Subgroups — C₇×Q₈×C₂²⋊C₈ #13199724

Order 1792.

1356 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₂×C₇×C₈×Q₈	C₂	1	2	yes	maximal, complemented
3	C₇×#1984	C₂	1	6	yes	maximal, complemented
4	C₇×#1797	C₂	1	3	yes	maximal
5	C₄×C₇×C₂²⋊C₈	C₂	1	3	yes	maximal
6	C₂³×C₇×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×Q₈	—	2	8	no	—
8	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇×Q₈	C₄	1	3	yes	—
10	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	—
11	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₇×C₄⋊C₄	C₄	1	3	yes	—
13	C₂²×C₄²×C₇	C₂²	1	3	yes	—
14	C₇×C₈×Q₈	—	2	8	no	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	6	yes	—
16	C₂×C₇×C₂²⋊C₈	C₂²	1	3	yes	—
17	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	6	yes	—
19	C₇×#336	C₂²	1	6	yes	—
20	C₇×#336	C₂²	1	6	yes	—
21	Q₈×C₂²⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×Q₈	—	2	4	no	—
23	C₂²×C₇×Q₈	—	2	4	no	—
24	C₂²×C₇×Q₈	C₈	1	8	yes	complemented
25	C₂²×C₇×Q₈	C₂×C₄	1	3	yes	—
26	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	6	yes	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₄×C₇×Q₈	—	2	4	no	—
33	C₄×C₇×Q₈	—	2	12	no	—
34	C₄×C₇×Q₈	—	2	12	no	—
35	C₄×C₇×Q₈	D₄	1	8	yes	—
36	C₂×C₇×C₄⋊C₄	—	2	12	no	—
37	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₂×C₄²×C₇	—	2	12	no	—
41	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
43	C₂×C₄²×C₇	C₂³	1	3	yes	—
44	C₂×C₄²×C₇	C₂³	1	3	yes	—
45	C₇×C₄⋊C₈	—	2	12	no	—
46	C₇×C₂²⋊C₈	—	2	6	no	—
47	C₇×C₂²⋊C₈	Q₈	1	4	yes	direct product factor
48	C₄×C₇×C₈	—	2	12	no	—
49	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
50	C₂×C₈×Q₈	C₂×C₇	1	2	yes	complemented
51	#1984	C₂×C₇	1	6	yes	complemented
52	#1797	C₂×C₇	1	3	yes	—
53	C₄×C₂²⋊C₈	C₂×C₇	1	3	yes	—
54	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×Q₈	—	2	32	no	—
56	C₂×C₇×Q₈	—	2	6	no	—
57	C₂×C₇×Q₈	—	2	6	no	—
58	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
59	C₂×C₇×Q₈	C₂×C₈	1	12	yes	—
60	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
61	C₂×C₇×Q₈	M₁₆	1	4	yes	—
62	C₂²×C₄×C₇	—	2	6	no	—
63	C₂²×C₄×C₇	—	2	6	no	—
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	—	2	2	no	—
66	C₂²×C₄×C₇	—	2	6	no	—
67	C₂²×C₄×C₇	—	2	6	no	—
68	C₂²×C₄×C₇	—	4	3	no	—
69	C₂²×C₄×C₇	C₂×C₈	1	12	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇	#41	1	2	yes	—
77	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	6	no	—
81	C₂×C₇×C₈	—	2	6	no	—
82	C₂×C₇×C₈	—	2	6	no	—
83	C₂×C₇×C₈	C₂×Q₈	1	4	yes	complemented
84	C₇×C₄⋊C₄	—	2	12	no	—
85	C₇×C₄⋊C₄	—	2	6	no	—
86	C₇×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
87	C₄²×C₇	—	2	6	no	—
88	C₄²×C₇	—	2	6	no	—
89	C₄²×C₇	—	2	6	no	—
90	C₄²×C₇	C₂×D₄	1	12	yes	—
91	C₂³×Q₈	C₄×C₇	1	1	yes	characteristic
92	C₂×C₄×Q₈	—	2	8	no	—
93	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
94	C₂×C₄×Q₈	C₄×C₇	1	3	yes	—
95	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	—
96	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
97	C₂²×C₄⋊C₄	C₄×C₇	1	3	yes	—
98	C₂²×C₄²	C₂²×C₇	1	3	yes	—
99	C₈×Q₈	—	2	8	no	—
100	C₂×C₄⋊C₈	C₂²×C₇	1	6	yes	—
101	C₂×C₂²⋊C₈	C₂²×C₇	1	3	yes	—
102	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₇	1	6	yes	—
104	#336	C₂²×C₇	1	6	yes	—
105	#336	C₂²×C₇	1	6	yes	—
106	C₂³×C₇	—	2	1	no	—
107	C₂³×C₇	—	2	1	no	—
108	C₂³×C₇	—	2	1	no	—
