Subgroups — C₂×#22558962 #30824811

Order 17152.

834 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#22558962	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863148	C₂	1	2	yes	maximal, complemented
5	C₂×#16863146	C₂	1	2	yes	maximal, complemented
6	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₄	1	1	yes	characteristic
8	C₂²×#11761612	C₄	1	1	yes	characteristic
9	C₂³×C₆₇⋊C₈	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
14	C₄×#11761612	C₄	1	4	yes	—
15	C₄×#11761612	C₂²	1	4	yes	—
16	C₂×#14523954	C₄	1	2	yes	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	#16863148	C₂²	1	8	yes	complemented
19	#16863146	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#431×C₆₇	—	2	2	no	—
22	#431×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	—	2	1	no	—
27	C₂×#11761612	—	2	4	no	—
28	C₂×#11761612	C₈	1	8	yes	complemented
29	C₂×#11761612	C₂×C₄	1	4	yes	—
30	C₂×#11761612	C₂×C₄	1	4	yes	—
31	C₂²×C₆₇⋊C₈	—	2	4	no	—
32	C₂²×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×C₆₇⋊C₈	—	4	4	no	—
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	#14523954	C₂×C₄	1	8	yes	—
40	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	C₄⋊C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
59	C₂²⋊C₈×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	—	2	4	no	—
61	#11761612	—	2	8	no	—
62	#11761612	C₂×C₈	1	16	yes	complemented
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	4	8	no	—
66	C₂×C₆₇⋊C₈	—	4	4	no	—
67	C₂×C₆₇⋊C₈	C₂×C₈	1	8	yes	complemented
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
69	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
70	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
81	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	4	2	no	—
91	C₂×C₈×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
94	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×D₄	1	2	yes	—
99	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
100	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	C₄⋊C₈	1	4	yes	—
105	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
116	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
125	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₆₇	#137	1	2	yes	—
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₆₇	—	4	2	no	—
133	C₈×C₆₇	—	4	2	no	—
134	C₆₇⋊C₈	—	2	4	no	—
135	C₆₇⋊C₈	—	4	4	no	—
136	C₆₇⋊C₈	C₂²×C₈	1	8	yes	complemented
137	C₂²×C₆₇	—	2	8	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
141	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
142	C₂²×C₆₇	#428	1	2	yes	—
143	C₂²×C₆₇	#430	1	2	yes	—
144	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
150	C₄×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
151	C₄×C₆₇	#428	1	2	yes	—
152	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
156	C₂×#1565	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	2	4	no	—
158	C₂×C₆₇	#1565	1	4	yes	complemented
159	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
160	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
162	C₂×#431	D₆₇	1	1	yes	characteristic
163	C₂²×C₄⋊C₈	—	67	1	no	—
164	C₂×C₄×M₁₆	—	67	1	no	—
165	#1565	—	67	8	no	—
166	C₂×#335	—	67	2	no	—
167	C₂×#334	—	67	2	no	—
168	C₆₇	C₂×#1565	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	67	1	no	—
170	C₂²×M₁₆	—	67	1	no	—
171	C₂³×C₈	—	134	1	no	—
172	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
173	#431	—	2	2	no	—
174	#431	C₂×D₆₇	1	4	yes	—
175	C₂×C₄⋊C₈	—	67	2	no	—
176	C₂×C₄⋊C₈	—	67	4	no	—
177	C₂×C₄⋊C₈	—	134	2	no	—
178	C₂×C₄⋊C₈	—	134	1	no	—
179	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
181	C₂×C₂²⋊C₈	—	2	1	no	—
182	C₄×M₁₆	—	67	4	no	—
183	C₄×M₁₆	—	67	4	no	—
184	C₂×#97	—	67	2	no	—
185	C₂×C₄×C₈	—	2	1	no	—
186	C₂×C₄×C₈	—	67	2	no	—
187	#335	—	67	8	no	—
188	#334	—	67	8	no	—
189	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
190	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂×M₁₆	—	67	8	no	—
