Subgroups — C₂²×C₇×#476 #13237690

Order 1792.

2790 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#476	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#106	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#102	C₂	1	2	yes	maximal
7	C₂³×C₇×Q₈	—	2	1	no	—
8	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₇	—	2	1	no	—
10	C₂³×C₇×C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₇×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₄×C₇×D₄	—	2	6	no	—
14	C₂×C₄×C₇×D₄	C₂²	1	12	yes	—
15	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
19	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₇×#476	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#106	C₂²	1	24	yes	complemented
22	C₂×C₇×C₄⋊C₈	C₂²	1	12	yes	complemented
23	C₂×C₇×#102	C₂²	1	24	yes	—
24	C₂²×#476	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₇	—	2	1	no	—
26	C₂²×C₇×Q₈	—	2	12	no	—
27	C₂²×C₇×Q₈	—	4	8	no	—
28	C₂²×C₇×D₄	—	2	3	no	—
29	C₂²×C₇×D₄	—	2	4	no	—
30	C₂²×C₇×D₄	Q₈	1	8	yes	complemented
31	C₂²×C₇×D₄	C₂³	1	6	yes	—
32	C₂³×C₄×C₇	—	2	2	no	—
33	C₂³×C₄×C₇	—	2	1	no	—
34	C₂³×C₄×C₇	—	2	12	no	—
35	C₂³×C₄×C₇	—	4	8	no	—
36	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	—	2	8	no	—
40	C₂²×C₇×C₈	C₂³	1	12	yes	—
41	C₇×C₄⋊Q₈	—	2	24	no	—
42	C₇×C₄⋊Q₈	C₂³	1	16	yes	complemented
43	C₄×C₇×D₄	—	2	24	no	—
44	C₄×C₇×D₄	C₂³	1	16	yes	—
45	C₂×C₇×C₄⋊C₄	—	2	12	no	—
46	C₂×C₇×C₄⋊C₄	—	2	3	no	—
47	C₂×C₇×C₄⋊C₄	—	2	6	no	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
50	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
51	C₂×C₄²×C₇	D₄	1	6	yes	—
52	C₂×C₄²×C₇	C₂³	1	6	yes	—
53	C₇×#106	C₂³	1	32	yes	complemented
54	C₇×C₄⋊C₈	C₂³	1	16	yes	complemented
55	C₇×#102	C₂³	1	32	yes	—
56	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂²×C₄×D₄	C₂×C₇	1	1	yes	characteristic
58	C₂×#476	C₂×C₇	1	24	yes	direct product factor
59	C₂²×#106	C₂×C₇	1	2	yes	complemented
60	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂²×#102	C₂×C₇	1	2	yes	—
62	C₂⁴×C₇	—	2	8	no	—
63	C₂⁴×C₇	—	2	6	no	—
64	C₂⁴×C₇	—	4	4	no	—
65	C₂⁴×C₇	—	4	4	no	—
66	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×Q₈	—	2	16	no	—
68	C₂×C₇×Q₈	—	4	48	no	—
69	C₂×C₇×D₄	—	2	24	no	—
70	C₂×C₇×D₄	—	2	6	no	—
71	C₂×C₇×D₄	—	2	12	no	—
72	C₂×C₇×D₄	C₂×Q₈	1	24	yes	complemented
73	C₂×C₇×D₄	C₂⁴	1	4	yes	—
74	C₂²×C₄×C₇	—	2	4	no	—
75	C₂²×C₄×C₇	—	2	6	no	—
76	C₂²×C₄×C₇	—	2	12	no	—
77	C₂²×C₄×C₇	—	2	16	no	—
78	C₂²×C₄×C₇	—	4	4	no	—
79	C₂²×C₄×C₇	—	4	8	no	—
80	C₂²×C₄×C₇	—	4	48	no	—
81	C₂²×C₄×C₇	SD₁₆	1	8	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
85	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
86	C₂²×C₄×C₇	#41	1	4	yes	—
87	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
88	C₂×C₇×C₈	—	2	24	no	—
89	C₂×C₇×C₈	C₂⁴	1	8	yes	—
90	C₇×C₄⋊C₄	—	2	16	no	—
91	C₇×C₄⋊C₄	—	2	6	no	—
92	C₇×C₄⋊C₄	—	2	12	no	—
93	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
94	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
95	C₇×C₂²⋊C₄	—	2	16	no	—
96	C₄²×C₇	C₂×D₄	1	12	yes	—
97	C₄²×C₇	C₂⁴	1	4	yes	—
98	C₂³×Q₈	—	2	1	no	—
99	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
100	C₂⁴×C₄	—	2	1	no	—
101	C₂³×C₈	C₂²×C₇	1	2	yes	—
102	C₂×C₄⋊Q₈	—	2	6	no	—
103	C₂×C₄⋊Q₈	C₂²×C₇	1	12	yes	complemented
104	C₂×C₄×D₄	—	2	6	no	—
105	C₂×C₄×D₄	C₂²×C₇	1	12	yes	—
106	C₂²×C₄⋊C₄	—	2	1	no	—
107	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
108	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
109	C₂²×C₂²⋊C₄	—	2	1	no	—
