Subgroups — C₂×C₇×#1675 #13196320

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 306 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×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#1675	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#333	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×C₈	—	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×C₈×D₄	—	2	2	no	—
15	C₇×C₈×D₄	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×#335	C₂²	1	4	yes	complemented
25	C₇×#333	C₂²	1	8	yes	complemented
26	C₂×#1675	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	C₂²×C₇×C₈	—	2	1	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	4	no	—
33	C₂²×C₇×C₈	—	4	4	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×C₈×D₄	C₂×C₇	1	1	yes	characteristic
63	#1675	C₂×C₇	1	8	yes	direct product factor
64	C₂×#335	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	2	no	—
72	C₂×C₇×D₄	Q₁₆	1	4	yes	—
73	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇	—	2	1	no	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	2	2	no	—
77	C₂²×C₄×C₇	—	4	2	no	—
78	C₂²×C₄×C₇	—	4	2	no	—
79	C₂²×C₄×C₇	—	4	2	no	—
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	#41	1	1	yes	characteristic
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	—	4	8	no	—
89	C₂×C₇×C₈	SD₁₆	1	4	yes	—
90	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
92	C₇×C₄⋊C₄	—	2	1	no	—
93	C₇×C₄⋊C₄	—	2	4	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	4	2	no	—
96	C₇×C₄⋊C₄	—	4	2	no	—
97	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
98	C₇×C₂²⋊C₄	—	2	2	no	—
99	C₇×C₂²⋊C₄	—	2	2	no	—
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇	#41	1	1	yes	characteristic
103	C₄²×C₇	C₂⁴	1	1	yes	characteristic
104	C₂³×C₈	—	2	1	no	—
105	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
106	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
107	#500	—	2	2	no	—
108	#500	C₂²×C₇	1	4	yes	complemented
109	#476	—	2	4	no	—
110	#476	C₂²×C₇	1	8	yes	—
111	C₈×D₄	—	2	2	no	—
112	C₈×D₄	C₂²×C₇	1	4	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	—	2	1	no	—
115	C₂×#106	—	2	2	no	—
116	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
118	C₂×#102	—	2	2	no	—
119	C₂×C₂²⋊C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
121	#335	C₂²×C₇	1	4	yes	complemented
122	#333	C₂²×C₇	1	8	yes	complemented
123	C₂³×C₇	—	2	2	no	—
124	C₂³×C₇	—	2	4	no	—
125	C₂³×C₇	—	4	2	no	—
126	C₂³×C₇	—	4	2	no	—
127	C₂³×C₇	#121	1	1	yes	characteristic
128	C₇×Q₈	—	8	4	no	—
129	C₇×Q₈	—	8	4	no	—
130	C₇×D₄	—	2	2	no	—
131	C₇×D₄	—	2	4	no	—
132	C₇×D₄	C₂×Q₁₆	1	4	yes	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	4	2	no	—
136	C₂×C₄×C₇	—	4	2	no	—
137	C₂×C₄×C₇	—	4	2	no	—
138	C₂×C₄×C₇	—	4	4	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	—	8	4	no	—
141	C₂×C₄×C₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
146	C₂×C₄×C₇	#135	1	2	yes	—
147	C₂×C₄×C₇	#137	1	2	yes	—
148	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
151	C₇×C₈	—	2	2	no	—
152	C₇×C₈	—	4	2	no	—
153	C₇×C₈	—	4	4	no	—
154	C₇×C₈	C₂×SD₁₆	1	4	yes	—
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
157	C₂³×C₄	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	4	no	—
161	C₂²×C₈	—	4	4	no	—
162	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
163	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₇×D₄	1	2	yes	—
171	C₄×D₄	C₂³×C₇	1	2	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	2	1	no	—
176	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
177	#107	—	2	4	no	—
178	#107	—	2	4	no	—
179	#106	—	2	8	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂³×C₇	1	2	yes	—
