Subgroups — C₂×#22558292 #30770689

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
3	#22558292	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₆₇	C₄	1	1	yes	characteristic
7	C₂×#14523983	—	2	1	no	—
8	C₂×#14523983	—	2	2	no	—
9	C₂×#14523983	C₄	1	4	yes	complemented
10	C₂×#14523983	C₂²	1	2	yes	—
11	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
12	C₂×#14523953	—	2	1	no	—
13	C₂×#14523953	—	2	1	no	—
14	C₂×#14523953	—	2	1	no	—
15	C₂×#14523953	—	2	1	no	—
16	C₂×#14523953	C₄	1	2	yes	—
17	C₂×#14523953	C₂²	1	2	yes	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₆₇	—	2	2	no	—
20	C₂×M₃₂×C₆₇	—	2	1	no	—
21	C₂×M₃₂×C₆₇	C₄	1	4	yes	complemented
22	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₆₇	—	2	1	no	—
24	C₂³×Dic₆₇	C₈	1	2	yes	—
25	C₂²×C₆₇⋊C₈	—	2	1	no	—
26	C₂²×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	C₈	1	4	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₇	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	—	2	1	no	—
33	C₂×C₄×Dic₆₇	—	2	1	no	—
34	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
36	#14523983	—	2	4	no	—
37	#14523983	—	2	2	no	—
38	#14523983	C₂×C₄	1	4	yes	complemented
39	C₂×C₆₇⋊C₁₆	—	2	4	no	—
40	C₂×C₆₇⋊C₁₆	—	2	2	no	—
41	#14523953	—	2	2	no	—
42	#14523953	—	2	2	no	—
43	#14523953	—	2	2	no	—
44	#14523953	—	2	2	no	—
45	C₂³×C₄×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	2	yes	—
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	D₄	1	2	yes	—
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	M₃₂×C₆₇	—	2	4	no	—
57	M₃₂×C₆₇	—	2	2	no	—
58	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₆₇	—	2	4	no	—
60	C₂×C₁₆×C₆₇	—	2	2	no	—
61	C₂²×Dic₆₇	—	2	2	no	—
62	C₂²×Dic₆₇	—	2	2	no	—
63	C₂²×Dic₆₇	—	4	4	no	—
64	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	—	2	2	no	—
72	C₄×Dic₆₇	—	2	2	no	—
73	C₄×Dic₆₇	—	2	2	no	—
74	C₆₇⋊C₁₆	—	2	4	no	—
75	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
83	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₄²	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₁₆×C₆₇	—	2	4	no	—
105	C₂³×C₆₇	—	2	4	no	—
106	C₂³×C₆₇	C₄×C₈	1	2	yes	—
107	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
109	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
110	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#95	1	2	yes	—
112	C₂×C₄×C₆₇	#95	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₆₇	#97	1	2	yes	—
115	C₂×C₄×C₆₇	#97	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₆₇	—	2	4	no	—
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	4	4	no	—
132	C₂×Dic₆₇	—	4	4	no	—
133	C₈×C₆₇	#95	1	2	yes	—
134	C₈×C₆₇	#95	1	2	yes	—
135	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₆₇⋊C₈	—	2	4	no	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	4	no	—
144	C₂²×C₆₇	#336	1	2	yes	—
145	C₂²×C₆₇	#336	1	2	yes	—
146	C₂²×C₆₇	#349	1	4	yes	—
147	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₆₇	#336	1	2	yes	—
151	C₄×C₆₇	#336	1	2	yes	—
152	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
154	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
156	Dic₆₇	—	4	4	no	—
157	C₂×#1377	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	—	2	1	no	—
161	C₂×C₆₇	#1377	1	4	yes	complemented
162	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#349	1	2	yes	—
164	C₂²×M₃₂	—	67	1	no	—
165	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
166	C₂²×#97	—	67	1	no	—
167	#1377	—	67	4	no	—
168	C₆₇	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	67	1	no	—
170	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
171	C₂²×C₄²	—	67	1	no	—
172	C₂×M₃₂	—	2	2	no	—
173	C₂×M₃₂	—	2	1	no	—
174	C₂×M₃₂	—	67	4	no	—
175	C₂×M₃₂	—	67	2	no	—
176	C₂×M₃₂	—	134	1	no	—
177	C₂×M₃₂	—	134	2	no	—
178	C₂×M₃₂	Dic₆₇	1	4	yes	complemented
179	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
180	C₂²×C₁₆	—	2	1	no	—
181	C₂²×C₁₆	—	134	1	no	—
182	C₂×#97	—	67	2	no	—
183	C₂×#97	—	67	2	no	—
184	C₂×#97	—	134	1	no	—
185	C₂×#97	—	134	1	no	—
186	C₂×#97	—	134	1	no	—
187	C₂×#97	—	134	1	no	—
188	C₂³×C₄	—	67	2	no	—
189	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₈	—	67	4	no	—
191	C₂²×C₈	—	67	2	no	—
192	C₂²×C₈	—	134	1	no	—
193	C₂²×C₈	—	134	1	no	—
194	C₂²×C₈	—	134	2	no	—
195	C₂²×C₈	Dic₁₃₄	1	2	yes	—
196	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₆₇	1	2	yes	—
198	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
199	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
202	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
205	C₂×C₄²	—	67	2	no	—
206	C₂×C₄²	—	67	2	no	—
207	C₂×C₄²	—	134	1	no	—
208	C₂×C₄²	—	134	1	no	—
209	C₂×C₄²	—	134	1	no	—
