Subgroups — C₂×#22561813 #31008488

Order 17152.

738 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	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
3	#22561813	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863173	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₄	1	1	yes	characteristic
7	C₂×#14524002	C₄	1	1	yes	characteristic
8	C₂×#14523983	C₄	1	2	yes	complemented
9	C₂×#14523983	C₄	1	4	yes	complemented
10	C₂×#14523983	C₂²	1	2	yes	—
11	C₂×#14523983	C₂²	1	4	yes	—
12	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
13	#16863351	C₄	1	4	yes	—
14	#16863351	C₂²	1	4	yes	—
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
17	#16863173	C₂²	1	16	yes	complemented
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₆₇	C₄	1	2	yes	complemented
20	C₂×M₃₂×C₆₇	C₄	1	4	yes	complemented
21	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
24	C₂×#11761612	—	2	1	no	—
25	C₂×#11761612	—	2	1	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
29	#14524002	—	2	2	no	—
30	#14524002	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
33	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
34	#14523983	C₂×C₄	1	4	yes	—
35	#14523983	C₂×C₄	1	8	yes	complemented
36	#14523983	C₂³	1	4	yes	—
37	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
38	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
39	#14523953	C₂×C₄	1	4	yes	—
40	#14523953	C₂³	1	4	yes	—
41	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
54	M₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
55	M₃₂×C₆₇	C₂³	1	4	yes	—
56	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×C₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
58	C₂²×Dic₆₇	—	2	2	no	—
59	#11761612	—	2	4	no	—
60	#11761612	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₂²⋊Dic₆₇	—	2	4	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
68	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	—
69	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
73	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
75	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₄²	1	2	yes	—
82	C₂×C₈×C₆₇	C₄²	1	2	yes	—
83	C₂×C₈×C₆₇	C₄²	1	2	yes	—
84	C₂×C₈×C₆₇	C₄²	1	2	yes	—
85	C₂×C₈×C₆₇	C₄²	1	2	yes	—
86	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
99	C₂³×C₆₇	—	4	2	no	—
100	C₂³×C₆₇	#97	1	1	yes	characteristic
101	C₂³×C₆₇	#97	1	1	yes	characteristic
102	C₂³×C₆₇	#97	1	2	yes	—
103	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₆₇	#97	1	2	yes	—
106	C₂×C₄×C₆₇	#97	1	2	yes	—
107	C₂×C₄×C₆₇	#97	1	2	yes	—
108	C₂×C₄×C₆₇	#97	1	2	yes	—
109	C₂×C₄×C₆₇	#97	1	2	yes	—
110	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₆₇	—	2	4	no	—
124	C₂×Dic₆₇	—	4	4	no	—
125	C₈×C₆₇	C₂×C₄²	1	2	yes	—
126	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
127	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
128	C₈×C₆₇	C₂×C₄²	1	2	yes	—
129	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	C₆₇⋊C₈	—	2	4	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	—	4	2	no	—
137	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
138	C₂²×C₆₇	C₂×#97	1	2	yes	—
139	C₂²×C₆₇	C₂×#97	1	2	yes	—
140	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
141	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
143	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
144	C₄×C₆₇	C₂×#97	1	2	yes	—
145	C₄×C₆₇	C₂×#97	1	2	yes	—
146	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
148	Dic₆₇	—	4	4	no	—
149	C₂×#2110	—	67	1	no	maximal, Sylow
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	—	2	2	no	—
154	C₂×C₆₇	#2110	1	2	yes	complemented
155	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
156	C₂²×M₃₂	—	67	1	no	—
157	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
158	C₂×#405	—	67	1	no	—
159	#2110	—	67	8	no	—
160	C₂×#347	—	67	4	no	—
161	C₆₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	67	1	no	—
163	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#117	—	67	1	no	—
165	C₂×M₃₂	—	67	2	no	—
166	C₂×M₃₂	—	67	4	no	—
167	C₂×M₃₂	—	67	2	no	—
168	C₂×M₃₂	—	67	4	no	—
169	C₂×M₃₂	Dic₆₇	1	2	yes	complemented
170	C₂×M₃₂	Dic₆₇	1	4	yes	complemented
171	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
172	C₂×M₃₂	C₂×D₆₇	1	4	yes	—
173	C₂²×C₁₆	—	67	2	no	—
174	C₂²×C₁₆	Dic₆₇	1	2	yes	complemented
175	#405	—	67	4	no	—
176	#405	—	67	4	no	—
177	C₂×#97	—	67	2	no	—
178	C₂×C₄×C₈	—	67	2	no	—
179	#347	—	67	16	no	—
180	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
181	C₂×M₁₆	—	67	4	no	—
182	C₂×M₁₆	—	134	1	no	—
183	C₂×M₁₆	—	134	1	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂²×C₈	—	67	2	no	—
186	C₂²×C₈	C₄×D₆₇	1	2	yes	—
187	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₆₇	1	2	yes	—
