Subgroups — C₂×#22558976 #30824849

Order 17152.

758 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863453	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558976	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863149	C₂	1	2	yes	maximal, complemented
5	C₂×#16863147	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	—	2	1	no	—
8	#16863453	—	2	2	no	—
9	#16863453	C₂²	1	4	yes	—
10	#16863420	—	2	2	no	—
11	#16863420	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
17	C₂×#14523954	—	2	1	no	—
18	#16863149	C₂²	1	8	yes	complemented
19	#16863147	C₂²	1	8	yes	complemented
20	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
23	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
24	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
25	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#11761612	—	2	4	no	—
28	C₂×#11761612	—	4	4	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	C₂²×C₆₇⋊C₈	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
37	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
40	#14523954	—	2	4	no	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	8	no	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
56	#97×C₆₇	C₂×C₄	1	4	yes	—
57	#97×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₆₇	C₂³	1	4	yes	—
60	#11761612	—	4	8	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	4	no	—
66	C₂×C₆₇⋊C₈	—	4	2	no	—
67	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
84	M₁₆×C₆₇	—	2	16	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	M₁₆	1	8	yes	complemented
87	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
97	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	4	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	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₆₇	#137	1	2	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₆₇	—	2	4	no	—
128	C₈×C₆₇	C₂×M₁₆	1	8	yes	complemented
129	C₆₇⋊C₈	—	4	2	no	—
130	C₆₇⋊C₈	—	4	4	no	—
131	C₆₇⋊C₈	—	4	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	#423	1	2	yes	—
137	C₂²×C₆₇	#428	1	2	yes	—
138	C₂²×C₆₇	#430	1	2	yes	—
139	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	#423	1	2	yes	—
146	C₄×C₆₇	#428	1	2	yes	—
147	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
148	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
150	C₂×#1569	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	4	2	no	—
152	C₂×C₆₇	#1569	1	4	yes	complemented
153	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
155	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
156	C₂×#432	—	67	1	no	—
157	C₂×#423	—	67	1	no	—
158	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
159	#1569	—	67	8	no	—
160	C₂×#335	—	67	2	no	—
161	C₂×#334	—	67	2	no	—
162	C₆₇	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	134	1	no	—
164	C₂²×M₁₆	Dic₆₇	1	2	yes	—
165	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
166	#432	—	67	4	no	—
167	#432	—	134	2	no	—
168	#423	—	67	4	no	—
169	#423	—	134	2	no	—
170	C₂×C₄⋊C₈	—	67	1	no	—
171	C₂×C₄⋊C₈	—	67	2	no	—
172	C₂×C₄⋊C₈	—	67	1	no	—
173	C₂×C₄⋊C₈	—	134	1	no	—
174	C₂×C₂²⋊C₈	—	134	1	no	—
175	C₄×M₁₆	Dic₆₇	1	4	yes	—
176	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
177	C₂×#97	—	134	1	no	—
178	C₂×#97	Dic₆₇	1	1	yes	characteristic
179	C₂×#97	Dic₆₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
181	#335	—	67	8	no	—
182	#334	—	67	8	no	—
183	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
184	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂×M₁₆	—	2	8	no	—
187	C₂×M₁₆	—	268	4	no	—
188	C₂×M₁₆	—	134	4	no	—
189	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
190	C₂²×C₈	—	134	2	no	—
191	C₂²×C₈	—	134	1	no	—
192	C₂²×C₈	—	134	1	no	—
193	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
194	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
195	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
196	C₂×C₄²	Dic₁₃₄	1	2	yes	—
197	C₂×C₄²	D₂₆₈	1	2	yes	—
198	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
201	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	C₄⋊C₈	—	67	2	no	—
204	C₄⋊C₈	—	67	4	no	—
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	134	2	no	—
207	C₄⋊C₈	—	134	1	no	—
208	C₄⋊C₈	—	134	1	no	—
209	C₄⋊C₈	—	134	4	no	—
210	C₂²⋊C₈	—	134	4	no	—
211	#97	—	134	4	no	—
212	#97	C₂×Dic₆₇	1	4	yes	—
213	#97	C₂×Dic₆₇	1	4	yes	—
214	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
215	C₄×C₈	C₂²×D₆₇	1	4	yes	—
216	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
223	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
225	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
230	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
231	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
233	M₁₆	—	2	16	no	—
234	M₁₆	—	268	8	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	268	4	no	—
237	C₂×C₈	—	268	2	no	—
238	C₂×C₈	—	268	2	no	—
239	C₂×C₈	—	134	2	no	—
240	C₂×C₈	—	134	2	no	—
241	C₂×C₈	—	134	4	no	—
242	C₂×C₈	#11761612	1	8	yes	complemented
243	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
244	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
245	C₄²	C₄⋊Dic₆₇	1	2	yes	—
246	C₄²	C₄⋊Dic₆₇	1	2	yes	—
247	C₄²	C₄⋊Dic₆₇	1	2	yes	—
248	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
249	C₄²	C₂²×Dic₆₇	1	2	yes	—
250	C₄²	C₂×D₂₆₈	1	2	yes	—
251	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
252	C₄²	C₂×Dic₁₃₄	1	2	yes	—
253	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
254	C₄²	C₂³×D₆₇	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
259	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#11761612	1	4	yes	—
278	C₂×C₄	#14524071	1	2	yes	—
279	C₂×C₄	#14524076	1	2	yes	—
280	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
282	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	268	4	no	—
285	C₈	—	268	2	no	—
286	C₈	—	268	2	no	—
287	C₈	C₂×#11761612	1	8	yes	complemented
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#16863420	1	2	yes	—
293	C₂²	#16863438	1	2	yes	—
294	C₂²	#16863445	1	2	yes	—
295	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	#16863420	1	2	yes	—
302	C₄	#16863438	1	2	yes	—
303	C₄	C₂²×#11761612	1	2	yes	—
304	C₄	C₂×#14524071	1	1	yes	characteristic
305	C₄	C₂×#14524076	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	#22558976	1	4	yes	direct product factor
308	C₂	C₂×#16863420	1	1	yes	characteristic
309	C₂	C₂×#16863438	1	1	yes	characteristic
310	C₂	C₂×#16863445	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₂×#1569
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22558976	C₂	direct (N × H)
4. C₂×#16863149	C₂	semidirect
5. C₂×#16863147	C₂	semidirect
18. #16863149	C₂²	semidirect
19. #16863147	C₂²	semidirect
86. C₂×C₈×C₆₇	M₁₆	semidirect
128. C₈×C₆₇	C₂×M₁₆	semidirect
152. C₂×C₆₇	#1569	semidirect
162. C₆₇	C₂×#1569	semidirect
242. C₂×C₈	#11761612	semidirect
287. C₈	C₂×#11761612	semidirect
307. C₂	#22558976	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×#16863453	2
C₂×#16863420	2
#22558976	2
C₂×#16863149	2
C₂×#16863147	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#1569	67

Subgroups — C2×#22558976 #30824849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558976 #30824849