Subgroups — C₂×#22562616 #31014128

Order 17152.

3854 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×#14524002	C₂	1	2	yes	maximal
2	#22562616	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863266	C₂	1	2	yes	maximal, complemented
4	C₂×#16863262	C₂	1	2	yes	maximal, complemented
5	C₂×#16863256	C₂	1	4	yes	maximal
6	C₂×C₄×C₄⋊Dic₆₇	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524002	—	2	16	no	—
9	C₂×#14524002	C₂²	1	16	yes	—
10	C₂²×C₄⋊Dic₆₇	C₂²	1	8	yes	—
11	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
12	C₂²×C₂²⋊Dic₆₇	C₂²	1	4	yes	—
13	C₂²×C₄×Dic₆₇	C₂²	1	4	yes	—
14	#16863266	C₂²	1	16	yes	complemented
15	#16863262	C₂²	1	16	yes	complemented
16	#16863256	C₂²	1	32	yes	—
17	C₄×C₄⋊Dic₆₇	C₂²	1	32	yes	complemented
18	C₂×C₆₇⋊#95	C₂²	1	8	yes	—
19	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
20	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	C₄	1	16	yes	complemented
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	8	yes	—
26	C₂³×Dic₆₇	C₂³	1	8	yes	—
27	#14524002	—	2	64	no	—
28	C₂×C₄⋊Dic₆₇	—	2	16	no	—
29	C₂×C₄⋊Dic₆₇	—	2	32	no	—
30	C₂×C₄⋊Dic₆₇	C₂³	1	32	yes	—
31	C₂×C₄⋊Dic₆₇	C₂³	1	16	yes	—
32	C₂×C₂²⋊Dic₆₇	—	2	16	no	—
33	C₂×C₂²⋊Dic₆₇	C₂³	1	16	yes	—
34	C₂×C₄×Dic₆₇	—	2	16	no	—
35	C₂×C₄×Dic₆₇	C₂³	1	16	yes	—
36	C₆₇⋊#95	C₂³	1	32	yes	—
37	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	—	2	16	no	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	16	yes	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₆₇	D₄	1	8	yes	—
42	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	16	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	64	yes	complemented
53	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
56	C₂²×Dic₆₇	—	2	16	no	—
57	C₂²×Dic₆₇	—	2	32	no	—
58	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
59	C₄⋊Dic₆₇	—	2	32	no	—
60	C₄⋊Dic₆₇	—	2	64	no	—
61	C₂²⋊Dic₆₇	—	2	32	no	—
62	C₄×Dic₆₇	—	2	32	no	—
63	C₂⁴×C₆₇	—	2	8	no	—
64	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
66	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
67	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
68	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	8	no	—
70	C₂²×C₄×C₆₇	—	2	64	no	—
71	C₂²×C₄×C₆₇	—	2	32	no	—
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	64	yes	complemented
73	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	32	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	—
85	C₂²×C₄×C₆₇	#41	1	16	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
91	C₄²×C₆₇	C₂²×C₄	1	32	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
93	C₂³×C₆₇	—	2	16	no	—
94	C₂³×C₆₇	—	2	16	no	—
95	C₂³×C₆₇	—	2	16	no	—
96	C₂³×C₆₇	—	2	8	no	—
97	C₂³×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
98	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
99	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
101	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
102	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
103	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
104	C₂³×C₆₇	C₂×#41	1	8	yes	—
105	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	16	no	—
107	C₂×C₄×C₆₇	—	2	8	no	—
108	C₂×C₄×C₆₇	—	2	64	no	—
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	64	yes	complemented
112	C₂×C₄×C₆₇	#117	1	32	yes	complemented
113	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
117	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
120	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	16	yes	—
122	C₂×Dic₆₇	—	2	16	no	—
123	C₂×Dic₆₇	—	2	32	no	—
124	C₂×Dic₆₇	—	4	32	no	—
125	C₂²×C₆₇	—	2	8	no	—
126	C₂²×C₆₇	—	2	8	no	—
127	C₂²×C₆₇	—	2	8	no	—
128	C₂²×C₆₇	—	2	32	no	—
