Subgroups — C₂×C₇×#1984 #13198514

Order 1792.

1502 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#1984	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#336	C₂	1	4	yes	maximal
6	C₂³×C₇×Q₈	C₄	1	1	yes	characteristic
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂³×C₇×C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₇×Q₈	—	2	4	no	—
10	C₂×C₄×C₇×Q₈	—	2	4	no	—
11	C₂×C₄×C₇×Q₈	C₄	1	4	yes	—
12	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
13	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₂²×C₇×C₄⋊C₄	C₄	1	2	yes	—
15	C₂²×C₄²×C₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₈	—	2	4	no	—
18	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₄×C₇×C₈	—	2	4	no	—
20	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
21	C₇×#336	C₂²	1	16	yes	—
22	C₂×#1984	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₇×Q₈	—	2	4	no	—
24	C₂²×C₇×Q₈	—	2	4	no	—
25	C₂²×C₇×Q₈	C₈	1	8	yes	complemented
26	C₂²×C₇×Q₈	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂²×C₇×C₈	—	2	4	no	—
33	C₂²×C₇×C₈	—	2	4	no	—
34	C₂²×C₇×C₈	—	2	4	no	—
35	C₂²×C₇×C₈	—	2	8	no	—
36	C₂²×C₇×C₈	Q₈	1	8	yes	complemented
37	C₂²×C₇×C₈	C₂³	1	4	yes	—
38	C₂²×C₇×C₈	C₂³	1	4	yes	—
39	C₄×C₇×Q₈	—	2	16	no	—
40	C₄×C₇×Q₈	—	2	8	no	—
41	C₄×C₇×Q₈	—	2	8	no	—
42	C₂×C₇×C₄⋊C₄	—	2	8	no	—
43	C₂×C₇×C₄⋊C₄	—	2	4	no	—
44	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	8	yes	—
46	C₂×C₄²×C₇	—	2	4	no	—
47	C₂×C₄²×C₇	—	2	4	no	—
48	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇	D₄	1	4	yes	—
51	C₂×C₄²×C₇	D₄	1	4	yes	—
52	C₂×C₄²×C₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₇	C₂³	1	2	yes	—
54	C₇×C₄⋊C₈	—	2	8	no	—
55	C₄×C₇×C₈	—	2	8	no	—
56	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
57	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
58	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
59	#1984	C₂×C₇	1	8	yes	direct product factor
60	C₂×#336	C₂×C₇	1	4	yes	—
61	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×Q₈	—	2	32	no	—
63	C₂×C₇×Q₈	—	2	8	no	—
64	C₂×C₇×Q₈	—	2	8	no	—
65	C₂×C₇×Q₈	C₂×C₈	1	16	yes	complemented
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	—	2	4	no	—
68	C₂²×C₄×C₇	—	2	4	no	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	—	2	8	no	—
71	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
74	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
75	C₂²×C₄×C₇	M₁₆	1	4	yes	—
76	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₇	#41	1	4	yes	—
85	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₇×C₈	—	2	16	no	—
87	C₂×C₇×C₈	—	2	4	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	2	4	no	—
90	C₂×C₇×C₈	—	2	8	no	—
91	C₂×C₇×C₈	—	4	8	no	—
92	C₂×C₇×C₈	C₂×Q₈	1	8	yes	complemented
93	C₇×C₄⋊C₄	—	2	16	no	—
94	C₇×C₄⋊C₄	—	2	8	no	—
95	C₄²×C₇	—	2	8	no	—
96	C₄²×C₇	—	2	4	no	—
97	C₄²×C₇	—	2	4	no	—
98	C₄²×C₇	C₂²⋊C₄	1	8	yes	—
99	C₄²×C₇	C₂×D₄	1	4	yes	—
100	C₄²×C₇	C₂×D₄	1	4	yes	—
101	C₂³×Q₈	C₄×C₇	1	1	yes	characteristic
102	C₂³×C₈	C₂²×C₇	1	2	yes	—
103	C₂³×C₈	C₂²×C₇	1	2	yes	—
104	C₂×C₄×Q₈	—	2	4	no	—
105	C₂×C₄×Q₈	—	2	4	no	—
106	C₂×C₄×Q₈	C₄×C₇	1	4	yes	—
107	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
108	C₂²×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
109	C₂²×C₄⋊C₄	C₄×C₇	1	2	yes	—
