Subgroups — C₂×C₇×#2089 #13200888

Order 1792.

810 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×#428	C₂	1	2	yes	maximal
2	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#424	C₂	1	1	yes	maximal, characteristic
4	C₇×#2089	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#343	C₂	1	2	yes	maximal
6	C₂×C₇×#341	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₇×#428	—	2	4	no	—
13	C₇×#428	C₂²	1	8	yes	—
14	C₂×C₇×#107	—	2	2	no	—
15	C₂×C₇×#107	—	2	2	no	—
16	C₂×C₇×#107	—	2	2	no	—
17	C₂×C₇×#107	C₂²	1	4	yes	—
18	C₂×C₇×#106	—	2	2	no	—
19	C₇×#424	—	2	2	no	—
20	C₇×#424	C₂²	1	4	yes	—
21	C₂×C₇×C₄⋊C₈	—	2	2	no	—
22	C₇×#343	C₂²	1	8	yes	—
23	C₇×#341	C₂²	1	4	yes	—
24	C₂×#2089	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₇	—	2	2	no	—
26	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	—	2	1	no	—
30	C₂×C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×M₁₆	Q₈	1	8	yes	—
32	C₂×C₇×M₁₆	C₂³	1	2	yes	—
33	C₂×C₇×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₇×C₈	—	2	2	no	—
35	C₂²×C₇×C₈	—	2	4	no	—
36	C₂²×C₇×C₈	—	2	1	no	—
37	C₂²×C₇×C₈	D₄	1	2	yes	—
38	C₂²×C₇×C₈	D₄	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₇×#117	—	2	1	no	—
41	C₇×#117	—	2	1	no	—
42	C₇×#117	D₄	1	2	yes	—
43	C₇×#117	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	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	—	2	2	no	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
53	C₂×C₄²×C₇	—	2	1	no	—
54	C₇×#107	—	2	8	no	—
55	C₇×#107	—	2	8	no	—
56	C₇×#106	—	2	8	no	—
57	C₇×C₄⋊C₈	—	2	8	no	—
58	C₂×#428	C₂×C₇	1	2	yes	—
59	C₂²×#107	C₂×C₇	1	1	yes	characteristic
60	C₂×#424	C₂×C₇	1	1	yes	characteristic
61	#2089	C₂×C₇	1	8	yes	direct product factor
62	C₂×#343	C₂×C₇	1	2	yes	—
63	C₂×#341	C₂×C₇	1	1	yes	characteristic
64	C₂⁴×C₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₇	—	2	1	no	—
66	C₂²×C₄×C₇	—	2	1	no	—
67	C₂²×C₄×C₇	—	2	4	no	—
68	C₂²×C₄×C₇	—	4	2	no	—
69	C₂²×C₄×C₇	—	4	2	no	—
70	C₂²×C₄×C₇	—	8	4	no	—
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₇	#41	1	4	yes	—
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×M₁₆	—	2	8	no	—
80	C₇×M₁₆	—	2	4	no	—
81	C₇×M₁₆	—	2	4	no	—
82	C₇×M₁₆	—	4	2	no	—
83	C₇×M₁₆	C₂×Q₈	1	8	yes	—
84	C₂×C₇×C₈	—	2	8	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₂×C₇×C₈	—	2	4	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	4	2	no	—
91	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
92	C₇×C₄⋊C₄	—	2	1	no	—
93	C₇×C₄⋊C₄	—	2	1	no	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	4	2	no	—
97	C₇×C₄⋊C₄	—	4	2	no	—
98	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
99	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
100	C₇×C₂²⋊C₄	—	2	2	no	—
101	C₇×C₂²⋊C₄	—	2	2	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
105	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
106	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
107	C₂×#117	C₂²×C₇	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
109	#428	—	2	4	no	—
110	#428	C₂²×C₇	1	8	yes	—
111	C₂×#107	—	2	2	no	—
112	C₂×#107	—	2	2	no	—
113	C₂×#107	—	2	2	no	—
114	C₂×#107	C₂²×C₇	1	4	yes	—
115	C₂×#106	—	2	2	no	—
116	#424	—	2	2	no	—
117	#424	C₂²×C₇	1	4	yes	—
118	C₂×C₄⋊C₈	—	2	2	no	—
119	#343	C₂²×C₇	1	8	yes	—
120	#341	C₂²×C₇	1	4	yes	—
121	C₂³×C₇	—	2	2	no	—
122	C₂³×C₇	—	2	2	no	—
123	C₂³×C₇	#121	1	1	yes	characteristic
124	C₂³×C₇	#121	1	1	yes	characteristic
125	C₂³×C₇	#125	1	4	yes	—
126	C₂³×C₇	C₂×#41	1	1	yes	characteristic
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	2	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	8	4	no	—
133	C₂×C₄×C₇	—	8	4	no	—
134	C₂×C₄×C₇	—	8	4	no	—
135	C₂×C₄×C₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
138	C₂×C₄×C₇	C₂²⋊Q₈	1	4	yes	—
139	C₂×C₄×C₇	#123	1	2	yes	—
140	C₂×C₄×C₇	#123	1	2	yes	—
141	C₂×C₄×C₇	#125	1	4	yes	—
142	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
