Subgroups — C₂×#22558315 #30770746

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
3	#22558315	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	—	2	1	no	—
6	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
7	#16863434	—	2	1	no	—
8	#16863434	—	2	1	no	—
9	#16863434	C₄	1	2	yes	complemented
10	#16863434	C₂²	1	2	yes	—
11	C₂×#14523977	C₄	1	2	yes	complemented
12	C₂×#14523975	—	2	1	no	—
13	C₂×#14523975	—	2	2	no	—
14	C₂×#14523975	—	2	1	no	—
15	C₂×#14523975	C₄	1	4	yes	complemented
16	C₂×#14523975	C₂²	1	2	yes	—
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₆₇	—	2	2	no	—
19	C₂×M₃₂×C₆₇	—	2	1	no	—
20	C₂×M₃₂×C₆₇	C₄	1	4	yes	complemented
21	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₆₇	—	2	1	no	—
23	C₂³×Dic₆₇	—	4	1	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	#14524002	—	4	1	no	—
27	#14524002	—	4	1	no	—
28	C₂×C₄⋊Dic₆₇	—	2	4	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	4	1	no	—
32	C₂×C₄⋊Dic₆₇	—	4	2	no	—
33	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
34	C₂×C₄×Dic₆₇	—	4	1	no	—
35	#14523977	—	2	2	no	—
36	#14523977	C₂×C₄	1	4	yes	complemented
37	#14523975	—	2	4	no	—
38	#14523975	—	2	2	no	—
39	#14523975	—	2	2	no	—
40	#14523975	—	2	2	no	—
41	#14523975	C₂×C₄	1	4	yes	complemented
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	M₃₂×C₆₇	—	2	4	no	—
54	M₃₂×C₆₇	—	2	2	no	—
55	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆×C₆₇	—	2	4	no	—
57	C₂×C₁₆×C₆₇	—	2	2	no	—
58	C₂²×Dic₆₇	—	4	2	no	—
59	C₂²×Dic₆₇	—	4	4	no	—
60	C₂²×Dic₆₇	—	4	2	no	—
61	C₂²×Dic₆₇	—	8	4	no	—
62	C₄⋊Dic₆₇	—	2	4	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	4	no	—
65	C₄⋊Dic₆₇	—	4	2	no	—
66	C₄⋊Dic₆₇	—	4	2	no	—
67	C₂²⋊Dic₆₇	—	4	2	no	—
68	C₂²⋊Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	C₄×Dic₆₇	—	4	2	no	—
71	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₄²	1	2	yes	—
83	C₂×C₈×C₆₇	C₄²	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
100	C₁₆×C₆₇	—	2	4	no	—
101	C₂³×C₆₇	—	2	4	no	—
102	C₂³×C₆₇	#102	1	1	yes	characteristic
103	C₂³×C₆₇	#102	1	2	yes	—
104	C₂³×C₆₇	#102	1	1	yes	characteristic
105	C₂³×C₆₇	#107	1	2	yes	—
106	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#95	1	2	yes	—
108	C₂×C₄×C₆₇	#95	1	2	yes	—
109	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#102	1	2	yes	—
111	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#103	1	2	yes	—
113	C₂×C₄×C₆₇	#103	1	2	yes	—
114	C₂×C₄×C₆₇	#103	1	2	yes	—
115	C₂×C₄×C₆₇	#103	1	2	yes	—
116	C₂×C₄×C₆₇	#106	1	2	yes	—
117	C₂×C₄×C₆₇	#106	1	2	yes	—
118	C₂×C₄×C₆₇	#107	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×Dic₆₇	—	4	4	no	—
126	C₂×Dic₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	8	4	no	—
128	C₂×Dic₆₇	—	8	8	no	—
129	C₂×Dic₆₇	—	8	4	no	—
130	C₈×C₆₇	#95	1	2	yes	—
131	C₈×C₆₇	#95	1	2	yes	—
132	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	—	2	4	no	—
140	C₂²×C₆₇	#340	1	2	yes	—
141	C₂²×C₆₇	#340	1	2	yes	—
142	C₂²×C₆₇	#360	1	4	yes	—
143	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
146	C₄×C₆₇	#340	1	2	yes	—
147	C₄×C₆₇	#340	1	2	yes	—
148	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
152	Dic₆₇	—	8	4	no	—
153	Dic₆₇	—	8	4	no	—
154	C₂×#1386	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	#1386	1	4	yes	complemented
159	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
160	C₂×C₆₇	C₂×#360	1	2	yes	—
161	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
162	C₂×#427	—	67	1	no	—
163	C₂²×#106	—	67	1	no	—
164	#1386	—	67	4	no	—
165	C₆₇	C₂×#1386	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
167	C₂×#117	—	134	1	no	—
168	C₂²×C₄⋊C₄	—	134	1	no	—
169	C₂×M₃₂	—	2	2	no	—
170	C₂×M₃₂	—	2	1	no	—
171	C₂×M₃₂	Dic₆₇	1	4	yes	complemented
172	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
173	C₂²×C₁₆	—	2	1	no	—
174	#427	—	67	2	no	—
175	#427	—	67	2	no	—
176	#427	—	134	1	no	—
177	#427	—	134	1	no	—
178	C₂×#107	—	67	2	no	—
179	C₂×#106	—	67	4	no	—
180	C₂×#106	—	67	2	no	—
181	C₂×#106	—	134	1	no	—
182	C₂×#106	—	134	2	no	—
183	C₂×#106	—	134	1	no	—
184	C₂³×C₄	—	268	1	no	—
185	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
186	C₂²×C₈	Dic₁₃₄	1	2	yes	—
187	C₂²×C₈	C₄×D₆₇	1	2	yes	—
188	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
194	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
196	#117	—	268	1	no	—
197	#117	—	268	1	no	—
198	#117	—	134	2	no	—
199	#117	—	134	2	no	—
200	C₂×C₄⋊C₄	—	268	1	no	—
201	C₂×C₄⋊C₄	—	268	2	no	—
