Subgroups — C₂×C₇×#3184 #13237907

Order 1792.

2514 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#532	C₂	1	2	yes	maximal, complemented
2	C₇×#3184	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#486	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#469	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×#442	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×#41	—	2	2	no	—
10	C₂²×C₇×D₈	C₂²	1	4	yes	—
11	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
12	C₇×#532	—	2	8	no	—
13	C₇×#532	C₂²	1	16	yes	complemented
14	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₇×#127	C₂²	1	2	yes	—
16	C₂×C₇×#124	—	2	2	no	—
17	C₂×C₇×#121	—	2	2	no	—
18	C₂×C₇×#121	C₂²	1	4	yes	—
19	C₂×C₄×C₇×D₄	—	2	2	no	—
20	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
21	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#498	C₂²	1	8	yes	complemented
23	C₇×#493	C₂²	1	8	yes	complemented
24	C₇×#486	C₂²	1	16	yes	complemented
25	C₇×#469	C₂²	1	32	yes	complemented
26	C₇×#442	C₂²	1	32	yes	complemented
27	C₂×C₇×#106	C₂²	1	4	yes	—
28	C₂×C₇×#102	C₂²	1	8	yes	—
29	C₄×C₇×M₁₆	C₂²	1	8	yes	complemented
30	C₂×C₇×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
33	C₂×#3184	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	16	no	—
35	C₂×C₇×#41	—	4	16	no	—
36	C₂²×C₇×Q₈	—	2	2	no	—
37	C₂²×C₇×D₄	—	2	4	no	—
38	C₂²×C₇×D₄	—	2	2	no	—
39	C₂²×C₇×D₄	—	2	2	no	—
40	C₂²×C₇×D₄	C₂³	1	4	yes	—
41	C₂³×C₄×C₇	—	2	4	no	—
42	C₂³×C₄×C₇	D₄	1	2	yes	—
43	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₇×D₈	—	2	16	no	—
45	C₂×C₇×D₈	C₂³	1	16	yes	—
46	C₂×C₇×M₁₆	—	2	8	no	—
47	C₂×C₇×M₁₆	C₂³	1	8	yes	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	2	yes	—
50	C₇×C₄⋊Q₈	—	2	4	no	—
51	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
52	C₇×#127	—	2	4	no	—
53	C₇×#127	C₂³	1	8	yes	—
54	C₇×#124	—	2	16	no	—
55	C₇×#121	—	2	16	no	—
56	C₇×#121	—	2	8	no	—
57	C₇×#121	C₂³	1	16	yes	—
58	C₄×C₇×D₄	—	2	16	no	—
59	C₄×C₇×D₄	—	2	8	no	—
60	C₄×C₇×D₄	C₂³	1	16	yes	—
61	C₂×C₇×C₄⋊C₄	—	2	2	no	—
62	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
63	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
64	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
65	C₂×C₄²×C₇	D₄	1	2	yes	—
66	C₂×C₄²×C₇	D₄	1	4	yes	—
67	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₇	C₂³	1	4	yes	—
70	C₇×#106	C₂³	1	16	yes	—
71	C₇×#102	C₂³	1	32	yes	—
72	C₇×#97	C₂³	1	8	yes	—
73	C₄×C₇×C₈	C₂³	1	4	yes	—
74	C₄×C₇×C₈	C₂³	1	4	yes	—
75	C₂×#532	C₂×C₇	1	2	yes	complemented
76	#3184	C₂×C₇	1	16	yes	direct product factor
77	C₂×#498	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂×#493	C₂×C₇	1	1	yes	characteristic, complemented
79	C₂×#486	C₂×C₇	1	2	yes	complemented
80	C₂×#469	C₂×C₇	1	4	yes	complemented
81	C₂×#442	C₂×C₇	1	4	yes	complemented
82	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