109	C₂³×C₇	—	2	1	no	—
110	C₂³×C₇	C₄×Q₈	1	2	yes	—
111	C₂³×C₇	C₂²×C₈	1	2	yes	—
112	C₂³×C₇	#131	1	2	yes	—
113	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
114	C₇×Q₈	—	2	24	no	—
115	C₇×Q₈	C₂²⋊C₈	1	16	yes	direct product factor
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	3	no	—
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	—	2	3	no	—
120	C₂×C₄×C₇	—	2	3	no	—
121	C₂×C₄×C₇	—	2	3	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	24	no	—
125	C₂×C₄×C₇	—	2	3	no	—
126	C₂×C₄×C₇	—	2	3	no	—
127	C₂×C₄×C₇	—	2	3	no	—
128	C₂×C₄×C₇	—	4	12	no	—
129	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
130	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
131	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
133	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
134	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
135	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
139	C₇×C₈	—	2	4	no	—
140	C₇×C₈	—	4	6	no	—
141	C₂²×Q₈	—	2	4	no	—
142	C₂²×Q₈	—	2	4	no	—
143	C₂²×Q₈	C₇×C₈	1	8	yes	complemented
144	C₂²×Q₈	C₂×C₄×C₇	1	3	yes	—
145	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
146	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
147	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
148	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₇	1	6	yes	—
150	C₂²×C₈	C₂³×C₇	1	2	yes	—
151	C₄×Q₈	—	2	4	no	—
152	C₄×Q₈	—	2	12	no	—
153	C₄×Q₈	—	2	12	no	—
154	C₄×Q₈	C₇×D₄	1	8	yes	—
155	C₂×C₄⋊C₄	—	2	12	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
157	C₂×C₄⋊C₄	C₂×C₄×C₇	1	6	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
159	C₂×C₄²	—	2	12	no	—
160	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
161	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
162	C₂×C₄²	C₂³×C₇	1	3	yes	—
163	C₂×C₄²	C₂³×C₇	1	3	yes	—
164	C₄⋊C₈	—	2	12	no	—
165	C₂²⋊C₈	—	2	6	no	—
166	C₂²⋊C₈	C₇×Q₈	1	4	yes	direct product factor
167	C₄×C₈	—	2	12	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	C₈×Q₈	1	4	yes	complemented
175	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
176	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₇	#519	1	1	yes	characteristic
178	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
179	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
180	C₂²×C₇	C₂×#131	1	1	yes	characteristic
181	C₂²×C₇	#568	1	1	yes	characteristic
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	6	no	—
184	C₄×C₇	—	2	6	no	—
185	C₄×C₇	—	4	3	no	—
186	C₄×C₇	C₂×C₂²⋊C₈	1	12	yes	—
187	C₄×C₇	#548	1	2	yes	—
188	C₄×C₇	D₄×Q₈	1	2	yes	—
189	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
190	C₂×Q₈	—	2	32	no	—
191	C₂×Q₈	—	2	6	no	—
192	C₂×Q₈	—	2	6	no	—
193	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
194	C₂×Q₈	C₂×C₇×C₈	1	12	yes	—
195	C₂×Q₈	C₂×C₇×C₈	1	4	yes	—
196	C₂×Q₈	C₇×M₁₆	1	4	yes	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	—	2	6	no	—
203	C₂²×C₄	—	4	3	no	—
204	C₂²×C₄	C₂×C₇×C₈	1	12	yes	—
205	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
206	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
207	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
209	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
210	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
211	C₂²×C₄	C₇×#41	1	2	yes	—