110	C₂²×C₄²	C₂²×C₇	1	2	yes	—
111	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	2	4	no	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
116	#336	C₂²×C₇	1	16	yes	—
117	C₂³×C₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂³×C₇	C₄×Q₈	1	4	yes	—
119	C₂³×C₇	C₂²×C₈	1	2	yes	—
120	C₂³×C₇	C₂×M₁₆	1	2	yes	—
121	C₂³×C₇	#131	1	4	yes	—
122	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
123	C₇×Q₈	—	2	32	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	16	no	—
128	C₂×C₄×C₇	—	2	8	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	8	no	—
131	C₂×C₄×C₇	—	4	8	no	—
132	C₂×C₄×C₇	C₂²⋊C₈	1	16	yes	—
133	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
136	C₂×C₄×C₇	C₄×Q₈	1	4	yes	—
137	C₂×C₄×C₇	C₂²⋊Q₈	1	8	yes	—
138	C₂×C₄×C₇	#124	1	4	yes	—
139	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
140	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
141	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
142	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
143	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
146	C₂×C₄×C₇	C₂×#41	1	2	yes	—
147	C₇×C₈	—	2	8	no	—
148	C₇×C₈	—	4	4	no	—
149	C₂²×Q₈	—	2	4	no	—
150	C₂²×Q₈	—	2	4	no	—
151	C₂²×Q₈	C₇×C₈	1	8	yes	complemented
152	C₂²×Q₈	C₂×C₄×C₇	1	4	yes	—
153	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
154	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
156	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
157	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	—	2	4	no	—
161	C₂²×C₈	—	2	8	no	—
162	C₂²×C₈	C₇×Q₈	1	8	yes	complemented
163	C₂²×C₈	C₂³×C₇	1	4	yes	—
164	C₂²×C₈	C₂³×C₇	1	4	yes	—
165	C₄×Q₈	—	2	16	no	—
166	C₄×Q₈	—	2	8	no	—
167	C₄×Q₈	—	2	8	no	—
168	C₂×C₄⋊C₄	—	2	8	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₇	1	8	yes	—
172	C₂×C₄²	—	2	4	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
175	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
176	C₂×C₄²	C₇×D₄	1	4	yes	—
177	C₂×C₄²	C₇×D₄	1	4	yes	—
178	C₂×C₄²	C₂³×C₇	1	4	yes	—
179	C₂×C₄²	C₂³×C₇	1	2	yes	—
180	C₄⋊C₈	—	2	8	no	—
181	C₄×C₈	—	2	8	no	—
182	C₂²×C₇	#391	1	4	yes	—
183	C₂²×C₇	C₂×C₂²⋊C₈	1	4	yes	—
184	C₂²×C₇	#408	1	4	yes	—
185	C₂²×C₇	C₈×Q₈	1	8	yes	—
186	C₂²×C₇	#446	1	8	yes	—
187	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×C₇	C₂×C₄×Q₈	1	2	yes	—
189	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
190	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
191	C₂²×C₇	C₂×#131	1	2	yes	—
192	C₄×C₇	—	2	8	no	—
193	C₄×C₇	—	2	4	no	—
194	C₄×C₇	—	4	4	no	—
195	C₄×C₇	#391	1	4	yes	—
196	C₄×C₇	C₂×C₂²⋊C₈	1	8	yes	—
197	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
198	C₄×C₇	C₂×#124	1	1	yes	characteristic
199	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
200	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
201	C₂×Q₈	—	2	32	no	—
202	C₂×Q₈	—	2	8	no	—
203	C₂×Q₈	—	2	8	no	—
204	C₂×Q₈	C₂×C₇×C₈	1	16	yes	complemented
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
211	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
212	C₂²×C₄	C₂×C₇×C₈	1	8	yes	—
213	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
214	C₂²×C₄	C₇×M₁₆	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
220	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
