂²×C₇×Q₈	—	2	3	no	—
28	C₂²×C₇×Q₈	—	2	4	no	—
29	C₂²×C₇×Q₈	D₄	1	8	yes	complemented
30	C₂²×C₇×Q₈	C₂³	1	6	yes	—
31	C₂²×C₇×D₄	—	2	8	no	—
32	C₂²×C₇×D₄	—	2	12	no	—
33	C₂²×C₇×D₄	—	2	12	no	—
34	C₂²×C₇×D₄	—	2	6	no	—
35	C₂²×C₇×D₄	—	2	8	no	—
36	C₂²×C₇×D₄	—	4	8	no	—
37	C₂²×C₇×D₄	—	4	8	no	—
38	C₂²×C₇×D₄	C₂³	1	12	yes	—
39	C₂³×C₄×C₇	—	2	1	no	—
40	C₂³×C₄×C₇	—	2	1	no	—
41	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₇×SD₁₆	—	2	96	no	—
45	C₂×C₇×SD₁₆	C₂³	1	32	yes	complemented
46	C₂²×C₇×C₈	—	2	8	no	—
47	C₂²×C₇×C₈	C₂³	1	12	yes	—
48	C₇×#127	—	2	24	no	—
49	C₇×#127	C₂³	1	16	yes	—
50	C₄×C₇×Q₈	—	2	24	no	—
51	C₄×C₇×Q₈	C₂³	1	16	yes	complemented
52	C₂×C₇×C₄⋊C₄	—	2	12	no	—
53	C₂×C₇×C₄⋊C₄	—	2	3	no	—
54	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
55	C₂×C₄²×C₇	—	2	12	no	—
56	C₂×C₄²×C₇	D₄	1	6	yes	—
57	C₂×C₄²×C₇	C₂³	1	6	yes	—
58	C₇×C₄⋊C₈	C₂³	1	16	yes	complemented
59	C₇×#102	C₂³	1	32	yes	complemented
60	C₂³×SD₁₆	C₂×C₇	1	2	yes	complemented
61	C₂²×#127	C₂×C₇	1	1	yes	characteristic
62	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂×#460	C₂×C₇	1	24	yes	direct product factor
64	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂²×#102	C₂×C₇	1	2	yes	complemented
66	C₂⁴×C₇	—	2	12	no	—
67	C₂⁴×C₇	—	4	8	no	—
68	C₂⁴×C₇	—	4	8	no	—
69	C₂⁴×C₇	—	4	8	no	—
70	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₇×Q₈	—	2	24	no	—
72	C₂×C₇×Q₈	—	2	6	no	—
73	C₂×C₇×Q₈	—	2	12	no	—
74	C₂×C₇×Q₈	C₂×D₄	1	24	yes	complemented
75	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
76	C₂×C₇×D₄	—	2	24	no	—
77	C₂×C₇×D₄	—	2	12	no	—
78	C₂×C₇×D₄	—	2	16	no	—
79	C₂×C₇×D₄	—	2	16	no	—
80	C₂×C₇×D₄	—	2	24	no	—
81	C₂×C₇×D₄	—	4	48	no	—
82	C₂×C₇×D₄	—	4	24	no	—
83	C₂×C₇×D₄	—	4	48	no	—
84	C₂×C₇×D₄	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₇	—	2	8	no	—
86	C₂²×C₄×C₇	—	2	4	no	—
87	C₂²×C₄×C₇	—	2	6	no	—
88	C₂²×C₄×C₇	—	2	6	no	—
89	C₂²×C₄×C₇	—	4	4	no	—
90	C₂²×C₄×C₇	SD₁₆	1	8	yes	complemented
91	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
92	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
95	C₂²×C₄×C₇	#41	1	4	yes	—
96	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
97	C₇×SD₁₆	—	2	64	no	—
98	C₂×C₇×C₈	—	2	24	no	—
99	C₂×C₇×C₈	C₂⁴	1	8	yes	—
100	C₇×C₄⋊C₄	—	2	16	no	—
101	C₇×C₄⋊C₄	—	2	6	no	—
102	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
103	C₄²×C₇	—	2	16	no	—
104	C₄²×C₇	C₂×D₄	1	12	yes	—
105	C₄²×C₇	C₂⁴	1	4	yes	—
106	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
107	C₂³×D₄	—	2	1	no	—
108	C₂³×D₄	—	2	1	no	—
109	C₂³×D₄	C₂²×C₇	1	2	yes	—
110	C₂²×SD₁₆	—	2	16	no	—
111	C₂²×SD₁₆	C₂²×C₇	1	24	yes	complemented
112	C₂³×C₈	C₂²×C₇	1	2	yes	—