83	C₂⁴×C₇	—	2	4	no	—
84	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
85	C₇×#41	—	4	64	no	—
86	C₂×C₇×Q₈	—	2	8	no	—
87	C₂×C₇×Q₈	—	4	8	no	—
88	C₂×C₇×D₄	—	2	16	no	—
89	C₂×C₇×D₄	—	2	4	no	—
90	C₂×C₇×D₄	—	2	8	no	—
91	C₂×C₇×D₄	—	2	8	no	—
92	C₂×C₇×D₄	—	2	16	no	—
93	C₂×C₇×D₄	—	4	16	no	—
94	C₂×C₇×D₄	—	4	8	no	—
95	C₂×C₇×D₄	—	4	8	no	—
96	C₂×C₇×D₄	C₂⁴	1	8	yes	—
97	C₂²×C₄×C₇	—	2	8	no	—
98	C₂²×C₄×C₇	—	2	4	no	—
99	C₂²×C₄×C₇	—	2	4	no	—
100	C₂²×C₄×C₇	—	2	16	no	—
101	C₂²×C₄×C₇	—	4	16	no	—
102	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
104	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
108	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
109	C₇×D₈	—	2	32	no	—
110	C₇×M₁₆	—	2	16	no	—
111	C₂×C₇×C₈	—	2	4	no	—
112	C₂×C₇×C₈	#41	1	8	yes	complemented
113	C₂×C₇×C₈	C₂⁴	1	4	yes	—
114	C₂×C₇×C₈	C₂⁴	1	4	yes	—
115	C₇×C₄⋊C₄	—	2	8	no	—
116	C₇×C₄⋊C₄	—	2	4	no	—
117	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
118	C₇×C₂²⋊C₄	—	2	16	no	—
119	C₇×C₂²⋊C₄	—	2	16	no	—
120	C₄²×C₇	C₂×D₄	1	2	yes	—
121	C₄²×C₇	C₂×D₄	1	8	yes	—
122	C₄²×C₇	C₂×D₄	1	2	yes	—
123	C₄²×C₇	C₂⁴	1	2	yes	—
124	C₄²×C₇	C₂⁴	1	2	yes	—
125	C₂²×#41	—	2	2	no	—
126	C₂²×D₈	C₂²×C₇	1	4	yes	—
127	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
128	#532	—	2	8	no	—
129	#532	C₂²×C₇	1	16	yes	complemented
130	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
131	C₂×#127	C₂²×C₇	1	2	yes	—
132	C₂×#124	—	2	2	no	—
133	C₂×#121	—	2	2	no	—
134	C₂×#121	C₂²×C₇	1	4	yes	—
135	C₂×C₄×D₄	—	2	2	no	—
136	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
137	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
138	#498	C₂²×C₇	1	8	yes	complemented
139	#493	C₂²×C₇	1	8	yes	complemented
140	#486	C₂²×C₇	1	16	yes	complemented
141	#469	C₂²×C₇	1	32	yes	complemented
142	#442	C₂²×C₇	1	32	yes	complemented
143	C₂×#106	C₂²×C₇	1	4	yes	—
144	C₂×#102	C₂²×C₇	1	8	yes	—
145	C₄×M₁₆	C₂²×C₇	1	8	yes	complemented
146	C₂×#97	C₂²×C₇	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
149	C₂³×C₇	—	2	2	no	—
150	C₂³×C₇	—	2	4	no	—
151	C₂³×C₇	—	2	8	no	—
152	C₂³×C₇	—	4	8	no	—
153	C₂³×C₇	—	4	8	no	—
154	C₂³×C₇	—	4	8	no	—
155	C₂³×C₇	C₂²×D₄	1	2	yes	—
156	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
157	C₇×Q₈	—	4	16	no	—
158	C₇×D₄	—	2	16	no	—
159	C₇×D₄	—	4	32	no	—
160	C₇×D₄	—	4	8	no	—
161	C₇×D₄	—	4	16	no	—
162	C₇×D₄	—	4	16	no	—
163	C₂×C₄×C₇	—	2	2	no	—
164	C₂×C₄×C₇	—	2	4	no	—
165	C₂×C₄×C₇	—	2	4	no	—
166	C₂×C₄×C₇	—	2	16	no	—
167	C₂×C₄×C₇	—	2	4	no	—
168	C₂×C₄×C₇	—	2	8	no	—
169	C₂×C₄×C₇	—	4	32	no	—
170	C₂×C₄×C₇	—	4	8	no	—
171	C₂×C₄×C₇	AGL(1,ℤ/8)	1	8	yes	—
172	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
176	C₂×C₄×C₇	C₂×#41	1	4	yes	—
177	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
178	C₇×C₈	—	2	4	no	—
179	C₇×C₈	C₂×#41	1	8	yes	complemented
