Subgroups — C₂×C₇×#1668 #13196328

Order 1792.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1668	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×SD₁₆	—	2	2	no	—
8	C₂²×C₇×D₈	—	2	1	no	—
9	C₂²×C₇×D₈	—	2	1	no	—
10	C₂×C₇×#127	C₂²	1	2	yes	—
11	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₇×#493	—	2	2	no	—
13	C₇×#493	C₂²	1	4	yes	—
14	C₇×#460	—	2	4	no	—
15	C₇×#460	C₂²	1	8	yes	complemented
16	C₇×C₈×Q₈	—	2	2	no	—
17	C₇×C₈×Q₈	C₂²	1	4	yes	complemented
18	C₂×C₇×C₄⋊C₈	—	2	1	no	—
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₇×#102	—	2	2	no	—
22	C₂×C₄×C₇×C₈	—	2	1	no	—
23	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
24	C₇×#335	C₂²	1	4	yes	complemented
25	C₇×#331	C₂²	1	8	yes	complemented
26	C₂×#1668	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
28	C₂²×C₇×D₄	—	2	2	no	—
29	C₂²×C₇×D₄	—	2	2	no	—
30	C₂²×C₇×D₄	—	2	2	no	—
31	C₂×C₇×SD₁₆	—	2	8	no	—
32	C₂×C₇×SD₁₆	—	4	8	no	—
33	C₂×C₇×D₈	—	2	4	no	—
34	C₂×C₇×D₈	—	2	4	no	—
35	C₂×C₇×D₈	—	4	4	no	—
36	C₂×C₇×D₈	—	4	4	no	—
37	C₂²×C₇×C₈	—	2	1	no	—
38	C₂²×C₇×C₈	—	2	2	no	—
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₇×#127	—	2	2	no	—
42	C₇×#127	—	2	2	no	—
43	C₇×#127	—	2	2	no	—
44	C₇×#127	C₂³	1	4	yes	—
45	C₄×C₇×Q₈	—	2	1	no	—
46	C₄×C₇×Q₈	—	2	1	no	—
47	C₄×C₇×Q₈	D₄	1	2	yes	—
48	C₄×C₇×Q₈	C₂³	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	—	2	1	no	—
50	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₇	—	2	1	no	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×C₄⋊C₈	—	2	4	no	—
54	C₇×C₄⋊C₈	—	2	1	no	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×C₄⋊C₈	C₂³	1	2	yes	—
57	C₇×C₄⋊C₈	C₂³	1	4	yes	—
58	C₇×#102	—	2	8	no	—
59	C₄×C₇×C₈	—	2	4	no	—
60	C₄×C₇×C₈	D₄	1	2	yes	—
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	C₂×#493	C₂×C₇	1	1	yes	characteristic
63	C₂×#460	C₂×C₇	1	2	yes	complemented
64	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
65	#1668	C₂×C₇	1	8	yes	direct product factor
66	C₂×#335	C₂×C₇	1	1	yes	characteristic, complemented
67	C₂×#331	C₂×C₇	1	2	yes	complemented
68	C₂⁴×C₇	—	4	2	no	—
69	C₂×C₇×Q₈	—	2	1	no	—
70	C₂×C₇×Q₈	—	2	2	no	—
71	C₂×C₇×Q₈	D₈	1	4	yes	complemented
72	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
73	C₂×C₇×D₄	—	2	4	no	—
74	C₂×C₇×D₄	—	2	4	no	—
75	C₂×C₇×D₄	—	2	4	no	—
76	C₂×C₇×D₄	—	4	2	no	—
77	C₂×C₇×D₄	—	4	2	no	—
78	C₂×C₇×D₄	—	4	2	no	—
79	C₂×C₇×D₄	—	4	4	no	—
80	C₂×C₇×D₄	—	4	4	no	—
81	C₂×C₇×D₄	—	4	4	no	—
82	C₂×C₇×D₄	—	4	4	no	—
83	C₂×C₇×D₄	—	8	4	no	—
84	C₂²×C₄×C₇	—	2	1	no	—
85	C₂²×C₄×C₇	—	2	1	no	—
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	#41	1	1	yes	characteristic
89	C₇×SD₁₆	—	4	16	no	—
90	C₇×D₈	—	4	8	no	—
91	C₇×D₈	—	4	8	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	4	no	—
95	C₂×C₇×C₈	—	4	4	no	—
96	C₂×C₇×C₈	—	4	2	no	—
97	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
98	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	1	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
104	C₄²×C₇	—	2	2	no	—
105	C₄²×C₇	—	2	2	no	—
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	#41	1	1	yes	characteristic