190	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
193	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
196	#117	—	67	4	no	—
197	#117	—	134	2	no	—
198	C₂×C₄⋊C₄	—	134	1	no	—
199	C₂×C₂²⋊C₄	—	134	1	no	—
200	C₂×C₄²	—	67	2	no	—
201	M₃₂	—	67	4	no	—
202	M₃₂	—	67	8	no	—
203	M₃₂	—	67	4	no	—
204	M₃₂	C₂×Dic₆₇	1	4	yes	—
205	M₃₂	C₂×Dic₆₇	1	8	yes	complemented
206	M₃₂	C₂²×D₆₇	1	4	yes	—
207	C₂×C₁₆	—	67	4	no	—
208	C₂×C₁₆	—	67	8	no	—
209	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
210	C₂×C₁₆	C₂×Dic₆₇	1	8	yes	complemented
211	#97	—	67	4	no	—
212	#97	—	67	4	no	—
213	C₄×C₈	—	67	8	no	—
214	C₂⁴	#11761611	1	1	yes	characteristic
215	C₂²×C₄	—	134	2	no	—
216	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
217	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
219	C₂²×C₄	#11761611	1	1	yes	characteristic
220	C₂²×C₄	#11761611	1	2	yes	—
221	C₂²×C₄	#11761611	1	1	yes	characteristic
222	C₂²×C₄	#11761611	1	1	yes	characteristic
223	C₂²×C₄	#11761611	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
226	M₁₆	—	134	4	no	—
227	M₁₆	—	134	4	no	—
228	C₂×C₈	—	67	4	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
232	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
233	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
234	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
235	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
236	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
237	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
244	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
246	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
248	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
249	C₄⋊C₄	—	134	4	no	—
250	C₂²⋊C₄	—	134	4	no	—
251	C₄²	—	67	4	no	—
252	C₄²	—	134	2	no	—
253	C₁₆	—	67	8	no	—
254	C₁₆	C₂²×Dic₆₇	1	8	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#14523953	1	1	yes	characteristic
257	C₂³	#14523953	1	1	yes	characteristic
258	C₂³	#14523953	1	2	yes	—
259	C₂³	C₂×#11761611	1	1	yes	characteristic
260	C₂³	C₂×#11761611	1	2	yes	—
261	C₂×C₄	—	268	4	no	—
262	C₂×C₄	—	134	4	no	—
263	C₂×C₄	#14523953	1	2	yes	—
264	C₂×C₄	#14523953	1	2	yes	—
265	C₂×C₄	#14523953	1	2	yes	—
266	C₂×C₄	#14523953	1	2	yes	—
267	C₂×C₄	#14523953	1	2	yes	—
268	C₂×C₄	#14523953	1	1	yes	characteristic
269	C₂×C₄	#14523953	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
272	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
275	C₂×C₄	C₂×#11761611	1	2	yes	—
276	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
277	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
278	C₂×C₄	C₂×#11761611	1	2	yes	—
279	C₂×C₄	C₂×#11761611	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
281	C₈	—	134	4	no	—
282	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
283	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
284	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
285	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
286	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
287	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#14523953	1	1	yes	characteristic
294	C₂²	C₂×#14523953	1	2	yes	—
295	C₂²	C₂×#14523953	1	2	yes	—
296	C₂²	C₂×#14523953	1	1	yes	characteristic
297	C₂²	C₂²×#11761611	1	1	yes	characteristic
298	C₄	—	268	4	no	—
299	C₄	C₂×#14523953	1	1	yes	characteristic
300	C₄	C₂×#14523953	1	2	yes	—
301	C₄	C₂×#14523953	1	2	yes	—
302	C₄	C₂×#14523953	1	1	yes	characteristic
303	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
304	C₄	C₂²×#11761611	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#22561813	1	2	yes	direct product factor
310	C₂	C₂²×#14523953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#2110
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22561813	C₂	direct (N × H)
4. C₂×#16863173	C₂	semidirect
8. C₂×#14523983	C₄	semidirect
9. C₂×#14523983	C₄	semidirect
12. C₂²×C₆₇⋊C₁₆	C₄	semidirect
17. #16863173	C₂²	semidirect
19. C₂×M₃₂×C₆₇	C₄	semidirect
20. C₂×M₃₂×C₆₇	C₄	semidirect
23. C₂²×C₁₆×C₆₇	C₄	semidirect
35. #14523983	C₂×C₄	semidirect
38. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
54. M₃₂×C₆₇	C₂×C₄	semidirect
57. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
154. C₂×C₆₇	#2110	semidirect
161. C₆₇	C₂×#2110	semidirect
169. C₂×M₃₂	Dic₆₇	semidirect
170. C₂×M₃₂	Dic₆₇	semidirect
174. C₂²×C₁₆	Dic₆₇	semidirect
205. M₃₂	C₂×Dic₆₇	semidirect
210. C₂×C₁₆	C₂×Dic₆₇	semidirect
309. C₂	#22561813	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×#14523983	2
C₂×#16863351	2
#22561813	2
C₂×#16863173	2
C₂²×M₃₂×C₆₇	2
C₂×#2110	67

Subgroups — C2×#22561813 #31008488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561813 #31008488