212	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	6	no	—
216	C₂×C₈	—	2	6	no	—
217	C₂×C₈	—	2	6	no	—
218	C₂×C₈	C₂×C₇×Q₈	1	4	yes	complemented
219	C₄⋊C₄	—	2	12	no	—
220	C₄⋊C₄	—	2	6	no	—
221	C₄⋊C₄	C₇×C₂²⋊C₄	1	12	yes	—
222	C₄²	—	2	6	no	—
223	C₄²	—	2	6	no	—
224	C₄²	—	2	6	no	—
225	C₄²	C₂×C₇×D₄	1	12	yes	—
226	C₂×C₇	—	2	4	no	—
227	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
229	C₂×C₇	#2880	1	1	yes	characteristic
230	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
231	C₂×C₇	Q₈×M₁₆	1	1	yes	characteristic
232	C₂×C₇	#2982	1	1	yes	characteristic
233	C₂×C₇	#2984	1	1	yes	characteristic
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	C₄×C₇×Q₈	1	2	yes	—
239	C₂³	C₂²×C₇×C₈	1	2	yes	—
240	C₂³	C₇×#131	1	2	yes	—
241	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
242	Q₈	—	2	24	no	—
243	Q₈	C₇×C₂²⋊C₈	1	16	yes	direct product factor
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	3	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	3	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	4	12	no	—
257	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
258	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
259	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
260	C₂×C₄	C₂²×C₇×C₈	1	6	yes	—
261	C₂×C₄	C₂²×C₇×C₈	1	6	yes	—
262	C₂×C₄	C₂×C₇×M₁₆	1	6	yes	—
263	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
264	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
266	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
267	C₈	—	2	4	no	—
268	C₈	—	4	6	no	—
269	C₇	Q₈×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	C₇×C₈×Q₈	1	4	yes	complemented
277	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
278	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
279	C₂²	C₇×#519	1	1	yes	characteristic
280	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
281	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
282	C₂²	C₂×C₇×#131	1	1	yes	characteristic
283	C₂²	C₇×#568	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	6	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₇×C₂²⋊C₈	1	12	yes	—
289	C₄	C₇×#548	1	2	yes	—
290	C₄	C₇×D₄×Q₈	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
293	C₂	C₂²×C₇×C₂²⋊C₈	1	1	yes	characteristic
294	C₂	C₇×#2880	1	1	yes	characteristic
295	C₂	C₂×C₇×C₈×Q₈	1	1	yes	characteristic
296	C₂	C₇×Q₈×M₁₆	1	1	yes	characteristic
297	C₂	C₇×#2982	1	1	yes	characteristic
298	C₂	C₇×#2984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×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
2. C₂×C₇×C₈×Q₈	C₂	semidirect
3. C₇×#1984	C₂	semidirect
21. Q₈×C₂²⋊C₈	C₇	direct (N × H)
24. C₂²×C₇×Q₈	C₈	semidirect
47. C₇×C₂²⋊C₈	Q₈	direct (N × H)
50. C₂×C₈×Q₈	C₂×C₇	semidirect
51. #1984	C₂×C₇	semidirect
83. C₂×C₇×C₈	C₂×Q₈	semidirect
115. C₇×Q₈	C₂²⋊C₈	direct (N × H)
143. C₂²×Q₈	C₇×C₈	semidirect
166. C₂²⋊C₈	C₇×Q₈	direct (N × H)
174. C₂²×C₇	C₈×Q₈	semidirect
218. C₂×C₈	C₂×C₇×Q₈	semidirect
243. Q₈	C₇×C₂²⋊C₈	direct (N × H)
269. C₇	Q₈×C₂²⋊C₈	direct (N × H)
276. C₂²	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
C₂²×C₄×C₇×Q₈	2
C₂×C₇×C₈×Q₈	2
C₇×#1984	2
C₇×#1797	2
C₄×C₇×C₂²⋊C₈	2
Q₈×C₂²⋊C₈	7

Subgroups — C7×Q8×C22⋊C8 #13199724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×C₂²⋊C₈ #13199724