Subgroups — C₂×#347×D₆₇ #31008483

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₇	C₂	1	2	yes	maximal
2	C₂×#97×D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863174	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863173	C₂	1	2	yes	maximal, complemented
5	#347×D₆₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#347×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₆₇	C₄	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	2	yes	—
11	C₂×#14523982	C₄	1	4	yes	complemented
12	M₃₂×D₆₇	C₄	1	8	yes	complemented
13	M₃₂×D₆₇	C₂²	1	8	yes	—
14	C₂×C₁₆×D₆₇	C₄	1	4	yes	complemented
15	C₂×#14523954	C₄	1	1	yes	characteristic
16	C₂×#14523953	C₄	1	1	yes	characteristic
17	C₂×#14523953	C₂²	1	1	yes	characteristic
18	#97×D₆₇	C₄	1	4	yes	—
19	#97×D₆₇	C₂²	1	4	yes	—
20	#16863174	C₂²	1	4	yes	complemented
21	#16863173	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
24	#347×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₆₇	—	2	1	no	—
26	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₆₇	—	2	2	no	—
30	C₂×C₈×D₆₇	—	2	1	no	—
31	C₂×C₈×D₆₇	—	2	1	no	—
32	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₆₇	C₂³	1	2	yes	—
38	C₂×C₄×Dic₆₇	—	2	1	no	—
39	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×D₆₇	—	2	2	no	—
41	C₄²×D₆₇	C₂×C₄	1	4	yes	—
42	#14523983	C₂×C₄	1	4	yes	complemented
43	#14523983	C₂³	1	4	yes	—
44	#14523982	C₂×C₄	1	16	yes	complemented
45	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
46	C₁₆×D₆₇	C₂×C₄	1	16	yes	complemented
47	#14523954	C₂×C₄	1	4	yes	—
48	#14523953	C₂×C₄	1	2	yes	—
49	#14523953	C₂×C₄	1	4	yes	—
50	#14523953	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
55	M₃₂×C₆₇	C₂³	1	4	yes	—
56	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
57	#97×C₆₇	C₂×C₄	1	2	yes	—
58	#97×C₆₇	C₂³	1	2	yes	—
59	C₂³×D₆₇	M₁₆	1	1	yes	characteristic
60	C₂²×Dic₆₇	—	2	1	no	—
61	C₂²×Dic₆₇	M₁₆	1	1	yes	characteristic
62	C₂×C₄×D₆₇	—	2	4	no	—
63	C₂×C₄×D₆₇	—	4	4	no	—
64	C₂×C₄×D₆₇	C₄²	1	2	yes	—
65	C₂×C₄×D₆₇	C₄²	1	1	yes	characteristic
66	C₂×C₄×D₆₇	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
68	C₂×C₄×D₆₇	M₁₆	1	1	yes	characteristic
69	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
70	C₂×C₄×D₆₇	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
75	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
76	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
78	C₈×D₆₇	—	2	4	no	—
79	C₈×D₆₇	—	2	4	no	—
80	C₈×D₆₇	C₄²	1	4	yes	—
81	C₈×D₆₇	C₄²	1	4	yes	—
82	C₈×D₆₇	C₂²×C₄	1	4	yes	—
83	C₈×D₆₇	C₂²×C₄	1	4	yes	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	4	no	—
86	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
87	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
88	C₂²×C₄×C₆₇	—	2	1	no	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
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	2	yes	—
96	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
99	C₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
100	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₆₇	—	2	4	no	—