180	C₂×#41	—	2	16	no	—
181	C₂×#41	—	4	16	no	—
182	C₂²×Q₈	—	2	2	no	—
183	C₂²×D₄	—	2	4	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	C₂³×C₇	1	4	yes	—
187	C₂³×C₄	—	2	4	no	—
188	C₂³×C₄	C₇×D₄	1	2	yes	—
189	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
190	C₂×D₈	—	2	16	no	—
191	C₂×D₈	C₂³×C₇	1	16	yes	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	C₂³×C₇	1	8	yes	—
194	C₂²×C₈	C₂³×C₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₇	1	2	yes	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
198	#127	—	2	4	no	—
199	#127	C₂³×C₇	1	8	yes	—
200	#124	—	2	16	no	—
201	#121	—	2	16	no	—
202	#121	—	2	8	no	—
203	#121	C₂³×C₇	1	16	yes	—
204	C₄×D₄	—	2	16	no	—
205	C₄×D₄	—	2	8	no	—
206	C₄×D₄	C₂³×C₇	1	16	yes	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₄²	C₇×D₄	1	2	yes	—
212	C₂×C₄²	C₇×D₄	1	4	yes	—
213	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₇	1	4	yes	—
216	#106	C₂³×C₇	1	16	yes	—
217	#102	C₂³×C₇	1	32	yes	—
218	#97	C₂³×C₇	1	8	yes	—
219	C₄×C₈	C₂³×C₇	1	4	yes	—
220	C₄×C₈	C₂³×C₇	1	4	yes	—
221	C₂²×C₇	—	2	2	no	—
222	C₂²×C₇	—	2	4	no	—
223	C₂²×C₇	—	4	4	no	—
224	C₂²×C₇	—	4	8	no	—
225	C₂²×C₇	—	4	8	no	—
226	C₂²×C₇	—	4	8	no	—
227	C₂²×C₇	—	8	16	no	—
228	C₂²×C₇	#532	1	2	yes	—
229	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
230	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₇	—	2	2	no	—
232	C₄×C₇	—	2	4	no	—
233	C₄×C₇	—	4	8	no	—
234	C₄×C₇	#532	1	2	yes	—
235	C₄×C₇	C₂×AGL(1,ℤ/8)	1	8	yes	—
236	C₄×C₇	C₂²×#41	1	2	yes	—
237	C₂⁴	—	2	4	no	—
238	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
239	#41	—	4	64	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	4	8	no	—
242	C₂×D₄	—	2	16	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	8	no	—
245	C₂×D₄	—	2	8	no	—
246	C₂×D₄	—	2	16	no	—
247	C₂×D₄	—	4	16	no	—
248	C₂×D₄	—	4	8	no	—
249	C₂×D₄	—	4	8	no	—
250	C₂×D₄	C₂⁴×C₇	1	8	yes	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	16	no	—
255	C₂²×C₄	—	4	16	no	—
256	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
258	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
259	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
260	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
262	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
263	D₈	—	2	32	no	—
264	M₁₆	—	2	16	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	C₇×#41	1	8	yes	complemented
267	C₂×C₈	C₂⁴×C₇	1	4	yes	—
268	C₂×C₈	C₂⁴×C₇	1	4	yes	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
272	C₂²⋊C₄	—	2	16	no	—
273	C₂²⋊C₄	—	2	16	no	—
274	C₄²	C₂×C₇×D₄	1	2	yes	—
275	C₄²	C₂×C₇×D₄	1	8	yes	—
276	C₄²	C₂×C₇×D₄	1	2	yes	—
277	C₄²	C₂⁴×C₇	1	2	yes	—
278	C₄²	C₂⁴×C₇	1	2	yes	—
279	C₂×C₇	—	4	2	no	—
280	C₂×C₇	—	8	8	no	—
281	C₂×C₇	#3184	1	4	yes	direct product factor
282	C₂×C₇	C₂×#532	1	1	yes	characteristic