110	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
111	#476	C₂²×C₇	1	64	yes	direct product factor
112	C₂×#106	C₂²×C₇	1	24	yes	complemented
113	C₂×C₄⋊C₈	C₂²×C₇	1	12	yes	complemented
114	C₂×#102	C₂²×C₇	1	24	yes	—
115	C₂³×C₇	—	2	24	no	—
116	C₂³×C₇	—	2	4	no	—
117	C₂³×C₇	—	4	12	no	—
118	C₂³×C₇	—	4	12	no	—
119	C₂³×C₇	—	4	32	no	—
120	C₂³×C₇	C₂²⋊Q₈	1	4	yes	—
121	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
122	C₂³×C₇	AGL(1,ℤ/8)	1	4	yes	—
123	C₂³×C₇	C₂²×D₄	1	3	yes	—
124	C₇×Q₈	—	4	32	no	—
125	C₇×D₄	—	2	24	no	—
126	C₇×D₄	—	2	4	no	—
127	C₇×D₄	C₂²×Q₈	1	8	yes	complemented
128	C₂×C₄×C₇	—	2	12	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	8	no	—
131	C₂×C₄×C₇	—	4	12	no	—
132	C₂×C₄×C₇	—	4	24	no	—
133	C₂×C₄×C₇	—	4	32	no	—
134	C₂×C₄×C₇	C₂²⋊Q₈	1	12	yes	—
135	C₂×C₄×C₇	C₂×SD₁₆	1	24	yes	—
136	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
138	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₇	C₂²×Q₈	1	6	yes	—
140	C₂×C₄×C₇	C₂×#41	1	6	yes	—
141	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
142	C₇×C₈	—	2	8	no	—
143	C₂⁵	—	2	1	no	—
144	C₂²×Q₈	—	2	12	no	—
145	C₂²×Q₈	—	4	8	no	—
146	C₂²×D₄	—	2	3	no	—
147	C₂²×D₄	—	2	4	no	—
148	C₂²×D₄	C₇×Q₈	1	8	yes	complemented
149	C₂²×D₄	C₂³×C₇	1	6	yes	—
150	C₂³×C₄	—	2	1	no	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	—	2	12	no	—
153	C₂³×C₄	—	4	8	no	—
154	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
155	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
156	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
157	C₂²×C₈	—	2	8	no	—
158	C₂²×C₈	C₂³×C₇	1	12	yes	—
159	C₄⋊Q₈	—	2	24	no	—
160	C₄⋊Q₈	C₂³×C₇	1	16	yes	complemented
161	C₄×D₄	—	2	24	no	—
162	C₄×D₄	C₂³×C₇	1	16	yes	—
163	C₂×C₄⋊C₄	—	2	12	no	—
164	C₂×C₄⋊C₄	—	2	3	no	—
165	C₂×C₄⋊C₄	—	2	6	no	—
166	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
167	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
168	C₂×C₂²⋊C₄	—	2	12	no	—
169	C₂×C₄²	C₇×D₄	1	6	yes	—
170	C₂×C₄²	C₂³×C₇	1	6	yes	—
171	#106	C₂³×C₇	1	32	yes	complemented
172	C₄⋊C₈	C₂³×C₇	1	16	yes	complemented
173	#102	C₂³×C₇	1	32	yes	—
174	C₂²×C₇	—	2	8	no	—
175	C₂²×C₇	—	4	4	no	—
176	C₂²×C₇	—	4	4	no	—
177	C₂²×C₇	—	4	48	no	—
178	C₂²×C₇	#476	1	16	yes	direct product factor
179	C₂²×C₇	C₂×C₂²⋊Q₈	1	6	yes	—
180	C₂²×C₇	C₂²×SD₁₆	1	6	yes	—
181	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
182	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
183	C₄×C₇	—	2	4	no	—
184	C₄×C₇	—	4	4	no	—
185	C₄×C₇	—	4	8	no	—
186	C₄×C₇	C₂×C₂²⋊Q₈	1	6	yes	—
187	C₄×C₇	C₂²×SD₁₆	1	8	yes	—
188	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
189	C₄×C₇	C₂²×#41	1	1	yes	characteristic
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	6	no	—
192	C₂⁴	—	4	4	no	—
193	C₂⁴	—	4	4	no	—
194	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
195	C₂×Q₈	—	2	16	no	—
196	C₂×Q₈	—	4	48	no	—
197	C₂×D₄	—	2	24	no	—
198	C₂×D₄	—	2	6	no	—
199	C₂×D₄	—	2	12	no	—
200	C₂×D₄	C₂×C₇×Q₈	1	24	yes	complemented
201	C₂×D₄	C₂⁴×C₇	1	4	yes	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	—	2	12	no	—
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	48	no	—
209	C₂²×C₄	C₇×SD₁₆	1	8	yes	—
210	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
211	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
212	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
213	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
214	C₂²×C₄	C₇×#41	1	4	yes	—
215	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
216	C₂×C₈	—	2	24	no	—