144	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
147	C₂×C₄×C₇	C₂×#41	1	2	yes	—
148	C₇×C₈	—	2	8	no	—
149	C₇×C₈	—	2	4	no	—
150	C₇×C₈	—	4	2	no	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	C₇×Q₈	1	8	yes	—
158	C₂×M₁₆	C₂³×C₇	1	2	yes	—
159	C₂×M₁₆	C₂³×C₇	1	4	yes	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₇×D₄	1	2	yes	—
164	C₂²×C₈	C₇×D₄	1	2	yes	—
165	C₂²×C₈	C₂³×C₇	1	2	yes	—
166	#117	—	2	1	no	—
167	#117	—	2	1	no	—
168	#117	C₇×D₄	1	2	yes	—
169	#117	C₂³×C₇	1	2	yes	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
178	C₂×C₂²⋊C₄	—	2	1	no	—
179	C₂×C₄²	—	2	1	no	—
180	#107	—	2	8	no	—
181	#107	—	2	8	no	—
182	#106	—	2	8	no	—
183	C₄⋊C₈	—	2	8	no	—
184	C₂²×C₇	—	2	8	no	—
185	C₂²×C₇	—	2	1	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	—	2	1	no	—
188	C₂²×C₇	—	2	1	no	—
189	C₂²×C₇	#398	1	2	yes	—
190	C₂²×C₇	#398	1	2	yes	—
191	C₂²×C₇	#472	1	2	yes	—
192	C₂²×C₇	#473	1	2	yes	—
193	C₂²×C₇	C₂×#121	1	1	yes	characteristic
194	C₂²×C₇	C₂×#125	1	2	yes	—
195	C₄×C₇	—	4	2	no	—
196	C₄×C₇	—	4	2	no	—
197	C₄×C₇	—	8	4	no	—
198	C₄×C₇	#398	1	2	yes	—
199	C₄×C₇	#398	1	2	yes	—
200	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
201	C₄×C₇	C₂×#123	1	1	yes	characteristic
202	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
203	C₂⁴	C₇×#41	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	8	4	no	—
210	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
215	C₂²×C₄	C₇×#41	1	1	yes	characteristic
216	C₂²×C₄	C₇×#41	1	4	yes	—
217	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
218	M₁₆	—	2	8	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	C₂×C₇×Q₈	1	8	yes	—
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
231	C₄⋊C₄	—	2	1	no	—
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	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₂×C₇	—	2	2	no	—
244	C₂×C₇	—	2	2	no	—
245	C₂×C₇	#2089	1	4	yes	direct product factor
246	C₂×C₇	C₂×#398	1	1	yes	characteristic
247	C₂×C₇	C₂×#472	1	1	yes	characteristic
248	C₂×C₇	C₂×#473	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₇×#121	1	1	yes	characteristic
252	C₂³	C₇×#121	1	1	yes	characteristic
253	C₂³	C₇×#125	1	4	yes	—
254	C₂³	C₂×C₇×#41	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₇×#121	1	1	yes	characteristic
264	C₂×C₄	C₇×#121	1	1	yes	characteristic
265	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
266	C₂×C₄	C₇×C₂²⋊Q₈	1	4	yes	—
267	C₂×C₄	C₇×#123	1	2	yes	—
268	C₂×C₄	C₇×#123	1	2	yes	—
269	C₂×C₄	C₇×#125	1	4	yes	—
270	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
271	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
272	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
275	C₂×C₄	C₂×C₇×#41	1	2	yes	—
276	C₈	—	2	8	no	—
277	C₈	—	2	4	no	—
278	C₈	—	4	2	no	—
279	C₇	C₂×#2089	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	8	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₇×#398	1	2	yes	—
286	C₂²	C₇×#398	1	2	yes	—
287	C₂²	C₇×#472	1	2	yes	—
288	C₂²	C₇×#473	1	2	yes	—
289	C₂²	C₂×C₇×#121	1	1	yes	characteristic
290	C₂²	C₂×C₇×#125	1	2	yes	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	4	no	—
294	C₄	C₇×#398	1	2	yes	—
295	C₄	C₇×#398	1	2	yes	—
296	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
297	C₄	C₂×C₇×#123	1	1	yes	characteristic
298	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₇×#2089	1	4	yes	direct product factor
302	C₂	C₂×C₇×#398	1	1	yes	characteristic
303	C₂	C₂×C₇×#472	1	1	yes	characteristic
304	C₂	C₂×C₇×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#2089	C₂	direct (N × H)
24. C₂×#2089	C₇	direct (N × H)
61. #2089	C₂×C₇	direct (N × H)
245. C₂×C₇	#2089	direct (N × H)
279. C₇	C₂×#2089	direct (N × H)
301. C₂	C₇×#2089	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×#428	2
C₂²×C₇×#107	2
C₂×C₇×#424	2
C₇×#2089	2
C₂×C₇×#343	2
C₂×C₇×#341	2
C₂×#2089	7

Subgroups — C2×C7×#2089 #13200888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2089 #13200888