202	C₂×C₄⋊C₄	—	134	4	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₄⋊C₄	—	134	2	no	—
205	C₂×C₂²⋊C₄	—	268	1	no	—
206	C₂×C₄²	—	268	1	no	—
207	M₃₂	—	2	4	no	—
208	M₃₂	—	2	2	no	—
209	M₃₂	C₂×Dic₆₇	1	4	yes	complemented
210	C₂×C₁₆	—	2	4	no	—
211	C₂×C₁₆	—	2	2	no	—
212	#107	—	67	4	no	—
213	#107	—	134	2	no	—
214	#106	—	67	4	no	—
215	#106	—	134	4	no	—
216	#106	—	134	2	no	—
217	#106	—	134	2	no	—
218	#106	—	134	2	no	—
219	C₂⁴	D₅₃₆	1	1	yes	characteristic
220	C₂²×C₄	—	268	4	no	—
221	C₂²×C₄	—	268	2	no	—
222	C₂²×C₄	—	268	2	no	—
223	C₂²×C₄	—	536	4	no	—
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
228	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
229	C₂²×C₄	#11761617	1	2	yes	—
230	C₂²×C₄	#11761617	1	1	yes	characteristic
231	C₂²×C₄	#11761617	1	1	yes	characteristic
232	C₂²×C₄	Dic₂₆₈	1	2	yes	—
233	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
234	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
235	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
236	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
237	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
238	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
240	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
241	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
242	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
243	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
244	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
245	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
252	C₄⋊C₄	—	268	4	no	—
253	C₄⋊C₄	—	268	2	no	—
254	C₄⋊C₄	—	268	2	no	—
255	C₄⋊C₄	—	134	4	no	—
256	C₄⋊C₄	—	134	4	no	—
257	C₂²⋊C₄	—	268	2	no	—
258	C₂²⋊C₄	—	268	2	no	—
259	C₄²	—	268	2	no	—
260	C₄²	—	268	2	no	—
261	C₁₆	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	#14523965	1	1	yes	characteristic
264	C₂³	#14523965	1	2	yes	—
265	C₂³	#14523965	1	1	yes	characteristic
266	C₂³	#14523977	1	2	yes	—
267	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
268	C₂×C₄	—	268	4	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	536	8	no	—
271	C₂×C₄	—	536	4	no	—
272	C₂×C₄	—	536	4	no	—
273	C₂×C₄	C₆₇⋊#95	1	2	yes	—
274	C₂×C₄	C₆₇⋊#95	1	2	yes	—
275	C₂×C₄	#14523965	1	1	yes	characteristic
276	C₂×C₄	#14523965	1	2	yes	—
277	C₂×C₄	#14523965	1	1	yes	characteristic
278	C₂×C₄	#14523968	1	2	yes	—
279	C₂×C₄	#14523968	1	2	yes	—
280	C₂×C₄	#14523968	1	2	yes	—
281	C₂×C₄	#14523968	1	2	yes	—
282	C₂×C₄	#14523975	1	2	yes	—
283	C₂×C₄	#14523975	1	2	yes	—
284	C₂×C₄	#14523977	1	2	yes	—
285	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
289	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
290	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
291	C₈	C₆₇⋊#95	1	2	yes	—
292	C₈	C₆₇⋊#95	1	2	yes	—
293	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
294	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
295	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
296	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#16863159	1	2	yes	—
302	C₂²	#16863159	1	2	yes	—
303	C₂²	#16863207	1	4	yes	—
304	C₂²	C₂×#14523965	1	1	yes	characteristic
305	C₂²	C₂×#14523965	1	1	yes	characteristic
306	C₂²	C₂×#14523977	1	1	yes	characteristic
307	C₄	—	536	4	no	—
308	C₄	—	536	4	no	—
309	C₄	#16863159	1	2	yes	—
310	C₄	#16863159	1	2	yes	—
311	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
312	C₄	C₂×#14523968	1	1	yes	characteristic
313	C₄	C₂×#14523968	1	1	yes	characteristic
314	C₄	C₂×#14523975	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#22558315	1	4	yes	direct product factor
319	C₂	C₂×#16863159	1	1	yes	characteristic
320	C₂	C₂×#16863207	1	2	yes	—

	subgroup	G/H
center	C₂³	#14523965
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂³×C₆₇	#102
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#1386
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558315	C₂	direct (N × H)
9. #16863434	C₄	semidirect
11. C₂×#14523977	C₄	semidirect
15. C₂×#14523975	C₄	semidirect
20. C₂×M₃₂×C₆₇	C₄	semidirect
36. #14523977	C₂×C₄	semidirect
41. #14523975	C₂×C₄	semidirect
55. M₃₂×C₆₇	C₂×C₄	semidirect
158. C₂×C₆₇	#1386	semidirect
165. C₆₇	C₂×#1386	semidirect
171. C₂×M₃₂	Dic₆₇	semidirect
209. M₃₂	C₂×Dic₆₇	semidirect
318. C₂	#22558315	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×#16863434	2
C₂²×#14523975	2
#22558315	2
C₂²×M₃₂×C₆₇	2
C₂×#1386	67

Subgroups — C2×#22558315 #30770746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558315 #30770746