Subgroups — C₂²×C₇×#497 #13237850

Order 1792.

3350 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	2	yes	maximal, complemented
2	C₂³×C₇×SD₁₆	C₂	1	2	yes	maximal
3	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×#124	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×#497	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₇×Q₁₆	—	2	16	no	—
11	C₂²×C₇×Q₁₆	C₂²	1	24	yes	complemented
12	C₂²×C₇×SD₁₆	—	2	16	no	—
13	C₂²×C₇×SD₁₆	C₂²	1	24	yes	—
14	C₂³×C₇×C₈	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊Q₈	—	2	6	no	—
16	C₂×C₇×C₄⋊Q₈	C₂²	1	12	yes	complemented
17	C₂×C₇×#124	—	2	6	no	—
18	C₂×C₇×#124	C₂²	1	12	yes	—
19	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
21	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#497	C₂²	1	64	yes	direct product factor
23	C₂×C₇×#97	C₂²	1	12	yes	complemented
24	C₂²×#497	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₇	—	2	1	no	—
26	C₂²×C₇×Q₈	—	2	3	no	—
27	C₂²×C₇×Q₈	—	2	6	no	—
28	C₂²×C₇×Q₈	—	2	8	no	—
29	C₂²×C₇×Q₈	—	2	8	no	—
30	C₂²×C₇×Q₈	—	2	8	no	—
31	C₂²×C₇×Q₈	C₂³	1	6	yes	—
32	C₂²×C₇×Q₈	C₂³	1	12	yes	—
33	C₂²×C₇×D₄	—	2	8	no	—
34	C₂²×C₇×D₄	—	2	3	no	—
35	C₂²×C₇×D₄	C₂³	1	6	yes	—
36	C₂³×C₄×C₇	—	2	1	no	—
37	C₂³×C₄×C₇	—	2	2	no	—
38	C₂³×C₄×C₇	D₄	1	2	yes	—
39	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₇×Q₁₆	—	2	96	no	—
41	C₂×C₇×Q₁₆	C₂³	1	32	yes	complemented
42	C₂×C₇×SD₁₆	—	2	96	no	—
43	C₂×C₇×SD₁₆	C₂³	1	32	yes	—
44	C₂²×C₇×C₈	D₄	1	16	yes	—
45	C₂²×C₇×C₈	C₂³	1	12	yes	—
46	C₇×C₄⋊Q₈	—	2	24	no	—
47	C₇×C₄⋊Q₈	C₂³	1	16	yes	complemented
48	C₇×#124	—	2	24	no	—
49	C₇×#124	C₂³	1	16	yes	—
50	C₂×C₇×C₄⋊C₄	—	2	24	no	—
51	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
52	C₂×C₄²×C₇	D₄	1	6	yes	—
53	C₂×C₄²×C₇	C₂³	1	6	yes	—
54	C₇×#97	C₂³	1	16	yes	complemented
55	C₂³×Q₁₆	C₂×C₇	1	2	yes	complemented
56	C₂³×SD₁₆	C₂×C₇	1	2	yes	—
57	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂²×#124	C₂×C₇	1	1	yes	characteristic
59	C₂×#497	C₂×C₇	1	24	yes	direct product factor
60	C₂²×#97	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂⁴×C₇	—	2	6	no	—
62	C₂⁴×C₇	—	4	4	no	—
63	C₂⁴×C₇	—	4	8	no	—
64	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×Q₈	—	2	6	no	—
66	C₂×C₇×Q₈	—	2	12	no	—
67	C₂×C₇×Q₈	—	2	24	no	—
68	C₂×C₇×Q₈	—	2	24	no	—
69	C₂×C₇×Q₈	—	2	24	no	—
70	C₂×C₇×Q₈	—	4	12	no	—
71	C₂×C₇×Q₈	—	4	24	no	—
72	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
73	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
74	C₂×C₇×D₄	—	2	24	no	—
75	C₂×C₇×D₄	—	2	6	no	—
76	C₂×C₇×D₄	—	4	12	no	—
77	C₂×C₇×D₄	C₂⁴	1	4	yes	—
78	C₂²×C₄×C₇	—	2	8	no	—
79	C₂²×C₄×C₇	—	2	6	no	—
80	C₂²×C₄×C₇	—	2	12	no	—
81	C₂²×C₄×C₇	—	4	4	no	—
82	C₂²×C₄×C₇	—	4	8	no	—
83	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
86	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
87	C₇×Q₁₆	—	2	64	no	—
88	C₇×SD₁₆	—	2	64	no	—
89	C₂×C₇×C₈	C₂×D₄	1	48	yes	—
90	C₂×C₇×C₈	C₂⁴	1	8	yes	—
91	C₇×C₄⋊C₄	—	2	32	no	—
92	C₇×C₂²⋊C₄	—	2	32	no	—
93	C₄²×C₇	C₂×D₄	1	12	yes	—
94	C₄²×C₇	C₂⁴	1	4	yes	—
95	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
