Subgroups — C₂²×C₇×#461 #13237620

Order 1792.

3222 conjugacy classes of subgroups.

The subgroup lattice has 298 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
2	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#461	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×#103	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₇	—	2	1	no	—
10	C₂²×C₇×SD₁₆	—	2	16	no	—
11	C₂²×C₇×SD₁₆	C₂²	1	24	yes	—
12	C₂³×C₇×C₈	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊Q₈	—	2	6	no	—
14	C₂×C₇×C₄⋊Q₈	C₂²	1	12	yes	complemented
15	C₂×C₄×C₇×D₄	—	2	6	no	—
16	C₂×C₄×C₇×D₄	C₂²	1	12	yes	—
17	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
18	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
19	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
21	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#461	C₂²	1	64	yes	direct product factor
23	C₂×C₇×C₄⋊C₈	C₂²	1	12	yes	complemented
24	C₂×C₇×#103	C₂²	1	24	yes	complemented
25	C₂²×#461	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂⁵×C₇	—	2	1	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	C₂³	1	12	yes	—
31	C₂²×C₇×D₄	—	2	3	no	—
32	C₂²×C₇×D₄	—	2	4	no	—
33	C₂²×C₇×D₄	D₄	1	8	yes	—
34	C₂²×C₇×D₄	C₂³	1	6	yes	—
35	C₂³×C₄×C₇	—	2	2	no	—
36	C₂³×C₄×C₇	—	2	1	no	—
37	C₂³×C₄×C₇	—	2	12	no	—
38	C₂³×C₄×C₇	—	4	8	no	—
39	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₇×SD₁₆	—	2	96	no	—
43	C₂×C₇×SD₁₆	C₂³	1	32	yes	—
44	C₂²×C₇×C₈	—	2	8	no	—
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	C₄×C₇×D₄	—	2	24	no	—
49	C₄×C₇×D₄	C₂³	1	16	yes	—
50	C₂×C₇×C₄⋊C₄	—	2	12	no	—
51	C₂×C₇×C₄⋊C₄	—	2	12	no	—
52	C₂×C₇×C₄⋊C₄	—	2	3	no	—
53	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
54	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
55	C₂×C₄²×C₇	D₄	1	6	yes	—
56	C₂×C₄²×C₇	C₂³	1	6	yes	—
57	C₇×C₄⋊C₈	C₂³	1	16	yes	complemented
58	C₇×#103	C₂³	1	32	yes	complemented
59	C₂³×SD₁₆	C₂×C₇	1	2	yes	—
60	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂²×C₄×D₄	C₂×C₇	1	1	yes	characteristic
62	C₂×#461	C₂×C₇	1	24	yes	direct product factor
63	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
64	C₂²×#103	C₂×C₇	1	2	yes	complemented
65	C₂⁴×C₇	—	2	8	no	—
66	C₂⁴×C₇	—	2	6	no	—
67	C₂⁴×C₇	—	4	4	no	—
68	C₂⁴×C₇	—	4	4	no	—
69	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₇×Q₈	—	2	12	no	—
71	C₂×C₇×Q₈	—	2	24	no	—
72	C₂×C₇×Q₈	—	2	24	no	—
73	C₂×C₇×Q₈	—	4	24	no	—
74	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
75	C₂×C₇×D₄	—	2	24	no	—
76	C₂×C₇×D₄	—	2	6	no	—
77	C₂×C₇×D₄	—	2	12	no	—
78	C₂×C₇×D₄	C₂×D₄	1	24	yes	—
79	C₂×C₇×D₄	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₇	—	2	4	no	—
81	C₂²×C₄×C₇	—	2	6	no	—
82	C₂²×C₄×C₇	—	2	12	no	—
83	C₂²×C₄×C₇	—	2	16	no	—
84	C₂²×C₄×C₇	—	4	8	no	—
85	C₂²×C₄×C₇	—	4	4	no	—