129	C₂²×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
130	C₂²×C₆₇	C₂²×C₄⋊C₄	1	4	yes	—
131	C₂²×C₆₇	C₂×#117	1	8	yes	—
132	C₂²×C₆₇	#532	1	8	yes	—
133	C₂²×C₆₇	#533	1	8	yes	—
134	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
135	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
136	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
137	C₂²×C₆₇	C₂²×#41	1	4	yes	—
138	C₄×C₆₇	—	2	8	no	—
139	C₄×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
140	C₄×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
141	C₄×C₆₇	C₂×#117	1	16	yes	complemented
142	Dic₆₇	—	4	16	no	—
143	C₂×#2285	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	8	no	—
145	C₂×C₆₇	#2285	1	8	yes	complemented
146	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₆₇	C₂²×#117	1	2	yes	—
148	C₂×C₆₇	C₂×#532	1	2	yes	—
149	C₂×C₆₇	C₂×#533	1	2	yes	—
150	C₂²×#117	—	67	2	no	—
151	C₂³×C₄²	D₆₇	1	1	yes	characteristic
152	#2285	—	67	16	no	—
153	C₂×#384	—	67	2	no	—
154	C₂×#382	—	67	2	no	—
155	C₂×#380	—	67	4	no	—
156	C₂×C₄×C₄⋊C₄	—	67	4	no	—
157	C₆₇	C₂×#2285	1	1	yes	characteristic, complemented, Sylow
158	C₂⁴×C₄	C₂×D₆₇	1	2	yes	—
159	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
160	C₂×#117	—	67	16	no	—
161	C₂×#117	—	134	16	no	—
162	C₂²×C₄⋊C₄	—	67	4	no	—
163	C₂²×C₄⋊C₄	—	67	8	no	—
164	C₂²×C₂²⋊C₄	—	67	4	no	—
165	C₂²×C₄²	—	67	4	no	—
166	C₂²×C₄²	Dic₆₇	1	16	yes	complemented
167	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
168	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
169	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
170	C₂²×C₄²	C₂×D₆₇	1	8	yes	—
171	#384	—	67	16	no	—
172	#382	—	67	16	no	—
173	#380	—	67	32	no	—
174	C₄×C₄⋊C₄	—	67	32	no	—
175	C₂×#95	—	67	8	no	—
176	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
177	C₂³×C₄	—	2	16	no	—
178	C₂³×C₄	—	67	8	no	—
179	C₂³×C₄	Dic₁₃₄	1	8	yes	—
180	C₂³×C₄	D₂₆₈	1	8	yes	—
181	C₂³×C₄	C₂×Dic₆₇	1	16	yes	—
182	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
183	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
184	C₂³×C₄	C₂²×D₆₇	1	8	yes	—
185	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
187	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
188	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
189	#117	—	134	64	no	—
190	C₂×C₄⋊C₄	—	67	16	no	—
191	C₂×C₄⋊C₄	—	67	32	no	—
192	C₂×C₄⋊C₄	—	134	16	no	—
193	C₂×C₄⋊C₄	—	134	32	no	—
194	C₂×C₂²⋊C₄	—	67	16	no	—
195	C₂×C₂²⋊C₄	—	134	16	no	—
196	C₂×C₄²	—	67	16	no	—
197	C₂×C₄²	—	134	16	no	—
198	C₂×C₄²	C₂×Dic₆₇	1	8	yes	—
199	C₂×C₄²	C₂×Dic₆₇	1	16	yes	—
200	C₂×C₄²	C₂×Dic₆₇	1	8	yes	—
201	C₂×C₄²	C₂×Dic₆₇	1	64	yes	complemented
202	C₂×C₄²	C₂²×D₆₇	1	8	yes	—
203	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
204	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
205	#95	—	67	32	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	C₂²×Dic₆₇	1	4	yes	—
208	C₂⁴	C₂×D₂₆₈	1	4	yes	—
209	C₂⁴	C₂×Dic₁₃₄	1	4	yes	—
210	C₂⁴	C₂³×D₆₇	1	2	yes	—
211	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	64	no	—
214	C₂²×C₄	—	2	32	no	—
215	C₂²×C₄	—	67	16	no	—
216	C₂²×C₄	—	134	16	no	—
217	C₂²×C₄	—	134	32	no	—
218	C₂²×C₄	C₄⋊Dic₆₇	1	64	yes	complemented
219	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
220	C₂²×C₄	C₂²×Dic₆₇	1	32	yes	—
221	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
222	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₆₇	1	16	yes	—
224	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
225	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
226	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
227	C₂²×C₄	C₂×D₂₆₈	1	16	yes	—
228	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
229	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
