Subgroups — C₂×#22560238 #30855393

Order 17152.

1790 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523977	C₂	1	2	yes	maximal, complemented
2	C₂×#16863256	C₂	1	2	yes	maximal
3	#22560238	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863159	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
7	C₂²×C₂²⋊Dic₆₇	—	2	2	no	—
8	C₂×#14523977	—	2	8	no	—
9	C₂×#14523977	C₂²	1	8	yes	complemented
10	#16863256	—	2	4	no	—
11	#16863256	C₂²	1	8	yes	—
12	C₂×C₆₇⋊#95	—	2	2	no	—
13	#16863159	C₂²	1	8	yes	complemented
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	—	2	8	no	—
16	C₂³×C₈×C₆₇	C₄	1	8	yes	complemented
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×Dic₆₇	—	2	4	no	—
21	C₂×C₄⋊Dic₆₇	—	2	8	no	—
22	C₂×C₄⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	8	no	—
24	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
25	C₂×C₂²⋊Dic₆₇	—	2	8	no	—
26	C₂×C₂²⋊Dic₆₇	—	4	8	no	—
27	#14523977	—	2	16	no	—
28	C₆₇⋊#95	—	2	8	no	—
29	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	—	2	4	no	—
31	C₂³×C₄×C₆₇	—	2	4	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	D₄	1	2	yes	—
35	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	32	no	—
40	C₂²×C₈×C₆₇	—	2	16	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₆₇	D₄	1	8	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×Dic₆₇	—	2	8	no	—
48	C₂²×Dic₆₇	—	4	8	no	—
49	C₂²×Dic₆₇	—	4	8	no	—
50	C₂²×Dic₆₇	—	4	8	no	—
51	C₄⋊Dic₆₇	—	2	8	no	—
52	C₄⋊Dic₆₇	—	2	8	no	—
53	C₄⋊Dic₆₇	—	4	8	no	—
54	C₂²⋊Dic₆₇	—	4	16	no	—
55	C₂⁴×C₆₇	—	2	4	no	—
56	C₂⁴×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
58	C₂⁴×C₆₇	D₈	1	4	yes	—
59	C₂⁴×C₆₇	Q₁₆	1	4	yes	—
60	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
61	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	8	no	—
64	C₂²×C₄×C₆₇	—	2	8	no	—
65	C₂²×C₄×C₆₇	—	2	8	no	—
66	C₂²×C₄×C₆₇	—	2	16	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₆₇	#41	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	8	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	32	no	—
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	16	yes	complemented
84	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
87	C₂³×C₆₇	—	2	4	no	—
88	C₂³×C₆₇	—	2	4	no	—
89	C₂³×C₆₇	—	2	8	no	—
90	C₂³×C₆₇	—	2	8	no	—
91	C₂³×C₆₇	—	2	16	no	—
92	C₂³×C₆₇	—	2	16	no	—
93	C₂³×C₆₇	#107	1	16	yes	complemented
94	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₆₇	#121	1	4	yes	—
97	C₂³×C₆₇	C₂×D₈	1	4	yes	—
98	C₂³×C₆₇	C₂×D₈	1	2	yes	—
99	C₂³×C₆₇	C₂×Q₁₆	1	4	yes	—
100	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
101	C₂³×C₆₇	#135	1	4	yes	—
102	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	8	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	8	no	—
108	C₂×C₄×C₆₇	—	2	16	no	—
109	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₇	#117	1	4	yes	—
113	C₂×C₄×C₆₇	#121	1	4	yes	—
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
119	C₂×Dic₆₇	—	4	8	no	—
120	C₂×Dic₆₇	—	4	8	no	—
121	C₂×Dic₆₇	—	4	8	no	—
122	C₂×Dic₆₇	—	8	16	no	—
123	C₈×C₆₇	—	2	4	no	—
124	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	complemented
125	C₂²×C₆₇	—	2	32	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	—	2	8	no	—
129	C₂²×C₆₇	—	2	8	no	—
130	C₂²×C₆₇	#380	1	4	yes	—
131	C₂²×C₆₇	C₂×#107	1	4	yes	—
132	C₂²×C₆₇	C₂×#107	1	8	yes	complemented
133	C₂²×C₆₇	#427	1	4	yes	—
134	C₂²×C₆₇	#466	1	8	yes	—
135	C₂²×C₆₇	#467	1	8	yes	—
136	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#121	1	2	yes	—
138	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
139	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×#135	1	2	yes	—
141	C₄×C₆₇	—	2	4	no	—
142	C₄×C₆₇	#380	1	4	yes	—
143	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
145	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
146	Dic₆₇	—	8	8	no	—
147	C₂×#1808	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	—	2	8	no	—
149	C₂×C₆₇	#1808	1	8	yes	complemented
150	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
151	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
153	C₂×C₆₇	C₂×#466	1	2	yes	—
154	C₂×C₆₇	C₂×#467	1	2	yes	—
155	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
156	C₂²×#107	—	67	2	no	—
157	C₂×#380	—	67	2	no	—
158	#1808	—	67	8	no	—
159	C₂×#340	—	67	2	no	—
160	C₆₇	C₂×#1808	1	1	yes	characteristic, complemented, Sylow
161	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
162	C₂³×C₈	—	2	8	no	—
163	C₂³×C₈	Dic₆₇	1	8	yes	complemented
164	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
165	C₂³×C₈	C₂×D₆₇	1	4	yes	—
166	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
167	C₂²×C₄⋊C₄	—	67	4	no	—
168	C₂²×C₂²⋊C₄	—	134	2	no	—
169	C₂×#107	—	67	8	no	—
170	C₂×#107	—	134	8	no	—
171	#380	—	67	8	no	—
172	#380	—	134	4	no	—
173	C₂×#95	—	134	2	no	—
174	#340	—	67	8	no	—
175	C₂⁵	D₂₆₈	1	1	yes	characteristic
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	—	134	4	no	—
179	C₂³×C₄	Dic₁₃₄	1	4	yes	—
180	C₂³×C₄	D₂₆₈	1	2	yes	—
