Subgroups — C₂×C₇×#1695 #13196352

Order 1792.

694 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	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#476	C₂	1	2	yes	maximal
3	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic
4	C₇×#1695	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#333	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×M₁₆	—	2	1	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₇×#500	—	2	2	no	—
11	C₇×#500	C₂²	1	4	yes	complemented
12	C₇×#476	—	2	4	no	—
13	C₇×#476	C₂²	1	8	yes	—
14	C₇×#436	—	2	2	no	—
15	C₇×#436	C₂²	1	4	yes	—
16	C₂×C₇×#107	—	2	1	no	—
17	C₂×C₇×#107	—	2	1	no	—
18	C₂×C₇×#106	—	2	2	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×C₇×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
24	C₇×#334	C₂²	1	4	yes	complemented
25	C₇×#333	C₂²	1	8	yes	complemented
26	C₂×#1695	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₇×Q₈	—	2	2	no	—
28	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	—	2	1	no	—
30	C₂×C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×M₁₆	—	4	4	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₇×C₄⋊Q₈	—	2	2	no	—
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×C₄⋊Q₈	—	2	2	no	—
39	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄×C₇×D₄	—	2	1	no	—
41	C₄×C₇×D₄	—	2	1	no	—
42	C₄×C₇×D₄	D₄	1	2	yes	—
43	C₄×C₇×D₄	C₂³	1	2	yes	—
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×C₇×C₂²⋊C₄	—	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×#106	—	2	8	no	—
52	C₇×C₄⋊C₈	—	2	2	no	—
53	C₇×C₄⋊C₈	—	2	1	no	—
54	C₇×C₄⋊C₈	C₂³	1	4	yes	—
55	C₇×C₄⋊C₈	C₂³	1	2	yes	—
56	C₇×#102	—	2	8	no	—
57	C₇×C₂²⋊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, complemented
61	C₂×#476	C₂×C₇	1	2	yes	—
62	C₂×#436	C₂×C₇	1	1	yes	characteristic
63	#1695	C₂×C₇	1	8	yes	direct product factor
64	C₂×#334	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂×#333	C₂×C₇	1	2	yes	complemented
66	C₂⁴×C₇	—	2	1	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×D₄	—	2	1	no	—
71	C₂×C₇×D₄	—	2	4	no	—
72	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇	—	2	1	no	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	4	2	no	—
77	C₂²×C₄×C₇	—	4	2	no	—
78	C₂²×C₄×C₇	—	4	2	no	—
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	#41	1	1	yes	characteristic
82	C₇×M₁₆	—	4	8	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₂×C₇×C₈	—	4	2	no	—
87	C₂×C₇×C₈	—	4	4	no	—
88	C₂×C₇×C₈	SD₁₆	1	4	yes	—
89	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
91	C₇×C₄⋊C₄	—	2	1	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	C₇×C₂²⋊C₄	—	2	2	no	—
98	C₇×C₂²⋊C₄	—	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×M₁₆	—	2	1	no	—
104	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
105	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
106	#500	—	2	2	no	—
107	#500	C₂²×C₇	1	4	yes	complemented
108	#476	—	2	4	no	—
109	#476	C₂²×C₇	1	8	yes	—
110	#436	—	2	2	no	—
111	#436	C₂²×C₇	1	4	yes	—
112	C₂×#107	—	2	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	—	2	2	no	—
115	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
117	C₂×#102	—	2	2	no	—
118	C₂×C₂²⋊C₈	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
120	#334	C₂²×C₇	1	4	yes	complemented
121	#333	C₂²×C₇	1	8	yes	complemented
122	C₂³×C₇	—	2	2	no	—
123	C₂³×C₇	—	4	2	no	—
124	C₂³×C₇	—	4	2	no	—
125	C₂³×C₇	—	4	2	no	—
126	C₂³×C₇	#121	1	1	yes	characteristic
127	C₇×Q₈	—	8	4	no	—
128	C₇×Q₈	—	8	4	no	—
129	C₇×D₄	—	2	4	no	—
130	C₇×D₄	—	4	2	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	2	no	—
135	C₂×C₄×C₇	—	4	2	no	—
136	C₂×C₄×C₇	—	4	4	no	—
137	C₂×C₄×C₇	—	4	4	no	—
138	C₂×C₄×C₇	—	8	4	no	—
139	C₂×C₄×C₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₇	#137	1	2	yes	—
144	C₂×C₄×C₇	#137	1	4	yes	—
145	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
