Subgroups — C₂×C₇×#2078 #13200860

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#417	C₂	1	2	yes	maximal
3	C₂×C₇×#408	C₂	1	1	yes	maximal, characteristic
4	C₇×#2078	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#343	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₇×#107	—	2	2	no	—
13	C₂×C₇×#107	—	2	2	no	—
14	C₂×C₇×#107	C₂²	1	4	yes	complemented
15	C₇×#417	—	2	4	no	—
16	C₇×#417	C₂²	1	8	yes	—
17	C₂×C₇×#103	—	2	2	no	—
18	C₂×C₇×#102	—	2	2	no	—
19	C₇×#408	—	2	2	no	—
20	C₇×#408	C₂²	1	4	yes	—
21	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
22	C₇×#343	C₂²	1	8	yes	complemented
23	C₇×#341	C₂²	1	4	yes	complemented
24	C₂×#2078	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×#41	—	2	1	no	—
26	C₂×C₇×#41	—	2	1	no	—
27	C₂×C₇×#41	—	2	4	no	—
28	C₂×C₇×#41	—	2	4	no	—
29	C₂×C₇×#41	D₄	1	2	yes	—
30	C₂×C₇×#41	C₂³	1	2	yes	—
31	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×D₄	—	2	1	no	—
33	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₇	—	2	1	no	—
35	C₂³×C₄×C₇	—	2	2	no	—
36	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂×C₇×M₁₆	—	2	1	no	—
38	C₂×C₇×M₁₆	—	2	2	no	—
39	C₂×C₇×M₁₆	—	2	4	no	—
40	C₂×C₇×M₁₆	—	2	4	no	—
41	C₂×C₇×M₁₆	—	2	4	no	—
42	C₂×C₇×M₁₆	C₂³	1	4	yes	—
43	C₂×C₇×M₁₆	C₂³	1	2	yes	—
44	C₂²×C₇×C₈	—	2	4	no	—
45	C₂²×C₇×C₈	—	2	1	no	—
46	C₂²×C₇×C₈	—	2	2	no	—
47	C₂²×C₇×C₈	D₄	1	2	yes	—
48	C₂²×C₇×C₈	D₄	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	—	2	2	no	—
51	C₂×C₇×C₄⋊C₄	—	2	2	no	—
52	C₂×C₇×C₄⋊C₄	—	2	2	no	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₇×C₄⋊C₄	—	2	2	no	—
55	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
56	C₇×#107	—	2	8	no	—
57	C₇×#103	—	2	8	no	—
58	C₇×#102	—	2	8	no	—
59	C₇×C₂²⋊C₈	—	2	8	no	—
60	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂×#417	C₂×C₇	1	2	yes	—
62	C₂×#408	C₂×C₇	1	1	yes	characteristic
63	#2078	C₂×C₇	1	8	yes	direct product factor
64	C₂×#343	C₂×C₇	1	2	yes	complemented
65	C₂×#341	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂⁴×C₇	—	2	1	no	—
67	C₂⁴×C₇	#41	1	1	yes	characteristic
68	C₇×#41	—	2	8	no	—
69	C₇×#41	—	2	4	no	—
70	C₇×#41	—	2	4	no	—
71	C₇×#41	—	4	2	no	—
72	C₇×#41	—	4	2	no	—
73	C₇×#41	—	4	4	no	—
74	C₂×C₇×Q₈	—	2	1	no	—
75	C₂×C₇×Q₈	—	2	4	no	—
76	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
77	C₂×C₇×D₄	—	2	2	no	—
78	C₂×C₇×D₄	—	2	2	no	—
79	C₂×C₇×D₄	—	2	1	no	—
80	C₂×C₇×D₄	—	2	4	no	—
81	C₂×C₇×D₄	—	4	4	no	—
82	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	—	2	1	no	—
84	C₂²×C₄×C₇	—	2	2	no	—
85	C₂²×C₄×C₇	—	2	2	no	—
86	C₂²×C₄×C₇	—	2	4	no	—
87	C₂²×C₄×C₇	—	4	2	no	—
88	C₂²×C₄×C₇	—	4	2	no	—
89	C₂²×C₄×C₇	—	4	2	no	—
90	C₂²×C₄×C₇	—	4	2	no	—
91	C₂²×C₄×C₇	—	8	4	no	—
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	#41	1	1	yes	characteristic
97	C₂²×C₄×C₇	#41	1	4	yes	—
98	C₂²×C₄×C₇	#41	1	4	yes	—
99	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
100	C₇×M₁₆	—	2	4	no	—
101	C₇×M₁₆	—	2	4	no	—
102	C₇×M₁₆	—	2	4	no	—
103	C₇×M₁₆	—	4	4	no	—
104	C₇×M₁₆	—	4	2	no	—
105	C₂×C₇×C₈	—	2	4	no	—
106	C₂×C₇×C₈	—	2	4	no	—
107	C₂×C₇×C₈	—	2	4	no	—
108	C₂×C₇×C₈	—	2	2	no	—
109	C₂×C₇×C₈	—	2	4	no	—
110	C₂×C₇×C₈	—	4	4	no	—
