Subgroups — C₇×#23149 #13200365

Order 1792.

836 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1984	C₂	1	2	yes	maximal
4	C₇×#1939	C₂	1	2	yes	maximal
5	C₇×#1916	C₂	1	4	yes	maximal
6	C₇×C₈×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
11	C₇×C₈×Q₈	—	2	8	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
20	C₇×#391	C₄	1	2	yes	—
21	C₇×#389	C₄	1	2	yes	—
22	C₇×#389	C₄	1	2	yes	—
23	C₇×#384	C₄	1	1	yes	characteristic
24	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄³×C₇	C₂²	1	1	yes	characteristic
27	C₇×#336	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	4	yes	—
29	C₇×#336	C₂²	1	2	yes	—
30	C₇×#336	C₂²	1	2	yes	—
31	#23149	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	2	yes	—
37	C₇×C₄⋊Q₈	C₈	1	8	yes	complemented
38	C₄×C₇×Q₈	—	2	4	no	—
39	C₄×C₇×Q₈	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₇×C₄⋊C₈	—	2	8	no	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×C₄⋊C₈	—	2	2	no	—
57	C₇×C₄⋊C₈	D₄	1	4	yes	—
58	C₇×C₄⋊C₈	Q₈	1	4	yes	complemented
59	C₄×C₇×C₈	—	2	8	no	—
60	C₄×C₇×C₈	—	2	4	no	—
61	C₄×C₇×C₈	—	2	4	no	—
62	C₇×#95	C₂×C₄	1	4	yes	—
63	C₇×#95	C₂×C₄	1	4	yes	—
64	C₂×C₈×Q₈	C₂×C₇	1	2	yes	—
65	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
66	#1984	C₂×C₇	1	2	yes	—
67	#1939	C₂×C₇	1	2	yes	—
68	#1916	C₂×C₇	1	4	yes	—
69	C₈×C₄⋊C₄	C₂×C₇	1	2	yes	—
70	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
71	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
72	C₂×C₇×Q₈	—	2	4	no	—
73	C₂×C₇×Q₈	—	2	4	no	—
74	C₂×C₇×Q₈	C₂×C₈	1	8	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
95	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
96	C₇×C₄⋊C₄	—	2	8	no	—
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	4	2	no	—
100	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
101	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₄²×C₇	—	2	4	no	—
105	C₄²×C₇	—	2	4	no	—
106	C₄²×C₇	—	2	1	no	—
107	C₄²×C₇	—	2	1	no	—
108	C₄²×C₇	—	4	1	no	—
109	C₄²×C₇	C₂×C₈	1	4	yes	—
110	C₄²×C₇	C₂×D₄	1	2	yes	—
111	C₄²×C₇	C₂×Q₈	1	2	yes	—
112	C₄²×C₇	#41	1	2	yes	—
113	C₄²×C₇	#41	1	2	yes	—
114	C₂×C₄⋊Q₈	C₄×C₇	1	1	yes	characteristic
115	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
116	C₈×Q₈	—	2	8	no	—
117	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
118	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
123	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
125	#391	C₄×C₇	1	2	yes	—
126	#389	C₄×C₇	1	2	yes	—
127	#389	C₄×C₇	1	2	yes	—
128	#384	C₄×C₇	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
130	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
131	C₄³	C₂²×C₇	1	1	yes	characteristic
132	#336	C₂²×C₇	1	2	yes	—
133	#336	C₂²×C₇	1	2	yes	—
134	#336	C₂²×C₇	1	4	yes	—
135	#336	C₂²×C₇	1	4	yes	—
136	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
137	C₇×Q₈	—	2	16	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	1	no	—
141	C₂×C₄×C₇	—	2	1	no	—
142	C₂×C₄×C₇	—	2	4	no	—
143	C₂×C₄×C₇	—	2	1	no	—
144	C₂×C₄×C₇	—	2	1	no	—
145	C₂×C₄×C₇	—	2	1	no	—
146	C₂×C₄×C₇	—	2	4	no	—
147	C₂×C₄×C₇	—	2	1	no	—
148	C₂×C₄×C₇	—	2	1	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	4	no	—
151	C₂×C₄×C₇	—	2	4	no	—
152	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
155	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
158	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
159	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₇	#131	1	2	yes	—
161	C₂×C₄×C₇	#131	1	2	yes	—
162	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
168	C₇×C₈	—	2	4	no	—
169	C₇×C₈	—	2	4	no	—
170	C₇×C₈	—	4	2	no	—
171	C₇×C₈	—	4	2	no	—
172	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₇	1	2	yes	—
174	C₂²×C₈	C₂³×C₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₇	1	2	yes	—
176	C₂²×C₈	C₂³×C₇	1	2	yes	—
177	C₄⋊Q₈	C₇×C₈	1	8	yes	complemented
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
186	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₇	1	4	yes	—
191	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