283	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
284	C₂³	—	2	2	no	—
285	C₂³	—	2	4	no	—
286	C₂³	—	2	8	no	—
287	C₂³	—	4	8	no	—
288	C₂³	—	4	8	no	—
289	C₂³	—	4	8	no	—
290	C₂³	C₂²×C₇×D₄	1	2	yes	—
291	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
292	Q₈	—	4	16	no	—
293	D₄	—	2	16	no	—
294	D₄	—	4	32	no	—
295	D₄	—	4	8	no	—
296	D₄	—	4	16	no	—
297	D₄	—	4	16	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	16	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	4	32	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	C₇×AGL(1,ℤ/8)	1	8	yes	—
307	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
308	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
309	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
311	C₂×C₄	C₂×C₇×#41	1	4	yes	—
312	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
313	C₈	—	2	4	no	—
314	C₈	C₂×C₇×#41	1	8	yes	complemented
315	C₇	C₂×#3184	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	8	no	—
321	C₂²	—	4	8	no	—
322	C₂²	—	8	16	no	—
323	C₂²	C₇×#532	1	2	yes	—
324	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	4	yes	—
325	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	8	no	—
329	C₄	C₇×#532	1	2	yes	—
330	C₄	C₂×C₇×AGL(1,ℤ/8)	1	8	yes	—
331	C₄	C₂²×C₇×#41	1	2	yes	—
332	C₂	—	4	2	no	—
333	C₂	—	8	8	no	—
334	C₂	C₇×#3184	1	4	yes	direct product factor
335	C₂	C₂×C₇×#532	1	1	yes	characteristic
336	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#532	C₂	semidirect
2. C₇×#3184	C₂	direct (N × H)
3. C₂×C₇×#498	C₂	semidirect
4. C₂×C₇×#493	C₂	semidirect
5. C₂×C₇×#486	C₂	semidirect
6. C₂×C₇×#469	C₂	semidirect
7. C₂×C₇×#442	C₂	semidirect
8. C₂×C₄×C₇×M₁₆	C₂	semidirect
13. C₇×#532	C₂²	semidirect
22. C₇×#498	C₂²	semidirect
23. C₇×#493	C₂²	semidirect
24. C₇×#486	C₂²	semidirect
25. C₇×#469	C₂²	semidirect
26. C₇×#442	C₂²	semidirect
29. C₄×C₇×M₁₆	C₂²	semidirect
33. C₂×#3184	C₇	direct (N × H)
75. C₂×#532	C₂×C₇	semidirect
76. #3184	C₂×C₇	direct (N × H)
77. C₂×#498	C₂×C₇	semidirect
78. C₂×#493	C₂×C₇	semidirect
79. C₂×#486	C₂×C₇	semidirect
80. C₂×#469	C₂×C₇	semidirect
81. C₂×#442	C₂×C₇	semidirect
82. C₂×C₄×M₁₆	C₂×C₇	semidirect
112. C₂×C₇×C₈	#41	semidirect
129. #532	C₂²×C₇	semidirect
138. #498	C₂²×C₇	semidirect
139. #493	C₂²×C₇	semidirect
140. #486	C₂²×C₇	semidirect
141. #469	C₂²×C₇	semidirect
142. #442	C₂²×C₇	semidirect
145. C₄×M₁₆	C₂²×C₇	semidirect
179. C₇×C₈	C₂×#41	semidirect
266. C₂×C₈	C₇×#41	semidirect
281. C₂×C₇	#3184	direct (N × H)
314. C₈	C₂×C₇×#41	semidirect
315. C₇	C₂×#3184	direct (N × H)
334. C₂	C₇×#3184	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×#532	2
C₇×#3184	2
C₂×C₇×#498	2
C₂×C₇×#493	2
C₂×C₇×#486	2
C₂×C₇×#469	2
C₂×C₇×#442	2
C₂×C₄×C₇×M₁₆	2
C₂×#3184	7

Subgroups — C2×C7×#3184 #13237907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#3184 #13237907