Subgroups — C₂×#22558316 #30770747

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	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22558316	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	—	2	1	no	—
6	C₂×#14523983	—	2	1	no	—
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	C₄	1	4	yes	complemented
9	C₂×#14523983	C₂²	1	2	yes	—
10	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
11	#16863434	—	2	1	no	—
12	#16863434	—	2	1	no	—
13	#16863434	C₄	1	2	yes	complemented
14	#16863434	C₂²	1	2	yes	—
15	C₂×#14523977	C₄	1	2	yes	complemented
16	C₂×#14523975	—	2	1	no	—
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₂×#106×C₆₇	—	2	1	no	—
20	C₂×#106×C₆₇	—	2	2	no	—
21	C₂×#106×C₆₇	C₄	1	4	yes	complemented
22	C₂×#106×C₆₇	C₂²	1	2	yes	—
23	#14524002	—	2	2	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	4	1	no	—
26	#14524002	—	4	1	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
29	C₂×C₄×Dic₆₇	—	4	1	no	—
30	#14523983	—	2	4	no	—
31	#14523983	—	2	2	no	—
32	#14523983	C₂×C₄	1	4	yes	complemented
33	C₂×C₆₇⋊C₁₆	—	2	4	no	—
34	C₂×C₆₇⋊C₁₆	—	2	2	no	—
35	#14523977	—	2	2	no	—
36	#14523977	C₂×C₄	1	4	yes	complemented
37	#14523975	—	2	2	no	—
38	#14523975	—	2	2	no	—
39	C₂³×C₄×C₆₇	—	4	1	no	—
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×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	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	2	yes	—
49	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
55	#106×C₆₇	—	2	4	no	—
56	#106×C₆₇	—	2	2	no	—
57	#106×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₂²×Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₂²⋊Dic₆₇	—	4	2	no	—
62	C₂²⋊Dic₆₇	—	4	2	no	—
63	C₄×Dic₆₇	—	4	2	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₆₇⋊C₁₆	—	2	4	no	—
66	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	—	8	4	no	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₄²	1	2	yes	—
81	C₂×C₈×C₆₇	C₄²	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	4	4	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	—	2	4	no	—
102	C₂³×C₆₇	#102	1	1	yes	characteristic
103	C₂³×C₆₇	#102	1	1	yes	characteristic
104	C₂³×C₆₇	#102	1	2	yes	—
105	C₂³×C₆₇	#107	1	2	yes	—
106	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	8	4	no	—
109	C₂×C₄×C₆₇	—	8	8	no	—
110	C₂×C₄×C₆₇	#95	1	2	yes	—
111	C₂×C₄×C₆₇	#95	1	2	yes	—
112	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#102	1	2	yes	—
115	C₂×C₄×C₆₇	#103	1	2	yes	—
116	C₂×C₄×C₆₇	#103	1	2	yes	—
117	C₂×C₄×C₆₇	#103	1	2	yes	—
118	C₂×C₄×C₆₇	#103	1	2	yes	—
119	C₂×C₄×C₆₇	#106	1	2	yes	—
120	C₂×C₄×C₆₇	#106	1	2	yes	—
121	C₂×C₄×C₆₇	#107	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	4	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	—	8	4	no	—
147	C₄×C₆₇	#340	1	2	yes	—
148	C₄×C₆₇	#340	1	2	yes	—
149	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
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	—	67	1	no	—
162	C₂×#427	—	67	1	no	—
163	C₂²×#106	D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
169	C₂×M₃₂	—	67	4	no	—
170	C₂×M₃₂	—	67	2	no	—
171	C₂×M₃₂	—	134	1	no	—
172	C₂×M₃₂	—	134	2	no	—
173	C₂²×C₁₆	—	134	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	—	2	1	no	—
180	C₂×#106	—	2	2	no	—
181	C₂×#106	—	134	1	no	—
182	C₂×#106	Dic₆₇	1	4	yes	complemented
183	C₂×#106	C₂×D₆₇	1	2	yes	—
184	C₂³×C₄	—	4	1	no	—
185	C₂³×C₄	C₂²⋊D₆₇	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	D₂₆₈	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
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	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	4	1	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄⋊C₄	—	134	2	no	—
205	C₂×C₂²⋊C₄	—	268	1	no	—
206	C₂×C₄²	—	268	1	no	—
207	M₃₂	—	67	4	no	—
208	M₃₂	—	134	4	no	—
209	M₃₂	—	134	2	no	—
210	C₂×C₁₆	—	134	4	no	—