109	C₄²×C₇	C₂⁴	1	1	yes	characteristic
110	C₂²×SD₁₆	—	2	2	no	—
111	C₂²×D₈	—	2	1	no	—
112	C₂²×D₈	—	2	1	no	—
113	C₂×#127	C₂²×C₇	1	2	yes	—
114	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
115	#493	—	2	2	no	—
116	#493	C₂²×C₇	1	4	yes	—
117	#460	—	2	4	no	—
118	#460	C₂²×C₇	1	8	yes	complemented
119	C₈×Q₈	—	2	2	no	—
120	C₈×Q₈	C₂²×C₇	1	4	yes	complemented
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
124	C₂×#102	—	2	2	no	—
125	C₂×C₄×C₈	—	2	1	no	—
126	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
127	#335	C₂²×C₇	1	4	yes	complemented
128	#331	C₂²×C₇	1	8	yes	complemented
129	C₂³×C₇	—	4	4	no	—
130	C₂³×C₇	—	8	4	no	—
131	C₂³×C₇	—	8	4	no	—
132	C₂³×C₇	—	8	4	no	—
133	C₂³×C₇	#121	1	1	yes	characteristic
134	C₇×Q₈	—	2	2	no	—
135	C₇×Q₈	—	2	4	no	—
136	C₇×Q₈	C₂×D₈	1	4	yes	complemented
137	C₇×D₄	—	4	4	no	—
138	C₇×D₄	—	4	4	no	—
139	C₇×D₄	—	4	4	no	—
140	C₇×D₄	—	4	4	no	—
141	C₇×D₄	—	8	4	no	—
142	C₇×D₄	—	8	4	no	—
143	C₇×D₄	—	8	4	no	—
144	C₇×D₄	—	8	4	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	4	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	4	2	no	—
150	C₂×C₄×C₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₇	#135	1	2	yes	—
156	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
157	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	—	4	4	no	—
162	C₇×C₈	—	4	2	no	—
163	C₇×C₈	C₂×SD₁₆	1	4	yes	complemented
164	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
165	C₂²×D₄	—	2	2	no	—
166	C₂²×D₄	—	2	2	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂×SD₁₆	—	2	8	no	—
169	C₂×SD₁₆	—	4	8	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	4	no	—
172	C₂×D₈	—	4	4	no	—
173	C₂×D₈	—	4	4	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
177	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
178	#127	—	2	2	no	—
179	#127	—	2	2	no	—
180	#127	—	2	2	no	—
181	#127	C₂³×C₇	1	4	yes	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	C₇×D₄	1	2	yes	—
185	C₄×Q₈	C₂³×C₇	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂³×C₇	1	2	yes	—
194	C₄⋊C₈	C₂³×C₇	1	4	yes	—
195	#102	—	2	8	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	C₇×D₄	1	2	yes	—
198	C₄×C₈	C₂³×C₇	1	2	yes	—
199	C₂²×C₇	—	8	4	no	—
200	C₂²×C₇	—	8	4	no	—
201	C₂²×C₇	—	8	4	no	—
202	C₂²×C₇	—	16	8	no	—
203	C₂²×C₇	#460	1	2	yes	—
204	C₂²×C₇	#466	1	2	yes	—
205	C₂²×C₇	#472	1	2	yes	—
206	C₂²×C₇	C₂×#121	1	1	yes	characteristic
207	C₄×C₇	—	2	4	no	—
208	C₄×C₇	—	2	1	no	—
209	C₄×C₇	—	2	1	no	—
210	C₄×C₇	—	4	2	no	—
211	C₄×C₇	#460	1	2	yes	—
212	C₄×C₇	#466	1	2	yes	—
213	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
214	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
215	C₄×C₇	C₂×#135	1	1	yes	characteristic
216	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂⁴	—	4	2	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	C₇×D₈	1	4	yes	complemented
221	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	8	4	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
237	C₂²×C₄	C₇×#41	1	1	yes	characteristic
238	SD₁₆	—	4	16	no	—
239	D₈	—	4	8	no	—