230	C₂²×C₄	C₂×Dic₁₃₄	1	16	yes	—
231	C₂²×C₄	#11761628	1	16	yes	—
232	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
233	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
235	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
236	C₄⋊C₄	—	134	32	no	—
237	C₄⋊C₄	—	134	64	no	—
238	C₂²⋊C₄	—	134	32	no	—
239	C₄²	—	134	32	no	—
240	C₄²	C₂²×Dic₆₇	1	16	yes	—
241	C₄²	C₂²×Dic₆₇	1	32	yes	—
242	C₄²	C₂²×Dic₆₇	1	16	yes	—
243	C₂³	—	2	16	no	—
244	C₂³	—	2	16	no	—
245	C₂³	—	2	16	no	—
246	C₂³	—	2	8	no	—
247	C₂³	C₂×C₄⋊Dic₆₇	1	16	yes	—
248	C₂³	C₂³×Dic₆₇	1	4	yes	—
249	C₂³	C₂³×Dic₆₇	1	2	yes	—
250	C₂³	C₂²×D₂₆₈	1	4	yes	—
251	C₂³	C₂²×D₂₆₈	1	2	yes	—
252	C₂³	C₂²×Dic₁₃₄	1	4	yes	—
253	C₂³	C₂²×Dic₁₃₄	1	2	yes	—
254	C₂³	C₂×#11761628	1	8	yes	—
255	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	64	no	—
259	C₂×C₄	—	268	32	no	—
260	C₂×C₄	—	134	16	no	—
261	C₂×C₄	—	134	32	no	—
262	C₂×C₄	C₂×C₄⋊Dic₆₇	1	16	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₆₇	1	16	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₆₇	1	64	yes	complemented
265	C₂×C₄	#14524002	1	32	yes	complemented
266	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
267	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
268	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
269	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
270	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
271	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
272	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
273	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
274	C₂×C₄	C₂×#11761628	1	16	yes	—
275	C₂²	—	2	32	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	C₂²×C₄⋊Dic₆₇	1	8	yes	—
280	C₂²	C₂²×C₄⋊Dic₆₇	1	4	yes	—
281	C₂²	C₂×#14524002	1	8	yes	—
282	C₂²	#16863946	1	8	yes	—
283	C₂²	#16863953	1	8	yes	—
284	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
285	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
286	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
287	C₂²	C₂²×#11761628	1	4	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	268	16	no	—
290	C₄	C₂²×C₄⋊Dic₆₇	1	8	yes	—
291	C₄	C₂²×C₄⋊Dic₆₇	1	8	yes	—
292	C₄	C₂×#14524002	1	16	yes	complemented
293	C₂	—	2	8	no	—
294	C₂	#22562616	1	8	yes	direct product factor
295	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
296	C₂	C₂²×#14524002	1	2	yes	—
297	C₂	C₂×#16863946	1	2	yes	—
298	C₂	C₂×#16863953	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₆₇
Fitting	C₂³×C₄²×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂⁴
prime radical	C₂⁵×C₆₇	C₂³
cosocle	C₂³×C₆₇	C₂⁵
hypercenter	C₂³×C₄²	D₆₇
hypocenter	C₆₇	C₂×#2285
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. #22562616	C₂	direct (N × H)
3. C₂×#16863266	C₂	semidirect
4. C₂×#16863262	C₂	semidirect
6. C₂×C₄×C₄⋊Dic₆₇	C₂	semidirect
14. #16863266	C₂²	semidirect
15. #16863262	C₂²	semidirect
17. C₄×C₄⋊Dic₆₇	C₂²	semidirect
21. C₂²×C₄²×C₆₇	C₄	semidirect
52. C₂×C₄²×C₆₇	C₂×C₄	semidirect
72. C₂²×C₄×C₆₇	C₄⋊C₄	semidirect
111. C₂×C₄×C₆₇	C₂×C₄⋊C₄	semidirect
112. C₂×C₄×C₆₇	#117	semidirect
141. C₄×C₆₇	C₂×#117	semidirect
145. C₂×C₆₇	#2285	semidirect
157. C₆₇	C₂×#2285	semidirect
166. C₂²×C₄²	Dic₆₇	semidirect
201. C₂×C₄²	C₂×Dic₆₇	semidirect
218. C₂²×C₄	C₄⋊Dic₆₇	semidirect
264. C₂×C₄	C₂×C₄⋊Dic₆₇	semidirect
265. C₂×C₄	#14524002	semidirect
292. C₄	C₂×#14524002	semidirect
294. C₂	#22562616	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×#14524002	2
#22562616	2
C₂×#16863266	2
C₂×#16863262	2
C₂×#16863256	2
C₂×C₄×C₄⋊Dic₆₇	2
C₂³×C₄²×C₆₇	2
C₂×#2285	67

Subgroups — C2×#22562616 #31014128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562616 #31014128