86	C₂²×C₄×C₇	—	4	48	no	—
87	C₂²×C₄×C₇	SD₁₆	1	8	yes	complemented
88	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
89	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
90	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
92	C₂²×C₄×C₇	#41	1	4	yes	—
93	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
94	C₇×SD₁₆	—	2	64	no	—
95	C₂×C₇×C₈	—	2	24	no	—
96	C₂×C₇×C₈	C₂⁴	1	8	yes	—
97	C₇×C₄⋊C₄	—	2	16	no	—
98	C₇×C₄⋊C₄	—	2	16	no	—
99	C₇×C₄⋊C₄	—	2	6	no	—
100	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
101	C₇×C₂²⋊C₄	—	2	16	no	—
102	C₄²×C₇	C₂×D₄	1	12	yes	—
103	C₄²×C₇	C₂⁴	1	4	yes	—
104	C₂³×Q₈	C₂²×C₇	1	2	yes	—
105	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
106	C₂⁴×C₄	—	2	1	no	—
107	C₂²×SD₁₆	—	2	16	no	—
108	C₂²×SD₁₆	C₂²×C₇	1	24	yes	—
109	C₂³×C₈	C₂²×C₇	1	2	yes	—
110	C₂×C₄⋊Q₈	—	2	6	no	—
111	C₂×C₄⋊Q₈	C₂²×C₇	1	12	yes	complemented
112	C₂×C₄×D₄	—	2	6	no	—
113	C₂×C₄×D₄	C₂²×C₇	1	12	yes	—
114	C₂²×C₄⋊C₄	—	2	1	no	—
115	C₂²×C₄⋊C₄	—	2	1	no	—
116	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₂²×C₂²⋊C₄	—	2	1	no	—
118	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
119	#461	C₂²×C₇	1	64	yes	direct product factor
120	C₂×C₄⋊C₈	C₂²×C₇	1	12	yes	complemented
121	C₂×#103	C₂²×C₇	1	24	yes	complemented
122	C₂³×C₇	—	2	24	no	—
123	C₂³×C₇	—	2	4	no	—
124	C₂³×C₇	—	4	12	no	—
125	C₂³×C₇	—	4	12	no	—
126	C₂³×C₇	—	4	32	no	—
127	C₂³×C₇	#121	1	4	yes	—
128	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
129	C₂³×C₇	#137	1	4	yes	—
130	C₂³×C₇	C₂²×D₄	1	3	yes	—
131	C₇×Q₈	—	2	8	no	—
132	C₇×Q₈	—	2	8	no	—
133	C₇×Q₈	—	4	24	no	—
134	C₇×D₄	—	2	24	no	—
135	C₇×D₄	—	2	4	no	—
136	C₇×D₄	C₂²×D₄	1	8	yes	—
137	C₂×C₄×C₇	—	2	12	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	8	no	—
140	C₂×C₄×C₇	—	4	24	no	—
141	C₂×C₄×C₇	—	4	12	no	—
142	C₂×C₄×C₇	—	4	32	no	—
143	C₂×C₄×C₇	#121	1	12	yes	—
144	C₂×C₄×C₇	C₂×SD₁₆	1	24	yes	complemented
145	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
146	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
147	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
148	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₇	C₂×#41	1	6	yes	—
150	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
151	C₇×C₈	—	2	8	no	—
152	C₂⁵	—	2	1	no	—
153	C₂²×Q₈	—	2	6	no	—
154	C₂²×Q₈	—	2	8	no	—
155	C₂²×Q₈	—	2	8	no	—
156	C₂²×Q₈	C₂³×C₇	1	12	yes	—
157	C₂²×D₄	—	2	3	no	—
158	C₂²×D₄	—	2	4	no	—
159	C₂²×D₄	C₇×D₄	1	8	yes	—
160	C₂²×D₄	C₂³×C₇	1	6	yes	—
161	C₂³×C₄	—	2	1	no	—
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	—	2	12	no	—
164	C₂³×C₄	—	4	8	no	—
165	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
166	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	96	no	—
169	C₂×SD₁₆	C₂³×C₇	1	32	yes	—
170	C₂²×C₈	—	2	8	no	—