113	C₂×#127	—	2	6	no	—
114	C₂×#127	C₂²×C₇	1	12	yes	—
115	C₂×C₄×Q₈	—	2	6	no	—
116	C₂×C₄×Q₈	C₂²×C₇	1	12	yes	complemented
117	C₂²×C₄⋊C₄	—	2	1	no	—
118	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
119	C₂²×C₄²	—	2	1	no	—
120	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
121	#460	C₂²×C₇	1	64	yes	direct product factor
122	C₂×C₄⋊C₈	C₂²×C₇	1	12	yes	complemented
123	C₂×#102	C₂²×C₇	1	24	yes	complemented
124	C₂³×C₇	—	2	8	no	—
125	C₂³×C₇	—	4	24	no	—
126	C₂³×C₇	—	4	24	no	—
127	C₂³×C₇	—	4	24	no	—
128	C₂³×C₇	—	8	32	no	—
129	C₂³×C₇	#121	1	4	yes	—
130	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
131	C₂³×C₇	AGL(1,ℤ/8)	1	4	yes	—
132	C₂³×C₇	C₂²×D₄	1	3	yes	—
133	C₇×Q₈	—	2	24	no	—
134	C₇×Q₈	—	2	4	no	—
135	C₇×Q₈	C₂²×D₄	1	8	yes	complemented
136	C₇×D₄	—	2	8	no	—
137	C₇×D₄	—	2	8	no	—
138	C₇×D₄	—	4	32	no	—
139	C₇×D₄	—	4	24	no	—
140	C₇×D₄	—	4	32	no	—
141	C₂×C₄×C₇	—	2	24	no	—
142	C₂×C₄×C₇	—	2	12	no	—
143	C₂×C₄×C₇	—	2	4	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	—	4	12	no	—
146	C₂×C₄×C₇	#121	1	12	yes	—
147	C₂×C₄×C₇	C₂×SD₁₆	1	24	yes	complemented
148	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
150	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
151	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₇	C₂×#41	1	6	yes	—
153	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
154	C₇×C₈	—	2	8	no	—
155	C₂⁵	—	2	2	no	—
156	C₂²×Q₈	—	2	3	no	—
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	C₇×D₄	1	8	yes	complemented
159	C₂²×Q₈	C₂³×C₇	1	6	yes	—
160	C₂²×D₄	—	2	8	no	—
161	C₂²×D₄	—	2	6	no	—
162	C₂²×D₄	—	2	12	no	—
163	C₂²×D₄	—	2	12	no	—
164	C₂²×D₄	—	2	8	no	—
165	C₂²×D₄	—	4	8	no	—
166	C₂²×D₄	—	4	8	no	—
167	C₂²×D₄	C₂³×C₇	1	12	yes	—
168	C₂³×C₄	—	2	1	no	—
169	C₂³×C₄	—	2	1	no	—
170	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
171	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
173	C₂×SD₁₆	—	2	96	no	—
174	C₂×SD₁₆	C₂³×C₇	1	32	yes	complemented
175	C₂²×C₈	—	2	8	no	—
176	C₂²×C₈	C₂³×C₇	1	12	yes	—
177	#127	—	2	24	no	—
178	#127	C₂³×C₇	1	16	yes	—
179	C₄×Q₈	—	2	24	no	—
180	C₄×Q₈	C₂³×C₇	1	16	yes	complemented
181	C₂×C₄⋊C₄	—	2	12	no	—
182	C₂×C₄⋊C₄	—	2	3	no	—
183	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
184	C₂×C₄²	—	2	12	no	—
185	C₂×C₄²	C₇×D₄	1	6	yes	—
186	C₂×C₄²	C₂³×C₇	1	6	yes	—
187	C₄⋊C₈	C₂³×C₇	1	16	yes	complemented
188	#102	C₂³×C₇	1	32	yes	complemented
189	C₂²×C₇	—	4	8	no	—
190	C₂²×C₇	—	4	8	no	—
191	C₂²×C₇	—	4	8	no	—
192	C₂²×C₇	—	8	48	no	—
193	C₂²×C₇	#460	1	16	yes	direct product factor
194	C₂²×C₇	C₂×#121	1	6	yes	—
195	C₂²×C₇	C₂²×SD₁₆	1	6	yes	—
196	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