240	D₈	—	4	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
247	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
248	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
256	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
257	C₄²	C₇×#41	1	1	yes	characteristic
258	C₄²	C₂⁴×C₇	1	1	yes	characteristic
259	C₂×C₇	—	16	4	no	—
260	C₂×C₇	#1668	1	4	yes	direct product factor
261	C₂×C₇	C₂×#460	1	1	yes	characteristic
262	C₂×C₇	C₂×#466	1	1	yes	characteristic
263	C₂×C₇	C₂×#472	1	1	yes	characteristic
264	C₂³	—	4	4	no	—
265	C₂³	—	8	4	no	—
266	C₂³	—	8	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	C₇×#121	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	2	2	no	—
271	Q₈	C₂×C₇×D₈	1	4	yes	complemented
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₇×#121	1	1	yes	characteristic
286	C₂×C₄	C₇×#121	1	1	yes	characteristic
287	C₂×C₄	C₇×#121	1	1	yes	characteristic
288	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
289	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
290	C₂×C₄	C₇×#135	1	2	yes	—
291	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
292	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	4	no	—
297	C₈	—	4	2	no	—
298	C₈	C₂×C₇×SD₁₆	1	4	yes	complemented
299	C₇	C₂×#1668	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	4	no	—
303	C₂²	—	16	8	no	—
304	C₂²	C₇×#460	1	2	yes	—
305	C₂²	C₇×#466	1	2	yes	—
306	C₂²	C₇×#472	1	2	yes	—
307	C₂²	C₂×C₇×#121	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	C₇×#460	1	2	yes	—
313	C₄	C₇×#466	1	2	yes	—
314	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
315	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
316	C₄	C₂×C₇×#135	1	1	yes	characteristic
317	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂	—	16	4	no	—
319	C₂	C₇×#1668	1	4	yes	direct product factor
320	C₂	C₂×C₇×#460	1	1	yes	characteristic
321	C₂	C₂×C₇×#466	1	1	yes	characteristic
322	C₂	C₂×C₇×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#460	C₂	semidirect
3. C₂×C₇×C₈×Q₈	C₂	semidirect
4. C₇×#1668	C₂	direct (N × H)
5. C₂×C₇×#335	C₂	semidirect
6. C₂×C₇×#331	C₂	semidirect
15. C₇×#460	C₂²	semidirect
17. C₇×C₈×Q₈	C₂²	semidirect
24. C₇×#335	C₂²	semidirect
25. C₇×#331	C₂²	semidirect
26. C₂×#1668	C₇	direct (N × H)
63. C₂×#460	C₂×C₇	semidirect
64. C₂×C₈×Q₈	C₂×C₇	semidirect
65. #1668	C₂×C₇	direct (N × H)
66. C₂×#335	C₂×C₇	semidirect
67. C₂×#331	C₂×C₇	semidirect
71. C₂×C₇×Q₈	D₈	semidirect
97. C₂×C₇×C₈	SD₁₆	semidirect
118. #460	C₂²×C₇	semidirect
120. C₈×Q₈	C₂²×C₇	semidirect
127. #335	C₂²×C₇	semidirect
128. #331	C₂²×C₇	semidirect
136. C₇×Q₈	C₂×D₈	semidirect
163. C₇×C₈	C₂×SD₁₆	semidirect
220. C₂×Q₈	C₇×D₈	semidirect
246. C₂×C₈	C₇×SD₁₆	semidirect
260. C₂×C₇	#1668	direct (N × H)
271. Q₈	C₂×C₇×D₈	semidirect
298. C₈	C₂×C₇×SD₁₆	semidirect
299. C₇	C₂×#1668	direct (N × H)
319. C₂	C₇×#1668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#493	2
C₂×C₇×#460	2
C₂×C₇×C₈×Q₈	2
C₇×#1668	2
C₂×C₇×#335	2
C₂×C₇×#331	2
C₂×#1668	7

Subgroups — C2×C7×#1668 #13196328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1668 #13196328