217	C₂×C₈	C₂⁴×C₇	1	8	yes	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	6	no	—
220	C₄⋊C₄	—	2	12	no	—
221	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
222	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
223	C₂²⋊C₄	—	2	16	no	—
224	C₄²	C₂×C₇×D₄	1	12	yes	—
225	C₄²	C₂⁴×C₇	1	4	yes	—
226	C₂×C₇	—	4	8	no	—
227	C₂×C₇	C₂×#476	1	12	yes	direct product factor
228	C₂×C₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
230	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	2	24	no	—
233	C₂³	—	4	32	no	—
234	C₂³	—	4	12	no	—
235	C₂³	—	4	12	no	—
236	C₂³	C₇×C₂²⋊Q₈	1	4	yes	—
237	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
238	C₂³	C₇×AGL(1,ℤ/8)	1	4	yes	—
239	C₂³	C₂²×C₇×D₄	1	3	yes	—
240	Q₈	—	4	32	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	24	no	—
243	D₄	C₂²×C₇×Q₈	1	8	yes	complemented
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	12	no	—
248	C₂×C₄	—	4	24	no	—
249	C₂×C₄	—	4	32	no	—
250	C₂×C₄	C₇×C₂²⋊Q₈	1	12	yes	—
251	C₂×C₄	C₂×C₇×SD₁₆	1	24	yes	—
252	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
253	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
254	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
255	C₂×C₄	C₂²×C₇×Q₈	1	6	yes	—
256	C₂×C₄	C₂×C₇×#41	1	6	yes	—
257	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
258	C₈	—	2	8	no	—
259	C₇	C₂²×#476	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	48	no	—
264	C₂²	C₇×#476	1	16	yes	direct product factor
265	C₂²	C₂×C₇×C₂²⋊Q₈	1	6	yes	—
266	C₂²	C₂²×C₇×SD₁₆	1	6	yes	—
267	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
268	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	8	no	—
272	C₄	C₂×C₇×C₂²⋊Q₈	1	6	yes	—
273	C₄	C₂²×C₇×SD₁₆	1	8	yes	—
274	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
275	C₄	C₂²×C₇×#41	1	1	yes	characteristic
276	C₂	—	4	8	no	—
277	C₂	C₂×C₇×#476	1	12	yes	direct product factor
278	C₂	C₂²×C₇×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic
280	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	C₂³×C₇×D₄	C₂²
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. C₂²×C₇×C₄⋊Q₈	C₂	semidirect
3. C₂×C₇×#476	C₂	direct (N × H)
4. C₂²×C₇×#106	C₂	semidirect
5. C₂²×C₇×C₄⋊C₈	C₂	semidirect
12. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
20. C₇×#476	C₂²	direct (N × H)
21. C₂×C₇×#106	C₂²	semidirect
22. C₂×C₇×C₄⋊C₈	C₂²	semidirect
24. C₂²×#476	C₇	direct (N × H)
30. C₂²×C₇×D₄	Q₈	semidirect
42. C₇×C₄⋊Q₈	C₂³	semidirect
53. C₇×#106	C₂³	semidirect
54. C₇×C₄⋊C₈	C₂³	semidirect
56. C₂²×C₄⋊Q₈	C₂×C₇	semidirect
58. C₂×#476	C₂×C₇	direct (N × H)
59. C₂²×#106	C₂×C₇	semidirect
60. C₂²×C₄⋊C₈	C₂×C₇	semidirect
72. C₂×C₇×D₄	C₂×Q₈	semidirect
103. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
111. #476	C₂²×C₇	direct (N × H)
112. C₂×#106	C₂²×C₇	semidirect
113. C₂×C₄⋊C₈	C₂²×C₇	semidirect
127. C₇×D₄	C₂²×Q₈	semidirect
148. C₂²×D₄	C₇×Q₈	semidirect
160. C₄⋊Q₈	C₂³×C₇	semidirect
171. #106	C₂³×C₇	semidirect
172. C₄⋊C₈	C₂³×C₇	semidirect
178. C₂²×C₇	#476	direct (N × H)
200. C₂×D₄	C₂×C₇×Q₈	semidirect
227. C₂×C₇	C₂×#476	direct (N × H)
243. D₄	C₂²×C₇×Q₈	semidirect
259. C₇	C₂²×#476	direct (N × H)
264. C₂²	C₇×#476	direct (N × H)
277. C₂	C₂×C₇×#476	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×C₄⋊Q₈	2
C₂²×C₄×C₇×D₄	2
C₂×C₇×#476	2
C₂²×C₇×#106	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₇×#102	2
C₂²×#476	7

Subgroups — C22×C7×#476 #13237690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#476 #13237690