96	C₂³×Q₈	C₂²×C₇	1	2	yes	—
97	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
98	C₂²×Q₁₆	—	2	16	no	—
99	C₂²×Q₁₆	C₂²×C₇	1	24	yes	complemented
100	C₂²×SD₁₆	—	2	16	no	—
101	C₂²×SD₁₆	C₂²×C₇	1	24	yes	—
102	C₂³×C₈	C₂²×C₇	1	2	yes	—
103	C₂×C₄⋊Q₈	—	2	6	no	—
104	C₂×C₄⋊Q₈	C₂²×C₇	1	12	yes	complemented
105	C₂×#124	—	2	6	no	—
106	C₂×#124	C₂²×C₇	1	12	yes	—
107	C₂²×C₄⋊C₄	—	2	2	no	—
108	C₂²×C₂²⋊C₄	—	2	2	no	—
109	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
110	#497	C₂²×C₇	1	64	yes	direct product factor
111	C₂×#97	C₂²×C₇	1	12	yes	complemented
112	C₂³×C₇	—	2	4	no	—
113	C₂³×C₇	—	4	12	no	—
114	C₂³×C₇	—	4	24	no	—
115	C₂³×C₇	—	8	16	no	—
116	C₂³×C₇	#127	1	4	yes	—
117	C₂³×C₇	#137	1	8	yes	—
118	C₂³×C₇	C₂²×D₄	1	3	yes	—
119	C₇×Q₈	—	2	8	no	—
120	C₇×Q₈	—	2	8	no	—
121	C₇×Q₈	—	2	8	no	—
122	C₇×Q₈	—	4	12	no	—
123	C₇×Q₈	—	4	24	no	—
124	C₇×D₄	—	2	8	no	—
125	C₇×D₄	—	4	12	no	—
126	C₂×C₄×C₇	—	2	24	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	8	no	—
129	C₂×C₄×C₇	—	4	12	no	—
130	C₂×C₄×C₇	—	4	24	no	—
131	C₂×C₄×C₇	#127	1	12	yes	—
132	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
133	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
134	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
135	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
136	C₇×C₈	C₂²×D₄	1	16	yes	—
137	C₂⁵	—	2	1	no	—
138	C₂²×Q₈	—	2	3	no	—
139	C₂²×Q₈	—	2	6	no	—
140	C₂²×Q₈	—	2	8	no	—
141	C₂²×Q₈	—	2	8	no	—
142	C₂²×Q₈	—	2	8	no	—
143	C₂²×Q₈	C₂³×C₇	1	6	yes	—
144	C₂²×Q₈	C₂³×C₇	1	12	yes	—
145	C₂²×D₄	—	2	8	no	—
146	C₂²×D₄	—	2	3	no	—
147	C₂²×D₄	C₂³×C₇	1	6	yes	—
148	C₂³×C₄	—	2	1	no	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	C₇×D₄	1	2	yes	—
151	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₂×Q₁₆	—	2	96	no	—
153	C₂×Q₁₆	C₂³×C₇	1	32	yes	complemented
154	C₂×SD₁₆	—	2	96	no	—
155	C₂×SD₁₆	C₂³×C₇	1	32	yes	—
156	C₂²×C₈	C₇×D₄	1	16	yes	—
157	C₂²×C₈	C₂³×C₇	1	12	yes	—
158	C₄⋊Q₈	—	2	24	no	—
159	C₄⋊Q₈	C₂³×C₇	1	16	yes	complemented
160	#124	—	2	24	no	—
161	#124	C₂³×C₇	1	16	yes	—
162	C₂×C₄⋊C₄	—	2	24	no	—
163	C₂×C₂²⋊C₄	—	2	24	no	—
164	C₂×C₄²	C₇×D₄	1	6	yes	—
165	C₂×C₄²	C₂³×C₇	1	6	yes	—
166	#97	C₂³×C₇	1	16	yes	complemented
167	C₂²×C₇	—	4	4	no	—
168	C₂²×C₇	—	4	8	no	—
169	C₂²×C₇	—	8	24	no	—
170	C₂²×C₇	#497	1	16	yes	direct product factor
171	C₂²×C₇	C₂×#127	1	6	yes	—
172	C₂²×C₇	C₂×#137	1	12	yes	—
173	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
174	C₄×C₇	—	2	8	no	—
175	C₄×C₇	—	4	4	no	—
176	C₄×C₇	—	4	8	no	—
177	C₄×C₇	C₂×#127	1	6	yes	—
178	C₄×C₇	C₂³×D₄	1	2	yes	—
179	C₂⁴	—	2	6	no	—
180	C₂⁴	—	4	4	no	—
181	C₂⁴	—	4	8	no	—
182	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
183	C₂×Q₈	—	2	6	no	—
184	C₂×Q₈	—	2	12	no	—
185	C₂×Q₈	—	2	24	no	—
186	C₂×Q₈	—	2	24	no	—
187	C₂×Q₈	—	2	24	no	—
188	C₂×Q₈	—	4	12	no	—
189	C₂×Q₈	—	4	24	no	—
190	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
191	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
192	C₂×D₄	—	2	24	no	—
193	C₂×D₄	—	2	6	no	—
194	C₂×D₄	—	4	12	no	—