183	C₄⋊C₈	C₂³×C₇	1	4	yes	—
184	#102	—	2	8	no	—
185	C₂²⋊C₈	—	2	4	no	—
186	C₄×C₈	C₇×D₄	1	2	yes	—
187	C₄×C₈	C₂³×C₇	1	2	yes	—
188	C₂²×C₇	—	2	4	no	—
189	C₂²×C₇	—	4	8	no	—
190	C₂²×C₇	—	4	2	no	—
191	C₂²×C₇	—	4	2	no	—
192	C₂²×C₇	#461	1	2	yes	—
193	C₂²×C₇	#467	1	2	yes	—
194	C₂²×C₇	#472	1	2	yes	—
195	C₂²×C₇	C₂×#121	1	1	yes	characteristic
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	2	1	no	—
198	C₄×C₇	—	4	2	no	—
199	C₄×C₇	—	8	4	no	—
200	C₄×C₇	#461	1	2	yes	—
201	C₄×C₇	#467	1	2	yes	—
202	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
203	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
204	C₄×C₇	C₂×#135	1	1	yes	characteristic
205	C₄×C₇	C₂×#137	1	1	yes	characteristic
206	C₂⁴	—	2	1	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	8	4	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₇×Q₁₆	1	4	yes	—
213	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₇×#41	1	1	yes	characteristic
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	4	8	no	—
229	C₂×C₈	C₇×SD₁₆	1	4	yes	—
230	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
231	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
241	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
242	C₄²	C₇×#41	1	1	yes	characteristic
243	C₄²	C₂⁴×C₇	1	1	yes	characteristic
244	C₂×C₇	—	4	4	no	—
245	C₂×C₇	#1675	1	4	yes	direct product factor
246	C₂×C₇	C₂×#461	1	1	yes	characteristic
247	C₂×C₇	C₂×#467	1	1	yes	characteristic
248	C₂×C₇	C₂×#472	1	1	yes	characteristic
249	C₂³	—	2	4	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	C₇×#121	1	1	yes	characteristic
254	Q₈	—	8	4	no	—
255	Q₈	—	8	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	D₄	C₂×C₇×Q₁₆	1	4	yes	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	C₇×#121	1	1	yes	characteristic
268	C₂×C₄	C₇×#121	1	1	yes	characteristic
269	C₂×C₄	C₇×#121	1	1	yes	characteristic
270	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
271	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
272	C₂×C₄	C₇×#135	1	2	yes	—
273	C₂×C₄	C₇×#137	1	2	yes	—
274	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	—	4	4	no	—
280	C₈	C₂×C₇×SD₁₆	1	4	yes	—
281	C₇	C₂×#1675	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	2	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	8	no	—
286	C₂²	C₇×#461	1	2	yes	—
287	C₂²	C₇×#467	1	2	yes	—
288	C₂²	C₇×#472	1	2	yes	—
289	C₂²	C₂×C₇×#121	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	4	no	—
294	C₄	C₇×#461	1	2	yes	—
295	C₄	C₇×#467	1	2	yes	—
296	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
297	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
298	C₄	C₂×C₇×#135	1	1	yes	characteristic
299	C₄	C₂×C₇×#137	1	1	yes	characteristic
300	C₂	—	4	4	no	—
301	C₂	C₇×#1675	1	4	yes	direct product factor
302	C₂	C₂×C₇×#461	1	1	yes	characteristic
303	C₂	C₂×C₇×#467	1	1	yes	characteristic
304	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×#1675	C₂	direct (N × H)
5. C₂×C₇×#335	C₂	semidirect
6. C₂×C₇×#333	C₂	semidirect
11. C₇×#500	C₂²	semidirect
24. C₇×#335	C₂²	semidirect
25. C₇×#333	C₂²	semidirect
26. C₂×#1675	C₇	direct (N × H)
60. C₂×#500	C₂×C₇	semidirect
63. #1675	C₂×C₇	direct (N × H)
64. C₂×#335	C₂×C₇	semidirect
65. C₂×#333	C₂×C₇	semidirect
108. #500	C₂²×C₇	semidirect
121. #335	C₂²×C₇	semidirect
122. #333	C₂²×C₇	semidirect
245. C₂×C₇	#1675	direct (N × H)
281. C₇	C₂×#1675	direct (N × H)
301. C₂	C₇×#1675	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×C₈×D₄	2
C₇×#1675	2
C₂×C₇×#335	2
C₂×C₇×#333	2
C₂×#1675	7

Subgroups — C2×C7×#1675 #13196320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1675 #13196320