Subgroups — C₂×C₇×#1676 #13196333

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#477	C₂	1	2	yes	maximal
3	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#1676	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×#333	C₂	1	2	yes	maximal
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×#500	—	2	2	no	—
10	C₇×#500	C₂²	1	4	yes	—
11	C₇×#477	—	2	4	no	—
12	C₇×#477	C₂²	1	8	yes	—
13	C₇×C₈×Q₈	—	2	2	no	—
14	C₇×C₈×Q₈	C₂²	1	4	yes	—
15	C₂×C₇×#107	—	2	1	no	—
16	C₂×C₇×#107	—	2	1	no	—
17	C₂×C₇×#107	—	2	2	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×#333	C₂²	1	8	yes	—
26	C₂×#1676	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	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
33	C₇×C₄⋊Q₈	—	2	2	no	—
34	C₇×C₄⋊Q₈	—	2	2	no	—
35	C₇×C₄⋊Q₈	—	2	2	no	—
36	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
37	C₄×C₇×Q₈	—	2	1	no	—
38	C₄×C₇×Q₈	—	2	1	no	—
39	C₄×C₇×Q₈	D₄	1	2	yes	—
40	C₄×C₇×Q₈	C₂³	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	—	2	1	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇	—	2	1	no	—
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₇×#107	—	2	4	no	—
48	C₇×#107	—	2	4	no	—
49	C₇×#107	—	2	8	no	—
50	C₇×C₄⋊C₈	—	2	4	no	—
51	C₇×C₄⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	—	2	1	no	—
53	C₇×C₄⋊C₈	C₂³	1	4	yes	—
54	C₇×C₄⋊C₈	C₂³	1	2	yes	—
55	C₇×#103	—	2	8	no	—
56	C₄×C₇×C₈	—	2	4	no	—
57	C₄×C₇×C₈	D₄	1	2	yes	—
58	C₄×C₇×C₈	C₂³	1	2	yes	—
59	C₂×#500	C₂×C₇	1	1	yes	characteristic
60	C₂×#477	C₂×C₇	1	2	yes	—
61	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
62	#1676	C₂×C₇	1	8	yes	direct product factor
63	C₂×#335	C₂×C₇	1	1	yes	characteristic
64	C₂×#333	C₂×C₇	1	2	yes	—
65	C₂×C₇×Q₈	—	2	1	no	—
66	C₂×C₇×Q₈	—	2	2	no	—
67	C₂×C₇×Q₈	—	2	4	no	—
68	C₂×C₇×Q₈	—	4	2	no	—
69	C₂×C₇×Q₈	—	8	4	no	—
70	C₂×C₇×Q₈	Q₁₆	1	4	yes	—
71	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇	—	2	1	no	—
73	C₂²×C₄×C₇	—	2	1	no	—
74	C₂²×C₄×C₇	—	4	2	no	—
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	#41	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	4	4	no	—
82	C₂×C₇×C₈	—	4	2	no	—
83	C₂×C₇×C₈	Q₁₆	1	4	yes	—
84	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
86	C₇×C₄⋊C₄	—	2	2	no	—
87	C₇×C₄⋊C₄	—	2	1	no	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	4	2	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
94	C₄²×C₇	—	2	2	no	—
95	C₄²×C₇	—	2	2	no	—
96	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇	#41	1	1	yes	characteristic
99	C₄²×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
101	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
102	#500	—	2	2	no	—
103	#500	C₂²×C₇	1	4	yes	—
104	#477	—	2	4	no	—
105	#477	C₂²×C₇	1	8	yes	—
106	C₈×Q₈	—	2	2	no	—
107	C₈×Q₈	C₂²×C₇	1	4	yes	—
108	C₂×#107	—	2	1	no	—
109	C₂×#107	—	2	1	no	—
110	C₂×#107	—	2	2	no	—
111	C₂×C₄⋊C₈	—	2	1	no	—
112	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
114	C₂×#103	—	2	2	no	—
115	C₂×C₄×C₈	—	2	1	no	—
116	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
117	#335	C₂²×C₇	1	4	yes	—
118	#333	C₂²×C₇	1	8	yes	—
119	C₂³×C₇	#121	1	1	yes	characteristic
120	C₇×Q₈	—	2	2	no	—
121	C₇×Q₈	—	2	4	no	—
122	C₇×Q₈	—	8	4	no	—
123	C₇×Q₈	—	8	4	no	—
124	C₇×Q₈	C₂×Q₁₆	1	4	yes	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	8	4	no	—
132	C₂×C₄×C₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
137	C₂×C₄×C₇	#135	1	2	yes	—
138	C₂×C₄×C₇	#137	1	2	yes	—