111	C₂×C₇×C₈	—	4	2	no	—
112	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	—	2	4	no	—
115	C₇×C₄⋊C₄	—	4	2	no	—
116	C₇×C₄⋊C₄	—	4	2	no	—
117	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
118	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
119	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
120	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₂×#107	—	2	2	no	—
123	C₂×#107	—	2	2	no	—
124	C₂×#107	C₂²×C₇	1	4	yes	complemented
125	#417	—	2	4	no	—
126	#417	C₂²×C₇	1	8	yes	—
127	C₂×#103	—	2	2	no	—
128	C₂×#102	—	2	2	no	—
129	#408	—	2	2	no	—
130	#408	C₂²×C₇	1	4	yes	—
131	C₂×C₂²⋊C₈	—	2	2	no	—
132	#343	C₂²×C₇	1	8	yes	complemented
133	#341	C₂²×C₇	1	4	yes	complemented
134	C₂³×C₇	—	2	2	no	—
135	C₂³×C₇	—	2	2	no	—
136	C₂³×C₇	—	2	2	no	—
137	C₂³×C₇	—	4	2	no	—
138	C₂³×C₇	—	4	2	no	—
139	C₂³×C₇	—	4	2	no	—
140	C₂³×C₇	#121	1	1	yes	characteristic
141	C₂³×C₇	#121	1	1	yes	characteristic
142	C₂³×C₇	#126	1	4	yes	—
143	C₂³×C₇	C₂×#41	1	1	yes	characteristic
144	C₇×Q₈	—	2	4	no	—
145	C₇×Q₈	—	4	2	no	—
146	C₇×D₄	—	2	4	no	—
147	C₇×D₄	—	4	4	no	—
148	C₇×D₄	—	4	4	no	—
149	C₇×D₄	—	4	2	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	2	no	—
153	C₂×C₄×C₇	—	4	2	no	—
154	C₂×C₄×C₇	—	4	4	no	—
155	C₂×C₄×C₇	—	4	4	no	—
156	C₂×C₄×C₇	—	8	4	no	—
157	C₂×C₄×C₇	—	8	4	no	—
158	C₂×C₄×C₇	—	8	4	no	—
159	C₂×C₄×C₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₇	#121	1	1	yes	characteristic
161	C₂×C₄×C₇	#123	1	2	yes	—
162	C₂×C₄×C₇	#123	1	2	yes	—
163	C₂×C₄×C₇	#123	1	4	yes	—
164	C₂×C₄×C₇	#123	1	4	yes	—
165	C₂×C₄×C₇	#124	1	2	yes	—
166	C₂×C₄×C₇	#124	1	2	yes	—
167	C₂×C₄×C₇	#126	1	4	yes	—
168	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×#41	1	2	yes	—
171	C₂×C₄×C₇	C₂×#41	1	2	yes	—
172	C₇×C₈	—	2	4	no	—
173	C₇×C₈	—	4	4	no	—
174	C₇×C₈	—	4	2	no	—
175	C₂×#41	—	2	1	no	—
176	C₂×#41	—	2	1	no	—
177	C₂×#41	—	2	4	no	—
178	C₂×#41	—	2	4	no	—
179	C₂×#41	C₇×D₄	1	2	yes	—
180	C₂×#41	C₂³×C₇	1	2	yes	—
181	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	—	2	2	no	—
186	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	C₂³×C₇	1	4	yes	—
193	C₂×M₁₆	C₂³×C₇	1	2	yes	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	C₇×D₄	1	2	yes	—
198	C₂²×C₈	C₇×D₄	1	2	yes	—
199	C₂²×C₈	C₂³×C₇	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
206	#107	—	2	8	no	—
207	#103	—	2	8	no	—
208	#102	—	2	8	no	—
209	C₂²⋊C₈	—	2	8	no	—
210	C₂²×C₇	—	2	8	no	—
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	—	2	1	no	—
213	C₂²×C₇	—	2	1	no	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	4	2	no	—
216	C₂²×C₇	—	4	2	no	—
217	C₂²×C₇	—	4	2	no	—
218	C₂²×C₇	—	8	4	no	—
219	C₂²×C₇	#397	1	2	yes	—
220	C₂²×C₇	#397	1	2	yes	—
221	C₂²×C₇	#472	1	2	yes	—
222	C₂²×C₇	#473	1	2	yes	—
223	C₂²×C₇	C₂×#121	1	1	yes	characteristic
224	C₂²×C₇	C₂×#126	1	2	yes	—
225	C₄×C₇	—	4	2	no	—
226	C₄×C₇	—	8	4	no	—
227	C₄×C₇	#397	1	2	yes	—
228	C₄×C₇	#397	1	2	yes	—
229	C₄×C₇	C₂×#123	1	1	yes	characteristic
230	C₄×C₇	C₂×#123	1	2	yes	—
231	C₄×C₇	C₂×#124	1	1	yes	characteristic
232	C₂⁴	—	2	1	no	—
233	C₂⁴	C₇×#41	1	1	yes	characteristic
234	#41	—	2	8	no	—
235	#41	—	2	4	no	—
236	#41	—	2	4	no	—