211	C₂×C₁₆	—	134	2	no	—
212	#107	—	67	4	no	—
213	#107	—	134	2	no	—
214	#106	—	2	4	no	—
215	#106	—	2	2	no	—
216	#106	—	134	2	no	—
217	#106	—	134	2	no	—
218	#106	C₂×Dic₆₇	1	4	yes	complemented
219	C₂⁴	#11761613	1	1	yes	characteristic
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	—	268	2	no	—
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
227	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
228	C₂²×C₄	#11761613	1	1	yes	characteristic
229	C₂²×C₄	#11761615	1	1	yes	characteristic
230	C₂²×C₄	#11761615	1	1	yes	characteristic
231	C₂²×C₄	#11761616	1	2	yes	—
232	C₂²×C₄	#11761618	1	2	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₆₇	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	1	yes	characteristic
237	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
238	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
239	C₂×C₈	(C₂×D₆₇)⋊C₄	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	C₄⋊Dic₆₇	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×D₂₆₈	1	1	yes	characteristic
251	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	268	2	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	—	134	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	#14523963	1	1	yes	characteristic
264	C₂³	#14523963	1	1	yes	characteristic
265	C₂³	#14523964	1	2	yes	—
266	C₂³	#14523976	1	2	yes	—
267	C₂³	C₂×#11761613	1	1	yes	characteristic
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	8	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	268	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	#14523963	1	1	yes	characteristic
276	C₂×C₄	#14523963	1	1	yes	characteristic
277	C₂×C₄	#14523964	1	2	yes	—
278	C₂×C₄	#14523966	1	2	yes	—
279	C₂×C₄	#14523966	1	2	yes	—
280	C₂×C₄	#14523967	1	2	yes	—
281	C₂×C₄	#14523967	1	2	yes	—
282	C₂×C₄	#14523974	1	2	yes	—
283	C₂×C₄	#14523974	1	2	yes	—
284	C₂×C₄	#14523976	1	2	yes	—
285	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
289	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
290	C₂×C₄	C₂×#11761618	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×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
295	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
296	C₈	C₂×C₄⋊Dic₆₇	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	#16863158	1	2	yes	—
302	C₂²	#16863158	1	2	yes	—
303	C₂²	#16863205	1	4	yes	—
304	C₂²	C₂×#14523963	1	1	yes	characteristic
305	C₂²	C₂×#14523964	1	1	yes	characteristic
306	C₂²	C₂×#14523976	1	1	yes	characteristic
307	C₄	—	8	4	no	—
308	C₄	—	536	4	no	—
309	C₄	#16863158	1	2	yes	—
310	C₄	#16863158	1	2	yes	—
311	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
312	C₄	C₂×#14523966	1	1	yes	characteristic
313	C₄	C₂×#14523967	1	1	yes	characteristic
314	C₄	C₂×#14523974	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#22558316	1	4	yes	direct product factor
319	C₂	C₂×#16863158	1	1	yes	characteristic
320	C₂	C₂×#16863205	1	2	yes	—

	subgroup	G/H
center	C₂³	#14523963
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#106×C₆₇	C₂
socle	C₂³×C₆₇	#102
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#106	D₆₇
hypocenter	C₆₇	C₂×#1386
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. #22558316	C₂	direct (N × H)
8. C₂×#14523983	C₄	semidirect
13. #16863434	C₄	semidirect
15. C₂×#14523977	C₄	semidirect
21. C₂×#106×C₆₇	C₄	semidirect
32. #14523983	C₂×C₄	semidirect
36. #14523977	C₂×C₄	semidirect
57. #106×C₆₇	C₂×C₄	semidirect
158. C₂×C₆₇	#1386	semidirect
165. C₆₇	C₂×#1386	semidirect
182. C₂×#106	Dic₆₇	semidirect
218. #106	C₂×Dic₆₇	semidirect
318. C₂	#22558316	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×#16863434	2
#22558316	2
C₂²×#106×C₆₇	2
C₂×#1386	67

Subgroups — C2×#22558316 #30770747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558316 #30770747