Subgroups — C₂×C₇×#1674 #13196332

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#462	C₂	1	2	yes	maximal
3	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#1674	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×#332	C₂	1	2	yes	maximal
7	C₂²×C₇×Q₁₆	—	2	2	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₇×#500	—	2	2	no	—
11	C₇×#500	C₂²	1	4	yes	—
12	C₇×#462	—	2	4	no	—
13	C₇×#462	C₂²	1	8	yes	—
14	C₇×C₈×Q₈	—	2	2	no	—
15	C₇×C₈×Q₈	C₂²	1	4	yes	—
16	C₂×C₇×#107	—	2	1	no	—
17	C₂×C₇×#107	—	2	1	no	—
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×#103	—	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	—
25	C₇×#332	C₂²	1	8	yes	—
26	C₂×#1674	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₇×Q₈	—	2	2	no	—
28	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
29	C₂×C₇×Q₁₆	—	2	8	no	—
30	C₂×C₇×Q₁₆	—	4	8	no	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₇×C₄⋊Q₈	—	2	2	no	—
36	C₇×C₄⋊Q₈	—	2	2	no	—
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₄×C₇×Q₈	—	2	1	no	—
40	C₄×C₇×Q₈	—	2	1	no	—
41	C₄×C₇×Q₈	D₄	1	2	yes	—
42	C₄×C₇×Q₈	C₂³	1	2	yes	—
43	C₂×C₇×C₄⋊C₄	—	2	1	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	2	no	—
46	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	—	2	1	no	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×#107	—	2	4	no	—
50	C₇×#107	—	2	4	no	—
51	C₇×C₄⋊C₈	—	2	4	no	—
52	C₇×C₄⋊C₈	—	2	1	no	—
53	C₇×C₄⋊C₈	—	2	2	no	—
54	C₇×C₄⋊C₈	C₂³	1	2	yes	—
55	C₇×C₄⋊C₈	C₂³	1	4	yes	—
56	C₇×#103	—	2	8	no	—
57	C₄×C₇×C₈	—	2	4	no	—
58	C₄×C₇×C₈	D₄	1	2	yes	—
59	C₄×C₇×C₈	C₂³	1	2	yes	—
60	C₂×#500	C₂×C₇	1	1	yes	characteristic
61	C₂×#462	C₂×C₇	1	2	yes	—
62	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
63	#1674	C₂×C₇	1	8	yes	direct product factor
64	C₂×#335	C₂×C₇	1	1	yes	characteristic
65	C₂×#332	C₂×C₇	1	2	yes	—
66	C₂×C₇×Q₈	—	2	1	no	—
67	C₂×C₇×Q₈	—	2	2	no	—
68	C₂×C₇×Q₈	—	2	4	no	—
69	C₂×C₇×Q₈	—	4	2	no	—
70	C₂×C₇×Q₈	—	4	4	no	—
71	C₂×C₇×Q₈	—	4	4	no	—
72	C₂×C₇×Q₈	D₈	1	4	yes	—
73	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇	—	2	1	no	—
75	C₂²×C₄×C₇	—	2	1	no	—
76	C₂²×C₄×C₇	—	4	2	no	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	#41	1	1	yes	characteristic
80	C₇×Q₁₆	—	4	16	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	4	4	no	—
85	C₂×C₇×C₈	—	4	2	no	—
86	C₂×C₇×C₈	Q₁₆	1	4	yes	complemented
87	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
89	C₇×C₄⋊C₄	—	2	2	no	—
90	C₇×C₄⋊C₄	—	2	1	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	2	4	no	—
94	C₇×C₄⋊C₄	—	4	2	no	—
95	C₇×C₄⋊C₄	—	4	2	no	—
96	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇	#41	1	1	yes	characteristic
102	C₄²×C₇	C₂⁴	1	1	yes	characteristic
103	C₂²×Q₁₆	—	2	2	no	—
104	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
105	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
106	#500	—	2	2	no	—
107	#500	C₂²×C₇	1	4	yes	—
108	#462	—	2	4	no	—
109	#462	C₂²×C₇	1	8	yes	—
110	C₈×Q₈	—	2	2	no	—
111	C₈×Q₈	C₂²×C₇	1	4	yes	—
112	C₂×#107	—	2	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×C₄⋊C₈	—	2	1	no	—
115	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
117	C₂×#103	—	2	2	no	—
118	C₂×C₄×C₈	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
120	#335	C₂²×C₇	1	4	yes	—
121	#332	C₂²×C₇	1	8	yes	—
122	C₂³×C₇	#121	1	1	yes	characteristic
123	C₇×Q₈	—	2	2	no	—
124	C₇×Q₈	—	2	4	no	—
125	C₇×Q₈	—	4	4	no	—
126	C₇×Q₈	—	4	4	no	—
127	C₇×Q₈	—	8	4	no	—
128	C₇×Q₈	C₂×D₈	1	4	yes	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	4	2	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	8	4	no	—
136	C₂×C₄×C₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
141	C₂×C₄×C₇	#135	1	2	yes	—
142	C₂×C₄×C₇	#137	1	2	yes	—
143	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