195	C₂×D₄	C₂⁴×C₇	1	4	yes	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	8	no	—
201	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
202	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
203	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
204	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
205	Q₁₆	—	2	64	no	—
206	SD₁₆	—	2	64	no	—
207	C₂×C₈	C₂×C₇×D₄	1	48	yes	—
208	C₂×C₈	C₂⁴×C₇	1	8	yes	—
209	C₄⋊C₄	—	2	32	no	—
210	C₂²⋊C₄	—	2	32	no	—
211	C₄²	C₂×C₇×D₄	1	12	yes	—
212	C₄²	C₂⁴×C₇	1	4	yes	—
213	C₂×C₇	—	8	4	no	—
214	C₂×C₇	C₂×#497	1	12	yes	direct product factor
215	C₂×C₇	C₂²×#127	1	1	yes	characteristic
216	C₂×C₇	C₂²×#137	1	2	yes	—
217	C₂³	—	2	4	no	—
218	C₂³	—	4	12	no	—
219	C₂³	—	4	24	no	—
220	C₂³	—	8	16	no	—
221	C₂³	C₇×#127	1	4	yes	—
222	C₂³	C₇×#137	1	8	yes	—
223	C₂³	C₂²×C₇×D₄	1	3	yes	—
224	Q₈	—	2	8	no	—
225	Q₈	—	2	8	no	—
226	Q₈	—	2	8	no	—
227	Q₈	—	4	12	no	—
228	Q₈	—	4	24	no	—
229	D₄	—	2	8	no	—
230	D₄	—	4	12	no	—
231	C₂×C₄	—	2	24	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	24	no	—
236	C₂×C₄	C₇×#127	1	12	yes	—
237	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
238	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
239	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
240	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
241	C₈	C₂²×C₇×D₄	1	16	yes	—
242	C₇	C₂²×#497	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	4	8	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	8	24	no	—
246	C₂²	C₇×#497	1	16	yes	direct product factor
247	C₂²	C₂×C₇×#127	1	6	yes	—
248	C₂²	C₂×C₇×#137	1	12	yes	—
249	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×C₇×#127	1	6	yes	—
254	C₄	C₂³×C₇×D₄	1	2	yes	—
255	C₂	—	8	4	no	—
256	C₂	C₂×C₇×#497	1	12	yes	direct product factor
257	C₂	C₂²×C₇×#127	1	1	yes	characteristic
258	C₂	C₂²×C₇×#137	1	2	yes	—

	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. C₂³×C₇×Q₁₆	C₂	semidirect
3. C₂²×C₇×C₄⋊Q₈	C₂	semidirect
5. C₂×C₇×#497	C₂	direct (N × H)
6. C₂²×C₇×#97	C₂	semidirect
11. C₂²×C₇×Q₁₆	C₂²	semidirect
16. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
22. C₇×#497	C₂²	direct (N × H)
23. C₂×C₇×#97	C₂²	semidirect
24. C₂²×#497	C₇	direct (N × H)
41. C₂×C₇×Q₁₆	C₂³	semidirect
47. C₇×C₄⋊Q₈	C₂³	semidirect
54. C₇×#97	C₂³	semidirect
55. C₂³×Q₁₆	C₂×C₇	semidirect
57. C₂²×C₄⋊Q₈	C₂×C₇	semidirect
59. C₂×#497	C₂×C₇	direct (N × H)
60. C₂²×#97	C₂×C₇	semidirect
99. C₂²×Q₁₆	C₂²×C₇	semidirect
104. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
110. #497	C₂²×C₇	direct (N × H)
111. C₂×#97	C₂²×C₇	semidirect
153. C₂×Q₁₆	C₂³×C₇	semidirect
159. C₄⋊Q₈	C₂³×C₇	semidirect
166. #97	C₂³×C₇	semidirect
170. C₂²×C₇	#497	direct (N × H)
214. C₂×C₇	C₂×#497	direct (N × H)
242. C₇	C₂²×#497	direct (N × H)
246. C₂²	C₇×#497	direct (N × H)
256. C₂	C₂×C₇×#497	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×Q₁₆	2
C₂³×C₇×SD₁₆	2
C₂²×C₇×C₄⋊Q₈	2
C₂²×C₇×#124	2
C₂×C₇×#497	2
C₂²×C₇×#97	2
C₂²×#497	7

Subgroups — C22×C7×#497 #13237850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#497 #13237850