148	C₇×C₈	—	2	2	no	—
149	C₇×C₈	—	4	2	no	—
150	C₇×C₈	—	4	4	no	—
151	C₇×C₈	C₂×SD₁₆	1	4	yes	—
152	C₂²×Q₈	—	2	2	no	—
153	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
154	C₂³×C₄	—	2	1	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	4	4	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	C₇×D₄	1	2	yes	—
168	C₄×D₄	C₂³×C₇	1	2	yes	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
174	#107	—	2	4	no	—
175	#107	—	2	4	no	—
176	#106	—	2	8	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂³×C₇	1	2	yes	—
180	C₄⋊C₈	C₂³×C₇	1	4	yes	—
181	#102	—	2	8	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	C₇×D₄	1	2	yes	—
184	C₄×C₈	C₂³×C₇	1	2	yes	—
185	C₂²×C₇	—	4	2	no	—
186	C₂²×C₇	—	4	2	no	—
187	C₂²×C₇	—	4	2	no	—
188	C₂²×C₇	—	8	4	no	—
189	C₂²×C₇	#461	1	2	yes	—
190	C₂²×C₇	#470	1	2	yes	—
191	C₂²×C₇	#472	1	2	yes	—
192	C₂²×C₇	C₂×#121	1	1	yes	characteristic
193	C₄×C₇	—	2	1	no	—
194	C₄×C₇	—	2	1	no	—
195	C₄×C₇	—	4	2	no	—
196	C₄×C₇	—	8	4	no	—
197	C₄×C₇	#461	1	2	yes	—
198	C₄×C₇	#470	1	2	yes	—
199	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
200	C₄×C₇	C₂×#137	1	1	yes	characteristic
201	C₄×C₇	C₂×#137	1	2	yes	—
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	8	4	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₇×#41	1	1	yes	characteristic
218	M₁₆	—	4	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	C₇×SD₁₆	1	4	yes	—
225	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
226	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
236	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
237	C₄²	C₇×#41	1	1	yes	characteristic
238	C₄²	C₂⁴×C₇	1	1	yes	characteristic
239	C₂×C₇	—	8	2	no	—
240	C₂×C₇	#1695	1	4	yes	direct product factor
241	C₂×C₇	C₂×#461	1	1	yes	characteristic
242	C₂×C₇	C₂×#470	1	1	yes	characteristic
243	C₂×C₇	C₂×#472	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₇×#121	1	1	yes	characteristic
249	Q₈	—	8	4	no	—
250	Q₈	—	8	4	no	—
251	D₄	—	2	4	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₇×#121	1	1	yes	characteristic
262	C₂×C₄	C₇×#121	1	1	yes	characteristic
263	C₂×C₄	C₇×#121	1	1	yes	characteristic
264	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
265	C₂×C₄	C₇×#137	1	4	yes	—
266	C₂×C₄	C₇×#137	1	2	yes	—
267	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	4	no	—
273	C₈	C₂×C₇×SD₁₆	1	4	yes	—
274	C₇	C₂×#1695	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	8	4	no	—
279	C₂²	C₇×#461	1	2	yes	—
280	C₂²	C₇×#470	1	2	yes	—
281	C₂²	C₇×#472	1	2	yes	—
282	C₂²	C₂×C₇×#121	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₇×#461	1	2	yes	—
288	C₄	C₇×#470	1	2	yes	—
289	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
290	C₄	C₂×C₇×#137	1	1	yes	characteristic
291	C₄	C₂×C₇×#137	1	2	yes	—
292	C₂	—	8	2	no	—
293	C₂	C₇×#1695	1	4	yes	direct product factor
294	C₂	C₂×C₇×#461	1	1	yes	characteristic
295	C₂	C₂×C₇×#470	1	1	yes	characteristic
296	C₂	C₂×C₇×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	D₄
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
1. C₂×C₇×#500	C₂	semidirect
4. C₇×#1695	C₂	direct (N × H)
5. C₂×C₇×#334	C₂	semidirect
6. C₂×C₇×#333	C₂	semidirect
11. C₇×#500	C₂²	semidirect
24. C₇×#334	C₂²	semidirect
25. C₇×#333	C₂²	semidirect
26. C₂×#1695	C₇	direct (N × H)
60. C₂×#500	C₂×C₇	semidirect
63. #1695	C₂×C₇	direct (N × H)
64. C₂×#334	C₂×C₇	semidirect
65. C₂×#333	C₂×C₇	semidirect
107. #500	C₂²×C₇	semidirect
120. #334	C₂²×C₇	semidirect
121. #333	C₂²×C₇	semidirect
240. C₂×C₇	#1695	direct (N × H)
274. C₇	C₂×#1695	direct (N × H)
293. C₂	C₇×#1695	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×#476	2
C₂×C₇×#436	2
C₇×#1695	2
C₂×C₇×#334	2
C₂×C₇×#333	2
C₂×#1695	7

Subgroups — C2×C7×#1695 #13196352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1695 #13196352