146	C₇×C₈	—	2	2	no	—
147	C₇×C₈	—	4	4	no	—
148	C₇×C₈	—	4	2	no	—
149	C₇×C₈	C₂×Q₁₆	1	4	yes	complemented
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
152	C₂×Q₁₆	—	2	8	no	—
153	C₂×Q₁₆	—	4	8	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
157	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	C₇×D₄	1	2	yes	—
165	C₄×Q₈	C₂³×C₇	1	2	yes	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
172	#107	—	2	4	no	—
173	#107	—	2	4	no	—
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂³×C₇	1	2	yes	—
178	C₄⋊C₈	C₂³×C₇	1	4	yes	—
179	#103	—	2	8	no	—
180	C₄×C₈	—	2	4	no	—
181	C₄×C₈	C₇×D₄	1	2	yes	—
182	C₄×C₈	C₂³×C₇	1	2	yes	—
183	C₂²×C₇	#462	1	2	yes	—
184	C₂²×C₇	#466	1	2	yes	—
185	C₂²×C₇	#473	1	2	yes	—
186	C₂²×C₇	C₂×#121	1	1	yes	characteristic
187	C₄×C₇	—	2	4	no	—
188	C₄×C₇	—	2	1	no	—
189	C₄×C₇	—	2	1	no	—
190	C₄×C₇	—	4	2	no	—
191	C₄×C₇	—	8	4	no	—
192	C₄×C₇	#462	1	2	yes	—
193	C₄×C₇	#466	1	2	yes	—
194	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
195	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
196	C₄×C₇	C₂×#135	1	1	yes	characteristic
197	C₄×C₇	C₂×#137	1	1	yes	characteristic
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	C₇×D₈	1	4	yes	—
205	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₇×#41	1	1	yes	characteristic
212	Q₁₆	—	4	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₇×Q₁₆	1	4	yes	complemented
219	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
220	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
232	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
233	C₄²	C₇×#41	1	1	yes	characteristic
234	C₄²	C₂⁴×C₇	1	1	yes	characteristic
235	C₂×C₇	#1674	1	4	yes	direct product factor
236	C₂×C₇	C₂×#462	1	1	yes	characteristic
237	C₂×C₇	C₂×#466	1	1	yes	characteristic
238	C₂×C₇	C₂×#473	1	1	yes	characteristic
239	C₂³	C₇×#121	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	Q₈	C₂×C₇×D₈	1	4	yes	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	C₇×#121	1	1	yes	characteristic
254	C₂×C₄	C₇×#121	1	1	yes	characteristic
255	C₂×C₄	C₇×#121	1	1	yes	characteristic
256	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
257	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
258	C₂×C₄	C₇×#135	1	2	yes	—
259	C₂×C₄	C₇×#137	1	2	yes	—
260	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
262	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	4	4	no	—
265	C₈	—	4	2	no	—
266	C₈	C₂×C₇×Q₁₆	1	4	yes	complemented
267	C₇	C₂×#1674	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	C₇×#462	1	2	yes	—
269	C₂²	C₇×#466	1	2	yes	—
270	C₂²	C₇×#473	1	2	yes	—
271	C₂²	C₂×C₇×#121	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₇×#462	1	2	yes	—
278	C₄	C₇×#466	1	2	yes	—
279	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
280	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
281	C₄	C₂×C₇×#135	1	1	yes	characteristic
282	C₄	C₂×C₇×#137	1	1	yes	characteristic
283	C₂	C₇×#1674	1	4	yes	direct product factor
284	C₂	C₂×C₇×#462	1	1	yes	characteristic
285	C₂	C₂×C₇×#466	1	1	yes	characteristic
286	C₂	C₂×C₇×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
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
4. C₇×#1674	C₂	direct (N × H)
26. C₂×#1674	C₇	direct (N × H)
63. #1674	C₂×C₇	direct (N × H)
86. C₂×C₇×C₈	Q₁₆	semidirect
149. C₇×C₈	C₂×Q₁₆	semidirect
218. C₂×C₈	C₇×Q₁₆	semidirect
235. C₂×C₇	#1674	direct (N × H)
266. C₈	C₂×C₇×Q₁₆	semidirect
267. C₇	C₂×#1674	direct (N × H)
283. C₂	C₇×#1674	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×#500	2
C₂×C₇×#462	2
C₂×C₇×C₈×Q₈	2
C₇×#1674	2
C₂×C₇×#335	2
C₂×C₇×#332	2
C₂×#1674	7

Subgroups — C2×C7×#1674 #13196332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1674 #13196332