102	C₂²×D₆₇	#97	1	1	yes	characteristic
103	C₂²×D₆₇	#97	1	2	yes	—
104	C₂²×D₆₇	#97	1	1	yes	characteristic
105	C₂²×D₆₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₆₇	—	2	2	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	#97	1	2	yes	—
117	C₂×Dic₆₇	#97	1	1	yes	characteristic
118	C₂×Dic₆₇	#97	1	1	yes	characteristic
119	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
120	C₄×D₆₇	—	4	8	no	—
121	C₄×D₆₇	#97	1	2	yes	—
122	C₄×D₆₇	#97	1	2	yes	—
123	C₄×D₆₇	#97	1	2	yes	—
124	C₄×D₆₇	#97	1	2	yes	—
125	C₄×D₆₇	C₂×C₄²	1	2	yes	—
126	C₄×D₆₇	C₂×C₄²	1	2	yes	—
127	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
128	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	C₂×C₄²	1	2	yes	—
131	C₈×C₆₇	C₂³×C₄	1	2	yes	—
132	C₆₇⋊C₈	—	2	2	no	—
133	C₆₇⋊C₈	C₂×C₄²	1	4	yes	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
137	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
138	C₂×D₆₇	—	2	2	no	—
139	C₂×D₆₇	—	2	4	no	—
140	C₂×D₆₇	—	2	4	no	—
141	C₂×D₆₇	#347	1	4	yes	direct product factor
142	C₂×D₆₇	C₂×#97	1	2	yes	—
143	C₂×D₆₇	C₂×#97	1	2	yes	—
144	C₄×C₆₇	—	4	2	no	—
145	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
147	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
148	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
149	Dic₆₇	—	4	2	no	—
150	Dic₆₇	C₂×#97	1	2	yes	—
151	Dic₆₇	C₂×#97	1	2	yes	—
152	C₂²×#347	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	C₂×#347	1	2	yes	complemented
156	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
157	D₆₇	—	2	2	no	—
158	D₆₇	C₂×#347	1	4	yes	direct product factor
159	C₂²×M₃₂	—	67	2	no	—
160	C₂²×#97	—	67	1	no	—
161	C₂×#347	—	67	2	no	—
162	C₂×#347	—	67	1	no	—
163	C₂×#347	—	67	8	no	—
164	C₂×#347	D₆₇	1	1	yes	characteristic, direct product factor
165	C₆₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₈	—	67	1	no	—
167	C₂³×C₈	—	67	1	no	—
168	C₂²×C₄²	—	67	1	no	—
169	C₂×M₃₂	—	67	2	no	—
170	C₂×M₃₂	—	67	8	no	—
171	C₂×M₃₂	—	67	8	no	—
172	C₂×M₃₂	—	67	4	no	—
173	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
174	C₂²×C₁₆	—	67	4	no	—
175	C₂×#97	—	67	1	no	—
176	C₂×#97	—	67	1	no	—
177	C₂×#97	—	67	1	no	—
178	C₂×#97	—	67	4	no	—
179	C₂×#97	—	67	4	no	—
180	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
181	#347	—	67	8	no	—
182	#347	—	67	4	no	—
183	#347	C₂×D₆₇	1	4	yes	direct product factor
184	C₂³×C₄	—	67	1	no	—
185	C₂³×C₄	—	134	1	no	—
186	C₂²×C₈	—	67	1	no	—
187	C₂²×C₈	—	67	2	no	—
188	C₂²×C₈	—	67	2	no	—
189	C₂²×C₈	—	67	4	no	—
190	C₂²×C₈	—	67	2	no	—
191	C₂²×C₈	—	67	2	no	—
192	C₂²×C₈	—	67	1	no	—
193	C₂²×C₈	—	67	4	no	—
194	C₂²×C₈	—	134	1	no	—
195	C₂²×C₈	—	134	1	no	—
196	C₂²×C₈	—	134	2	no	—
197	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₂×C₄²	—	67	1	no	—
200	C₂×C₄²	—	67	4	no	—
201	C₂×C₄²	—	134	2	no	—
202	C₂×C₄²	—	134	1	no	—
203	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
204	M₃₂	—	67	4	no	—
205	M₃₂	—	67	4	no	—
206	M₃₂	—	67	16	no	—
207	M₃₂	C₄×D₆₇	1	4	yes	complemented
208	M₃₂	C₂²×D₆₇	1	4	yes	—
209	C₂×C₁₆	—	67	4	no	—
210	C₂×C₁₆	—	67	16	no	—
211	C₂×C₁₆	C₄×D₆₇	1	4	yes	complemented