171	C₂²×C₈	C₂³×C₇	1	12	yes	—
172	C₄⋊Q₈	—	2	24	no	—
173	C₄⋊Q₈	C₂³×C₇	1	16	yes	complemented
174	C₄×D₄	—	2	24	no	—
175	C₄×D₄	C₂³×C₇	1	16	yes	—
176	C₂×C₄⋊C₄	—	2	12	no	—
177	C₂×C₄⋊C₄	—	2	12	no	—
178	C₂×C₄⋊C₄	—	2	3	no	—
179	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
180	C₂×C₂²⋊C₄	—	2	12	no	—
181	C₂×C₄²	C₇×D₄	1	6	yes	—
182	C₂×C₄²	C₂³×C₇	1	6	yes	—
183	C₄⋊C₈	C₂³×C₇	1	16	yes	complemented
184	#103	C₂³×C₇	1	32	yes	complemented
185	C₂²×C₇	—	2	8	no	—
186	C₂²×C₇	—	4	48	no	—
187	C₂²×C₇	—	4	4	no	—
188	C₂²×C₇	—	4	4	no	—
189	C₂²×C₇	#461	1	16	yes	direct product factor
190	C₂²×C₇	C₂×#121	1	6	yes	—
191	C₂²×C₇	C₂²×SD₁₆	1	6	yes	—
192	C₂²×C₇	C₂×#137	1	6	yes	—
193	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
194	C₄×C₇	—	2	4	no	—
195	C₄×C₇	—	4	8	no	—
196	C₄×C₇	—	4	4	no	—
197	C₄×C₇	C₂×#121	1	6	yes	—
198	C₄×C₇	C₂²×SD₁₆	1	8	yes	complemented
199	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
200	C₄×C₇	C₂²×#41	1	1	yes	characteristic
201	C₂⁴	—	2	8	no	—
202	C₂⁴	—	2	6	no	—
203	C₂⁴	—	4	4	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
206	C₂×Q₈	—	2	12	no	—
207	C₂×Q₈	—	2	24	no	—
208	C₂×Q₈	—	2	24	no	—
209	C₂×Q₈	—	4	24	no	—
210	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
211	C₂×D₄	—	2	24	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	—	2	12	no	—
214	C₂×D₄	C₂×C₇×D₄	1	24	yes	—
215	C₂×D₄	C₂⁴×C₇	1	4	yes	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	6	no	—
218	C₂²×C₄	—	2	12	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	4	8	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	48	no	—
223	C₂²×C₄	C₇×SD₁₆	1	8	yes	complemented
224	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
225	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
226	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
227	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
228	C₂²×C₄	C₇×#41	1	4	yes	—
229	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
230	SD₁₆	—	2	64	no	—
231	C₂×C₈	—	2	24	no	—
232	C₂×C₈	C₂⁴×C₇	1	8	yes	—
233	C₄⋊C₄	—	2	16	no	—
234	C₄⋊C₄	—	2	16	no	—
235	C₄⋊C₄	—	2	6	no	—
236	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
237	C₂²⋊C₄	—	2	16	no	—
238	C₄²	C₂×C₇×D₄	1	12	yes	—
239	C₄²	C₂⁴×C₇	1	4	yes	—
240	C₂×C₇	—	4	8	no	—
241	C₂×C₇	C₂×#461	1	12	yes	direct product factor
242	C₂×C₇	C₂²×#121	1	1	yes	characteristic
243	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
244	C₂×C₇	C₂²×#137	1	1	yes	characteristic
245	C₂³	—	2	4	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	4	12	no	—
248	C₂³	—	4	12	no	—
249	C₂³	—	4	32	no	—
250	C₂³	C₇×#121	1	4	yes	—
251	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
252	C₂³	C₇×#137	1	4	yes	—
253	C₂³	C₂²×C₇×D₄	1	3	yes	—
254	Q₈	—	2	8	no	—
255	Q₈	—	2	8	no	—