139	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
142	C₇×C₈	—	2	2	no	—
143	C₇×C₈	—	4	4	no	—
144	C₇×C₈	—	4	2	no	—
145	C₇×C₈	C₂×Q₁₆	1	4	yes	—
146	C₂²×Q₈	—	2	2	no	—
147	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	—	2	1	no	—
158	C₄×Q₈	C₇×D₄	1	2	yes	—
159	C₄×Q₈	C₂³×C₇	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
166	#107	—	2	4	no	—
167	#107	—	2	4	no	—
168	#107	—	2	8	no	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	C₂³×C₇	1	2	yes	—
173	C₄⋊C₈	C₂³×C₇	1	4	yes	—
174	#103	—	2	8	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	C₇×D₄	1	2	yes	—
177	C₄×C₈	C₂³×C₇	1	2	yes	—
178	C₂²×C₇	#462	1	2	yes	—
179	C₂²×C₇	#467	1	2	yes	—
180	C₂²×C₇	#472	1	2	yes	—
181	C₂²×C₇	C₂×#121	1	1	yes	characteristic
182	C₄×C₇	—	2	4	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	1	no	—
185	C₄×C₇	—	4	2	no	—
186	C₄×C₇	—	8	4	no	—
187	C₄×C₇	#462	1	2	yes	—
188	C₄×C₇	#467	1	2	yes	—
189	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
190	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₇	C₂×#135	1	1	yes	characteristic
192	C₄×C₇	C₂×#137	1	1	yes	characteristic
193	C₂×Q₈	—	2	1	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	8	4	no	—
198	C₂×Q₈	C₇×Q₁₆	1	4	yes	—
199	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₇×#41	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	C₇×Q₁₆	1	4	yes	—
212	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
213	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
225	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
226	C₄²	C₇×#41	1	1	yes	characteristic
227	C₄²	C₂⁴×C₇	1	1	yes	characteristic
228	C₂×C₇	#1676	1	4	yes	direct product factor
229	C₂×C₇	C₂×#462	1	1	yes	characteristic
230	C₂×C₇	C₂×#467	1	1	yes	characteristic
231	C₂×C₇	C₂×#472	1	1	yes	characteristic
232	C₂³	C₇×#121	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	8	4	no	—
236	Q₈	—	8	4	no	—
237	Q₈	C₂×C₇×Q₁₆	1	4	yes	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	C₇×#121	1	1	yes	characteristic
246	C₂×C₄	C₇×#121	1	1	yes	characteristic
247	C₂×C₄	C₇×#121	1	1	yes	characteristic
248	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
249	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
250	C₂×C₄	C₇×#135	1	2	yes	—
251	C₂×C₄	C₇×#137	1	2	yes	—
252	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	2	no	—
258	C₈	C₂×C₇×Q₁₆	1	4	yes	—
259	C₇	C₂×#1676	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	C₇×#462	1	2	yes	—
261	C₂²	C₇×#467	1	2	yes	—
262	C₂²	C₇×#472	1	2	yes	—
263	C₂²	C₂×C₇×#121	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	4	no	—
269	C₄	C₇×#462	1	2	yes	—
270	C₄	C₇×#467	1	2	yes	—
271	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
272	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
273	C₄	C₂×C₇×#135	1	1	yes	characteristic
274	C₄	C₂×C₇×#137	1	1	yes	characteristic
275	C₂	C₇×#1676	1	4	yes	direct product factor
276	C₂	C₂×C₇×#462	1	1	yes	characteristic
277	C₂	C₂×C₇×#467	1	1	yes	characteristic
278	C₂	C₂×C₇×#472	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×#1676	C₂	direct (N × H)
26. C₂×#1676	C₇	direct (N × H)
62. #1676	C₂×C₇	direct (N × H)
228. C₂×C₇	#1676	direct (N × H)
259. C₇	C₂×#1676	direct (N × H)
275. C₂	C₇×#1676	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×#477	2
C₂×C₇×C₈×Q₈	2
C₇×#1676	2
C₂×C₇×#335	2
C₂×C₇×#333	2
C₂×#1676	7

Subgroups — C2×C7×#1676 #13196333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1676 #13196333