197	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
198	C₄×C₇	—	2	8	no	—
199	C₄×C₇	—	2	4	no	—
200	C₄×C₇	—	4	4	no	—
201	C₄×C₇	C₂×#121	1	6	yes	—
202	C₄×C₇	C₂²×SD₁₆	1	8	yes	complemented
203	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
204	C₄×C₇	C₂²×#41	1	1	yes	characteristic
205	C₂⁴	—	2	12	no	—
206	C₂⁴	—	4	8	no	—
207	C₂⁴	—	4	8	no	—
208	C₂⁴	—	4	8	no	—
209	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂×Q₈	—	2	24	no	—
211	C₂×Q₈	—	2	6	no	—
212	C₂×Q₈	—	2	12	no	—
213	C₂×Q₈	C₂×C₇×D₄	1	24	yes	complemented
214	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
215	C₂×D₄	—	2	24	no	—
216	C₂×D₄	—	2	12	no	—
217	C₂×D₄	—	2	16	no	—
218	C₂×D₄	—	2	16	no	—
219	C₂×D₄	—	2	24	no	—
220	C₂×D₄	—	4	48	no	—
221	C₂×D₄	—	4	24	no	—
222	C₂×D₄	—	4	48	no	—
223	C₂×D₄	C₂⁴×C₇	1	8	yes	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	C₇×SD₁₆	1	8	yes	complemented
230	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
231	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
232	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
233	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
234	C₂²×C₄	C₇×#41	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
236	SD₁₆	—	2	64	no	—
237	C₂×C₈	—	2	24	no	—
238	C₂×C₈	C₂⁴×C₇	1	8	yes	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
242	C₄²	—	2	16	no	—
243	C₄²	C₂×C₇×D₄	1	12	yes	—
244	C₄²	C₂⁴×C₇	1	4	yes	—
245	C₂×C₇	—	8	8	no	—
246	C₂×C₇	C₂×#460	1	12	yes	direct product factor
247	C₂×C₇	C₂²×#121	1	1	yes	characteristic
248	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
249	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂³	—	2	8	no	—
251	C₂³	—	4	24	no	—
252	C₂³	—	4	24	no	—
253	C₂³	—	4	24	no	—
254	C₂³	—	8	32	no	—
255	C₂³	C₇×#121	1	4	yes	—
256	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
257	C₂³	C₇×AGL(1,ℤ/8)	1	4	yes	—
258	C₂³	C₂²×C₇×D₄	1	3	yes	—
259	Q₈	—	2	4	no	—
260	Q₈	—	2	24	no	—
261	Q₈	C₂²×C₇×D₄	1	8	yes	complemented
262	D₄	—	2	8	no	—
263	D₄	—	2	8	no	—
264	D₄	—	4	32	no	—
265	D₄	—	4	24	no	—
266	D₄	—	4	32	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	24	no	—
269	C₂×C₄	—	2	12	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	12	no	—
272	C₂×C₄	C₇×#121	1	12	yes	—
273	C₂×C₄	C₂×C₇×SD₁₆	1	24	yes	complemented
274	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
275	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
276	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
277	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
278	C₂×C₄	C₂×C₇×#41	1	6	yes	—
279	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