256	Q₈	—	4	24	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	24	no	—
259	D₄	C₂²×C₇×D₄	1	8	yes	—
260	C₂×C₄	—	2	12	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	24	no	—
264	C₂×C₄	—	4	12	no	—
265	C₂×C₄	—	4	32	no	—
266	C₂×C₄	C₇×#121	1	12	yes	—
267	C₂×C₄	C₂×C₇×SD₁₆	1	24	yes	complemented
268	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
269	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
270	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
271	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
272	C₂×C₄	C₂×C₇×#41	1	6	yes	—
273	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₇	C₂²×#461	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	8	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	48	no	—
280	C₂²	C₇×#461	1	16	yes	direct product factor
281	C₂²	C₂×C₇×#121	1	6	yes	—
282	C₂²	C₂²×C₇×SD₁₆	1	6	yes	—
283	C₂²	C₂×C₇×#137	1	6	yes	—
284	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	8	no	—
288	C₄	C₂×C₇×#121	1	6	yes	—
289	C₄	C₂²×C₇×SD₁₆	1	8	yes	complemented
290	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
291	C₄	C₂²×C₇×#41	1	1	yes	characteristic
292	C₂	—	4	8	no	—
293	C₂	C₂×C₇×#461	1	12	yes	direct product factor
294	C₂	C₂²×C₇×#121	1	1	yes	characteristic
295	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic
296	C₂	C₂²×C₇×#137	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
2. C₂²×C₇×C₄⋊Q₈	C₂	semidirect
4. C₂×C₇×#461	C₂	direct (N × H)
5. C₂²×C₇×C₄⋊C₈	C₂	semidirect
6. C₂²×C₇×#103	C₂	semidirect
14. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
22. C₇×#461	C₂²	direct (N × H)
23. C₂×C₇×C₄⋊C₈	C₂²	semidirect
24. C₂×C₇×#103	C₂²	semidirect
25. C₂²×#461	C₇	direct (N × H)
47. C₇×C₄⋊Q₈	C₂³	semidirect
57. C₇×C₄⋊C₈	C₂³	semidirect
58. C₇×#103	C₂³	semidirect
60. C₂²×C₄⋊Q₈	C₂×C₇	semidirect
62. C₂×#461	C₂×C₇	direct (N × H)
63. C₂²×C₄⋊C₈	C₂×C₇	semidirect
64. C₂²×#103	C₂×C₇	semidirect
87. C₂²×C₄×C₇	SD₁₆	semidirect
111. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
119. #461	C₂²×C₇	direct (N × H)
120. C₂×C₄⋊C₈	C₂²×C₇	semidirect
121. C₂×#103	C₂²×C₇	semidirect
144. C₂×C₄×C₇	C₂×SD₁₆	semidirect
173. C₄⋊Q₈	C₂³×C₇	semidirect
183. C₄⋊C₈	C₂³×C₇	semidirect
184. #103	C₂³×C₇	semidirect
189. C₂²×C₇	#461	direct (N × H)
198. C₄×C₇	C₂²×SD₁₆	semidirect
223. C₂²×C₄	C₇×SD₁₆	semidirect
241. C₂×C₇	C₂×#461	direct (N × H)
267. C₂×C₄	C₂×C₇×SD₁₆	semidirect
275. C₇	C₂²×#461	direct (N × H)
280. C₂²	C₇×#461	direct (N × H)
289. C₄	C₂²×C₇×SD₁₆	semidirect
293. C₂	C₂×C₇×#461	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×C₄⋊Q₈	2
C₂²×C₄×C₇×D₄	2
C₂×C₇×#461	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₇×#103	2
C₂²×#461	7

Subgroups — C22×C7×#461 #13237620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#461 #13237620