181	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
182	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
183	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
184	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	2	32	no	—
187	C₂²×C₈	—	2	16	no	—
188	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
189	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
190	C₂²×C₈	C₂×Dic₆₇	1	16	yes	complemented
191	C₂²×C₈	C₂²⋊D₆₇	1	8	yes	—
192	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
193	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
194	C₂×C₄⋊C₄	—	67	8	no	—
195	C₂×C₄⋊C₄	—	134	4	no	—
196	C₂×C₄⋊C₄	—	134	8	no	—
197	C₂×C₄⋊C₄	—	134	8	no	—
198	C₂×C₂²⋊C₄	—	268	8	no	—
199	C₂×C₂²⋊C₄	—	134	8	no	—
200	#107	—	134	16	no	—
201	#95	—	134	8	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	2	4	no	—
204	C₂⁴	C₄⋊Dic₆₇	1	4	yes	—
205	C₂⁴	D₅₃₆	1	4	yes	—
206	C₂⁴	Dic₂₆₈	1	4	yes	—
207	C₂⁴	C₂×D₂₆₈	1	2	yes	—
208	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	268	8	no	—
217	C₂²×C₄	—	268	8	no	—
218	C₂²×C₄	—	268	8	no	—
219	C₂²×C₄	—	134	8	no	—
220	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
221	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
222	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
225	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
226	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
228	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
229	C₂²×C₄	#11761628	1	4	yes	—
230	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	32	no	—
234	C₂×C₈	C₂²⋊Dic₆₇	1	16	yes	complemented
235	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
236	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	8	yes	—
238	C₄⋊C₄	—	268	8	no	—
239	C₄⋊C₄	—	134	8	no	—
240	C₄⋊C₄	—	134	8	no	—
241	C₂²⋊C₄	—	268	16	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	16	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	16	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	8	no	—
248	C₂³	#14523977	1	16	yes	complemented
249	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
250	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
251	C₂³	#14524017	1	4	yes	—
252	C₂³	C₂×D₅₃₆	1	4	yes	—
253	C₂³	C₂×D₅₃₆	1	2	yes	—
254	C₂³	C₂×Dic₂₆₈	1	4	yes	—
255	C₂³	C₂×Dic₂₆₈	1	2	yes	—
256	C₂³	#14524065	1	4	yes	—
257	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	268	8	no	—
265	C₂×C₄	—	268	8	no	—
266	C₂×C₄	—	268	8	no	—
267	C₂×C₄	—	536	16	no	—
268	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	4	yes	—
269	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
270	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
271	C₂×C₄	#14524002	1	4	yes	—
272	C₂×C₄	#14524017	1	4	yes	—
273	C₂×C₄	#14524022	1	8	yes	—
274	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
276	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
277	C₂×C₄	C₂×#11761628	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	C₂×C₂²⋊Dic₆₇	1	8	yes	complemented
280	C₂²	—	2	4	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	32	no	—
285	C₂²	#16863256	1	4	yes	—
286	C₂²	C₂×#14523977	1	4	yes	—
287	C₂²	C₂×#14523977	1	8	yes	complemented
288	C₂²	#16863434	1	4	yes	—
289	C₂²	#16863632	1	8	yes	—
290	C₂²	#16863637	1	8	yes	—
291	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
292	C₂²	C₂×#14524017	1	2	yes	—
293	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
294	C₂²	C₂²×Dic₂₆₈	1	1	yes	characteristic
295	C₂²	C₂×#14524065	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	536	8	no	—
298	C₄	#16863256	1	4	yes	—
299	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
300	C₄	C₂×#14524002	1	1	yes	characteristic
301	C₄	C₂×#14524022	1	2	yes	—
302	C₂	—	2	8	no	—
303	C₂	#22560238	1	8	yes	direct product factor
304	C₂	C₂×#16863256	1	1	yes	characteristic
305	C₂	C₂²×#14523977	1	1	yes	characteristic
306	C₂	C₂×#16863434	1	1	yes	characteristic
307	C₂	C₂×#16863632	1	2	yes	—
308	C₂	C₂×#16863637	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂×D₄
prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂×#1808
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523977	C₂	semidirect
3. #22560238	C₂	direct (N × H)
4. C₂×#16863159	C₂	semidirect
9. C₂×#14523977	C₂²	semidirect
13. #16863159	C₂²	semidirect
16. C₂³×C₈×C₆₇	C₄	semidirect
43. C₂²×C₈×C₆₇	C₂×C₄	semidirect
83. C₂×C₈×C₆₇	C₂²⋊C₄	semidirect
93. C₂³×C₆₇	#107	semidirect
124. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
132. C₂²×C₆₇	C₂×#107	semidirect
149. C₂×C₆₇	#1808	semidirect
160. C₆₇	C₂×#1808	semidirect
163. C₂³×C₈	Dic₆₇	semidirect
190. C₂²×C₈	C₂×Dic₆₇	semidirect
234. C₂×C₈	C₂²⋊Dic₆₇	semidirect
248. C₂³	#14523977	semidirect
279. C₈	C₂×C₂²⋊Dic₆₇	semidirect
287. C₂²	C₂×#14523977	semidirect
303. C₂	#22560238	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×#14523977	2
C₂×#16863256	2
#22560238	2
C₂×#16863159	2
C₂⁴×C₈×C₆₇	2
C₂×#1808	67

Subgroups — C2×#22560238 #30855393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560238 #30855393