193	C₂×M₁₆	—	67	4	no	—
194	C₂×M₁₆	—	67	4	no	—
195	C₂×M₁₆	—	134	4	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	268	4	no	—
199	C₂²×C₈	—	67	2	no	—
200	C₂²×C₈	—	67	2	no	—
201	C₂²×C₈	—	134	4	no	—
202	C₂²×C₈	—	134	2	no	—
203	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
204	C₂×C₄²	Dic₁₃₄	1	2	yes	—
205	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₆₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
210	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	67	4	no	—
214	C₄⋊C₈	—	134	4	no	—
215	C₄⋊C₈	—	134	2	no	—
216	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
217	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
218	C₂²⋊C₈	—	2	4	no	—
219	#97	—	67	8	no	—
220	C₄×C₈	—	2	4	no	—
221	C₄×C₈	—	67	4	no	—
222	C₄×C₈	—	67	4	no	—
223	C₂⁴	Dic₆₇⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
229	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
231	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
232	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	C₈×D₆₇	1	4	yes	—
234	C₂²×C₄	#11761611	1	4	yes	—
235	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
241	M₁₆	—	67	16	no	—
242	M₁₆	—	134	8	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	268	8	no	—
248	C₂×C₈	—	268	4	no	—
249	C₂×C₈	—	67	8	no	—
250	C₂×C₈	—	67	4	no	—
251	C₂×C₈	—	67	4	no	—
252	C₂×C₈	—	134	4	no	—
253	C₂×C₈	—	134	4	no	—
254	C₄²	Dic₆₇⋊C₄	1	2	yes	—
255	C₄²	Dic₆₇⋊C₄	1	2	yes	—
256	C₄²	Dic₆₇⋊C₄	1	2	yes	—
257	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₆₇	1	2	yes	—
259	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
260	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
261	C₄²	C₂×Dic₁₃₄	1	2	yes	—
262	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
263	C₄²	C₂³×D₆₇	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	Dic₆₇⋊C₈	1	4	yes	—
267	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
268	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	Dic₆₇⋊C₈	1	8	yes	—
278	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
279	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
281	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
286	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
287	C₂×C₄	C₂×#11761611	1	2	yes	—
288	C₂×C₄	C₂×#11761611	1	4	yes	—
289	C₂×C₄	#14524068	1	2	yes	—
290	C₂×C₄	#14524073	1	2	yes	—
291	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	—	268	4	no	—
297	C₈	—	67	8	no	—
298	C₈	—	134	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	8	no	—
302	C₂²	C₂×Dic₆₇⋊C₈	1	4	yes	—
303	C₂²	C₂×Dic₆₇⋊C₈	1	2	yes	—
304	C₂²	#16863439	1	2	yes	—
305	C₂²	#16863444	1	2	yes	—
306	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	C₂×Dic₆₇⋊C₈	1	2	yes	—
312	C₄	C₂×Dic₆₇⋊C₈	1	4	yes	—
313	C₄	#16863439	1	2	yes	—
314	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
315	C₄	C₂²×#11761611	1	1	yes	characteristic
316	C₄	C₂×#14524068	1	1	yes	characteristic
317	C₄	C₂×#14524073	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	#22558962	1	4	yes	direct product factor
320	C₂	C₂²×Dic₆₇⋊C₈	1	1	yes	characteristic
321	C₂	C₂×#16863439	1	1	yes	characteristic
322	C₂	C₂×#16863444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	C₂×#1565
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22558962	C₂	direct (N × H)
4. C₂×#16863148	C₂	semidirect
5. C₂×#16863146	C₂	semidirect
18. #16863148	C₂²	semidirect
19. #16863146	C₂²	semidirect
28. C₂×#11761612	C₈	semidirect
62. #11761612	C₂×C₈	semidirect
67. C₂×C₆₇⋊C₈	C₂×C₈	semidirect
136. C₆₇⋊C₈	C₂²×C₈	semidirect
158. C₂×C₆₇	#1565	semidirect
168. C₆₇	C₂×#1565	semidirect
319. C₂	#22558962	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⋊(C₆₇⋊C₈)	2
C₂×C₄×#11761612	2
#22558962	2
C₂×#16863148	2
C₂×#16863146	2
C₂×#431×C₆₇	2
C₂×#1565	67

Subgroups — C2×#22558962 #30824811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558962 #30824811