194	C₄⋊C₈	—	2	8	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₇×D₄	1	4	yes	—
198	C₄⋊C₈	C₇×Q₈	1	4	yes	complemented
199	C₄×C₈	—	2	8	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	#95	C₂×C₄×C₇	1	4	yes	—
203	#95	C₂×C₄×C₇	1	4	yes	—
204	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
206	C₂²×C₇	#519	1	1	yes	characteristic
207	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
208	C₂²×C₇	C₂×#131	1	1	yes	characteristic
209	C₂²×C₇	C₂×#131	1	1	yes	characteristic
210	C₂²×C₇	#568	1	1	yes	characteristic
211	C₄×C₇	—	2	8	no	—
212	C₄×C₇	—	2	2	no	—
213	C₄×C₇	—	2	2	no	—
214	C₄×C₇	—	2	2	no	—
215	C₄×C₇	—	4	4	no	—
216	C₄×C₇	—	4	1	no	—
217	C₄×C₇	C₈×D₄	1	4	yes	—
218	C₄×C₇	C₈×Q₈	1	4	yes	complemented
219	C₄×C₇	#547	1	1	yes	characteristic
220	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
221	C₄×C₇	#556	1	1	yes	characteristic
222	C₄×C₇	#557	1	1	yes	characteristic
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	C₂×C₇×C₈	1	8	yes	—
226	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
228	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
246	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	—
252	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	4	1	no	—
260	C₄²	C₂×C₇×C₈	1	4	yes	—
261	C₄²	C₂×C₇×D₄	1	2	yes	—
262	C₄²	C₂×C₇×Q₈	1	2	yes	—
263	C₄²	C₇×#41	1	2	yes	—
264	C₄²	C₇×#41	1	2	yes	—
265	C₂×C₇	#2365	1	1	yes	characteristic
266	C₂×C₇	C₂×C₈×D₄	1	1	yes	characteristic
267	C₂×C₇	#2935	1	1	yes	characteristic
268	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
269	C₂×C₇	#2962	1	1	yes	characteristic
270	C₂×C₇	#2982	1	1	yes	characteristic
271	C₂×C₇	#2986	1	1	yes	characteristic
272	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
273	Q₈	—	2	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
289	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
290	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
291	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
292	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₇×C₈	1	8	yes	—
294	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
295	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
296	C₂×C₄	C₇×#131	1	2	yes	—
297	C₂×C₄	C₇×#131	1	2	yes	—
298	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
301	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
302	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
303	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	2	4	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	2	no	—
308	C₇	#23149	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
310	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
311	C₂²	C₇×#519	1	1	yes	characteristic
312	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
313	C₂²	C₂×C₇×#131	1	1	yes	characteristic
314	C₂²	C₂×C₇×#131	1	1	yes	characteristic
315	C₂²	C₇×#568	1	1	yes	characteristic
316	C₄	—	2	8	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	1	no	—
322	C₄	C₇×C₈×D₄	1	4	yes	—
323	C₄	C₇×C₈×Q₈	1	4	yes	complemented
324	C₄	C₇×#547	1	1	yes	characteristic
325	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
326	C₄	C₇×#556	1	1	yes	characteristic
327	C₄	C₇×#557	1	1	yes	characteristic
328	C₂	C₇×#2365	1	1	yes	characteristic
329	C₂	C₂×C₇×C₈×D₄	1	1	yes	characteristic
330	C₂	C₇×#2935	1	1	yes	characteristic
331	C₂	C₂×C₇×C₈×Q₈	1	1	yes	characteristic
332	C₂	C₇×#2962	1	1	yes	characteristic
333	C₂	C₇×#2982	1	1	yes	characteristic
334	C₂	C₇×#2986	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
31. #23149	C₇	direct (N × H)
37. C₇×C₄⋊Q₈	C₈	semidirect
58. C₇×C₄⋊C₈	Q₈	semidirect
177. C₄⋊Q₈	C₇×C₈	semidirect
198. C₄⋊C₈	C₇×Q₈	semidirect
218. C₄×C₇	C₈×Q₈	semidirect
308. C₇	#23149	direct (N × H)
323. C₄	C₇×C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
C₄×C₇×C₄⋊Q₈	2
C₇×#1984	2
C₇×#1939	2
C₇×#1916	2
C₇×C₈×C₄⋊C₄	2
C₄×C₇×C₄⋊C₈	2
#23149	7

Subgroups — C7×#23149 #13200365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23149 #13200365