210	C₂×C₄²	—	134	1	no	—
211	M₃₂	—	2	4	no	—
212	M₃₂	—	2	2	no	—
213	M₃₂	—	67	4	no	—
214	M₃₂	—	134	4	no	—
215	M₃₂	—	134	2	no	—
216	M₃₂	C₂×Dic₆₇	1	4	yes	complemented
217	C₂×C₁₆	—	2	4	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	—	134	4	no	—
220	C₂×C₁₆	—	134	2	no	—
221	#97	—	134	2	no	—
222	#97	—	134	2	no	—
223	#97	—	134	2	no	—
224	#97	—	134	2	no	—
225	C₂⁴	C₈×D₆₇	1	1	yes	characteristic
226	C₂²×C₄	—	268	4	no	—
227	C₂²×C₄	—	67	4	no	—
228	C₂²×C₄	—	134	2	no	—
229	C₂²×C₄	—	134	2	no	—
230	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
231	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
234	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₈×D₆₇	1	2	yes	—
236	C₂²×C₄	#11761611	1	2	yes	—
237	C₂²×C₄	#11761611	1	1	yes	characteristic
238	C₂²×C₄	#11761611	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
240	C₂×C₈	—	67	4	no	—
241	C₂×C₈	—	134	4	no	—
242	C₂×C₈	—	134	4	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	—	134	2	no	—
245	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
246	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
247	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
249	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
250	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
251	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
252	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
253	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
254	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
255	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
256	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
262	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
263	C₄²	—	134	2	no	—
264	C₄²	—	134	2	no	—
265	C₄²	—	134	2	no	—
266	C₄²	—	134	2	no	—
267	C₁₆	—	2	4	no	—
268	C₁₆	—	134	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₈×Dic₆₇	1	2	yes	—
271	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
272	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
273	C₂³	Dic₆₇⋊C₈	1	2	yes	—
274	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
275	C₂×C₄	—	268	4	no	—
276	C₂×C₄	—	268	4	no	—
277	C₂×C₄	—	268	4	no	—
278	C₂×C₄	—	134	4	no	—
279	C₂×C₄	C₆₇⋊#95	1	2	yes	—
280	C₂×C₄	C₆₇⋊#95	1	2	yes	—
281	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
282	C₂×C₄	#14523953	1	2	yes	—
283	C₂×C₄	#14523953	1	2	yes	—
284	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
285	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
286	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
287	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
288	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
289	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
290	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
291	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
297	C₈	—	134	4	no	—
298	C₈	C₆₇⋊#95	1	2	yes	—
299	C₈	C₆₇⋊#95	1	2	yes	—
300	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
301	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
302	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
303	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	#16863150	1	2	yes	—
309	C₂²	#16863150	1	2	yes	—
310	C₂²	#16863177	1	4	yes	—
311	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
312	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
313	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
314	C₄	—	268	4	no	—
315	C₄	#16863150	1	2	yes	—
316	C₄	#16863150	1	2	yes	—
317	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
318	C₄	C₂×#14523953	1	1	yes	characteristic
319	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
320	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	—	2	1	no	—
324	C₂	#22558292	1	4	yes	direct product factor
325	C₂	C₂×#16863150	1	1	yes	characteristic
326	C₂	C₂×#16863177	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	(C₂×D₆₇)⋊C₄
commutator	C₂²×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂³×C₆₇	C₂²⋊C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#1377
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558292	C₂	direct (N × H)
9. C₂×#14523983	C₄	semidirect
21. C₂×M₃₂×C₆₇	C₄	semidirect
38. #14523983	C₂×C₄	semidirect
58. M₃₂×C₆₇	C₂×C₄	semidirect
161. C₂×C₆₇	#1377	semidirect
168. C₆₇	C₂×#1377	semidirect
178. C₂×M₃₂	Dic₆₇	semidirect
216. M₃₂	C₂×Dic₆₇	semidirect
324. C₂	#22558292	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
C₂²×#14523953	2
#22558292	2
C₂²×M₃₂×C₆₇	2
C₂×#1377	67

Subgroups — C2×#22558292 #30770689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558292 #30770689