221	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
222	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
223	C₂²×C₄	C₇×#41	1	4	yes	—
224	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
225	C₂×C₈	—	2	16	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	4	8	no	—
231	C₂×C₈	C₂×C₇×Q₈	1	8	yes	complemented
232	C₄⋊C₄	—	2	16	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄²	—	2	8	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	C₇×C₂²⋊C₄	1	8	yes	—
238	C₄²	C₂×C₇×D₄	1	4	yes	—
239	C₄²	C₂×C₇×D₄	1	4	yes	—
240	C₂×C₇	#1984	1	8	yes	direct product factor
241	C₂×C₇	C₂×#391	1	1	yes	characteristic
242	C₂×C₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
243	C₂×C₇	C₂×#408	1	1	yes	characteristic
244	C₂×C₇	C₂×C₈×Q₈	1	2	yes	—
245	C₂×C₇	C₂×#446	1	2	yes	—
246	C₂³	C₂×C₇×C₂²⋊C₄	1	2	yes	—
247	C₂³	C₄×C₇×Q₈	1	4	yes	—
248	C₂³	C₂²×C₇×C₈	1	2	yes	—
249	C₂³	C₂×C₇×M₁₆	1	2	yes	—
250	C₂³	C₇×#131	1	4	yes	—
251	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
252	Q₈	—	2	32	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	16	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	C₇×C₂²⋊C₈	1	16	yes	—
262	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
263	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
265	C₂×C₄	C₄×C₇×Q₈	1	4	yes	—
266	C₂×C₄	C₇×C₂²⋊Q₈	1	8	yes	—
267	C₂×C₄	C₇×#124	1	4	yes	—
268	C₂×C₄	C₇×C₄⋊Q₈	1	4	yes	—
269	C₂×C₄	C₂²×C₇×C₈	1	8	yes	—
270	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
271	C₂×C₄	C₂×C₇×M₁₆	1	4	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	—	4	4	no	—
278	C₇	C₂×#1984	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	C₇×#391	1	4	yes	—
280	C₂²	C₂×C₇×C₂²⋊C₈	1	4	yes	—
281	C₂²	C₇×#408	1	4	yes	—
282	C₂²	C₇×C₈×Q₈	1	8	yes	—
283	C₂²	C₇×#446	1	8	yes	—
284	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
285	C₂²	C₂×C₄×C₇×Q₈	1	2	yes	—
286	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
287	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
288	C₂²	C₂×C₇×#131	1	2	yes	—
289	C₄	—	2	8	no	—
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	C₇×#391	1	4	yes	—
293	C₄	C₂×C₇×C₂²⋊C₈	1	8	yes	—
294	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
295	C₄	C₂×C₇×#124	1	1	yes	characteristic
296	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
297	C₂	C₇×#1984	1	8	yes	direct product factor
298	C₂	C₂×C₇×#391	1	1	yes	characteristic
299	C₂	C₂²×C₇×C₂²⋊C₈	1	1	yes	characteristic
300	C₂	C₂×C₇×#408	1	1	yes	characteristic
301	C₂	C₂×C₇×C₈×Q₈	1	2	yes	—
302	C₂	C₂×C₇×#446	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1984	C₂	direct (N × H)
22. C₂×#1984	C₇	direct (N × H)
25. C₂²×C₇×Q₈	C₈	semidirect
36. C₂²×C₇×C₈	Q₈	semidirect
59. #1984	C₂×C₇	direct (N × H)
65. C₂×C₇×Q₈	C₂×C₈	semidirect
92. C₂×C₇×C₈	C₂×Q₈	semidirect
151. C₂²×Q₈	C₇×C₈	semidirect
162. C₂²×C₈	C₇×Q₈	semidirect
204. C₂×Q₈	C₂×C₇×C₈	semidirect
231. C₂×C₈	C₂×C₇×Q₈	semidirect
240. C₂×C₇	#1984	direct (N × H)
278. C₇	C₂×#1984	direct (N × H)
297. C₂	C₇×#1984	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×C₄×C₇×Q₈	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₄×C₇×C₈	2
C₇×#1984	2
C₂×C₇×#336	2
C₂×#1984	7

Subgroups — C2×C7×#1984 #13198514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1984 #13198514