280	C₈	—	2	8	no	—
281	C₇	C₂²×#460	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	4	8	no	—
283	C₂²	—	4	8	no	—
284	C₂²	—	4	8	no	—
285	C₂²	—	8	48	no	—
286	C₂²	C₇×#460	1	16	yes	direct product factor
287	C₂²	C₂×C₇×#121	1	6	yes	—
288	C₂²	C₂²×C₇×SD₁₆	1	6	yes	—
289	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
290	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
291	C₄	—	2	4	no	—
292	C₄	—	2	8	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₇×#121	1	6	yes	—
295	C₄	C₂²×C₇×SD₁₆	1	8	yes	complemented
296	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
297	C₄	C₂²×C₇×#41	1	1	yes	characteristic
298	C₂	—	8	8	no	—
299	C₂	C₂×C₇×#460	1	12	yes	direct product factor
300	C₂	C₂²×C₇×#121	1	1	yes	characteristic
301	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic
302	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂²×C₇×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×SD₁₆	C₂	semidirect
3. C₂²×C₄×C₇×Q₈	C₂	semidirect
4. C₂×C₇×#460	C₂	direct (N × H)
5. C₂²×C₇×C₄⋊C₈	C₂	semidirect
6. C₂²×C₇×#102	C₂	semidirect
12. C₂²×C₇×SD₁₆	C₂²	semidirect
17. C₂×C₄×C₇×Q₈	C₂²	semidirect
22. C₇×#460	C₂²	direct (N × H)
23. C₂×C₇×C₄⋊C₈	C₂²	semidirect
24. C₂×C₇×#102	C₂²	semidirect
25. C₂²×#460	C₇	direct (N × H)
29. C₂²×C₇×Q₈	D₄	semidirect
45. C₂×C₇×SD₁₆	C₂³	semidirect
51. C₄×C₇×Q₈	C₂³	semidirect
58. C₇×C₄⋊C₈	C₂³	semidirect
59. C₇×#102	C₂³	semidirect
60. C₂³×SD₁₆	C₂×C₇	semidirect
62. C₂²×C₄×Q₈	C₂×C₇	semidirect
63. C₂×#460	C₂×C₇	direct (N × H)
64. C₂²×C₄⋊C₈	C₂×C₇	semidirect
65. C₂²×#102	C₂×C₇	semidirect
74. C₂×C₇×Q₈	C₂×D₄	semidirect
90. C₂²×C₄×C₇	SD₁₆	semidirect
111. C₂²×SD₁₆	C₂²×C₇	semidirect
116. C₂×C₄×Q₈	C₂²×C₇	semidirect
121. #460	C₂²×C₇	direct (N × H)
122. C₂×C₄⋊C₈	C₂²×C₇	semidirect
123. C₂×#102	C₂²×C₇	semidirect
135. C₇×Q₈	C₂²×D₄	semidirect
147. C₂×C₄×C₇	C₂×SD₁₆	semidirect
158. C₂²×Q₈	C₇×D₄	semidirect
174. C₂×SD₁₆	C₂³×C₇	semidirect
180. C₄×Q₈	C₂³×C₇	semidirect
187. C₄⋊C₈	C₂³×C₇	semidirect
188. #102	C₂³×C₇	semidirect
193. C₂²×C₇	#460	direct (N × H)
202. C₄×C₇	C₂²×SD₁₆	semidirect
213. C₂×Q₈	C₂×C₇×D₄	semidirect
229. C₂²×C₄	C₇×SD₁₆	semidirect
246. C₂×C₇	C₂×#460	direct (N × H)
261. Q₈	C₂²×C₇×D₄	semidirect
273. C₂×C₄	C₂×C₇×SD₁₆	semidirect
281. C₇	C₂²×#460	direct (N × H)
286. C₂²	C₇×#460	direct (N × H)
295. C₄	C₂²×C₇×SD₁₆	semidirect
299. C₂	C₂×C₇×#460	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×SD₁₆	2
C₂²×C₇×#127	2
C₂²×C₄×C₇×Q₈	2
C₂×C₇×#460	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₇×#102	2
C₂²×#460	7

Subgroups — C22×C7×#460 #13237622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#460 #13237622