237	#41	—	4	2	no	—
238	#41	—	4	2	no	—
239	#41	—	4	4	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	4	no	—
248	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	8	4	no	—
258	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
262	C₂²×C₄	C₇×#41	1	1	yes	characteristic
263	C₂²×C₄	C₇×#41	1	4	yes	—
264	C₂²×C₄	C₇×#41	1	4	yes	—
265	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
266	M₁₆	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	—	4	4	no	—
270	M₁₆	—	4	2	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	4	4	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₂×C₇	—	2	2	no	—
284	C₂×C₇	—	2	2	no	—
285	C₂×C₇	—	8	2	no	—
286	C₂×C₇	#2078	1	4	yes	direct product factor
287	C₂×C₇	C₂×#397	1	1	yes	characteristic
288	C₂×C₇	C₂×#472	1	1	yes	characteristic
289	C₂×C₇	C₂×#473	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₇×#121	1	1	yes	characteristic
297	C₂³	C₇×#121	1	1	yes	characteristic
298	C₂³	C₇×#126	1	4	yes	—
299	C₂³	C₂×C₇×#41	1	1	yes	characteristic
300	Q₈	—	2	4	no	—
301	Q₈	—	4	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₇×#121	1	1	yes	characteristic
316	C₂×C₄	C₇×#121	1	1	yes	characteristic
317	C₂×C₄	C₇×#123	1	2	yes	—
318	C₂×C₄	C₇×#123	1	2	yes	—
319	C₂×C₄	C₇×#123	1	4	yes	—
320	C₂×C₄	C₇×#123	1	4	yes	—
321	C₂×C₄	C₇×#124	1	2	yes	—
322	C₂×C₄	C₇×#124	1	2	yes	—
323	C₂×C₄	C₇×#126	1	4	yes	—
324	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂×C₇×#41	1	2	yes	—
327	C₂×C₄	C₂×C₇×#41	1	2	yes	—
328	C₈	—	2	4	no	—
329	C₈	—	4	4	no	—
330	C₈	—	4	2	no	—
331	C₇	C₂×#2078	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²	—	2	8	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	4	no	—
341	C₂²	C₇×#397	1	2	yes	—
342	C₂²	C₇×#397	1	2	yes	—
343	C₂²	C₇×#472	1	2	yes	—
344	C₂²	C₇×#473	1	2	yes	—
345	C₂²	C₂×C₇×#121	1	1	yes	characteristic
346	C₂²	C₂×C₇×#126	1	2	yes	—
347	C₄	—	4	2	no	—
348	C₄	—	8	4	no	—
349	C₄	C₇×#397	1	2	yes	—
350	C₄	C₇×#397	1	2	yes	—
351	C₄	C₂×C₇×#123	1	1	yes	characteristic
352	C₄	C₂×C₇×#123	1	2	yes	—
353	C₄	C₂×C₇×#124	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	—	8	2	no	—
357	C₂	C₇×#2078	1	4	yes	direct product factor
358	C₂	C₂×C₇×#397	1	1	yes	characteristic
359	C₂	C₂×C₇×#472	1	1	yes	characteristic
360	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	C₂²×C₇×#41	C₂²
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
1. C₂²×C₇×#107	C₂	semidirect
4. C₇×#2078	C₂	direct (N × H)
5. C₂×C₇×#343	C₂	semidirect
6. C₂×C₇×#341	C₂	semidirect
14. C₂×C₇×#107	C₂²	semidirect
22. C₇×#343	C₂²	semidirect
23. C₇×#341	C₂²	semidirect
24. C₂×#2078	C₇	direct (N × H)
60. C₂²×#107	C₂×C₇	semidirect
63. #2078	C₂×C₇	direct (N × H)
64. C₂×#343	C₂×C₇	semidirect
65. C₂×#341	C₂×C₇	semidirect
124. C₂×#107	C₂²×C₇	semidirect
132. #343	C₂²×C₇	semidirect
133. #341	C₂²×C₇	semidirect
286. C₂×C₇	#2078	direct (N × H)
331. C₇	C₂×#2078	direct (N × H)
357. C₂	C₇×#2078	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×#107	2
C₂×C₇×#417	2
C₂×C₇×#408	2
C₇×#2078	2
C₂×C₇×#343	2
C₂×C₇×#341	2
C₂×#2078	7

Subgroups — C2×C7×#2078 #13200860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2078 #13200860