212	#97	—	67	4	no	—
213	#97	—	67	2	no	—
214	#97	—	67	2	no	—
215	#97	—	67	4	no	—
216	#97	C₄×D₆₇	1	2	yes	—
217	#97	C₂²×D₆₇	1	2	yes	—
218	C₂⁴	—	67	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	268	4	no	—
221	C₂²×C₄	—	67	1	no	—
222	C₂²×C₄	—	67	2	no	—
223	C₂²×C₄	—	67	1	no	—
224	C₂²×C₄	—	67	1	no	—
225	C₂²×C₄	—	67	2	no	—
226	C₂²×C₄	—	67	1	no	—
227	C₂²×C₄	—	67	2	no	—
228	C₂²×C₄	—	67	1	no	—
229	C₂²×C₄	—	67	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	—	134	1	no	—
232	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	67	2	no	—
236	C₂×C₈	—	67	2	no	—
237	C₂×C₈	—	67	2	no	—
238	C₂×C₈	—	67	4	no	—
239	C₂×C₈	—	67	4	no	—
240	C₂×C₈	—	67	4	no	—
241	C₂×C₈	—	67	4	no	—
242	C₂×C₈	—	67	2	no	—
243	C₂×C₈	—	134	4	no	—
244	C₂×C₈	—	134	4	no	—
245	C₂×C₈	—	134	1	no	—
246	C₂×C₈	—	134	1	no	—
247	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
251	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
252	C₄²	—	2	1	no	—
253	C₄²	—	67	2	no	—
254	C₄²	—	134	4	no	—
255	C₄²	—	134	1	no	—
256	C₄²	C₂×C₄×D₆₇	1	2	yes	—
257	C₁₆	—	67	8	no	—
258	C₁₆	C₂×C₄×D₆₇	1	8	yes	complemented
259	C₂³	—	67	1	no	—
260	C₂³	—	67	2	no	—
261	C₂³	—	67	1	no	—
262	C₂³	—	67	2	no	—
263	C₂³	—	134	4	no	—
264	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	268	8	no	—
268	C₂×C₄	—	268	2	no	—
269	C₂×C₄	—	67	2	no	—
270	C₂×C₄	—	67	1	no	—
271	C₂×C₄	—	67	1	no	—
272	C₂×C₄	—	67	2	no	—
273	C₂×C₄	—	67	2	no	—
274	C₂×C₄	—	67	2	no	—
275	C₂×C₄	—	67	2	no	—
276	C₂×C₄	—	67	2	no	—
277	C₂×C₄	—	67	2	no	—
278	C₂×C₄	—	67	2	no	—
279	C₂×C₄	—	67	2	no	—
280	C₂×C₄	—	67	2	no	—
281	C₂×C₄	—	134	2	no	—
282	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
283	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
285	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	67	4	no	—
290	C₈	—	134	2	no	—
291	C₈	C₄²×D₆₇	1	2	yes	—
292	C₈	C₂²×C₄×D₆₇	1	2	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	67	4	no	—
295	C₂²	—	67	2	no	—
296	C₂²	—	67	2	no	—
297	C₂²	—	134	4	no	—
298	C₂²	—	134	4	no	—
299	C₂²	—	134	2	no	—
300	C₂²	#97×D₆₇	1	1	yes	characteristic
301	C₂²	#97×D₆₇	1	1	yes	characteristic
302	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
303	C₄	—	4	2	no	—
304	C₄	—	268	2	no	—
305	C₄	—	67	2	no	—
306	C₄	—	67	2	no	—
307	C₄	#97×D₆₇	1	1	yes	characteristic
308	C₄	#97×D₆₇	1	1	yes	characteristic
309	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
310	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	67	4	no	—
314	C₂	—	134	2	no	—
315	C₂	#347×D₆₇	1	2	yes	direct product factor
316	C₂	C₂×#97×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#347×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×D₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#347	D₆₇
hypocenter	C₆₇	C₂²×#347
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×#16863174	C₂	semidirect
4. C₂×#16863173	C₂	semidirect
5. #347×D₆₇	C₂	direct (N × H)
6. C₂×#347×C₆₇	C₂	semidirect
11. C₂×#14523982	C₄	semidirect
12. M₃₂×D₆₇	C₄	semidirect
14. C₂×C₁₆×D₆₇	C₄	semidirect
20. #16863174	C₂²	semidirect
21. #16863173	C₂²	semidirect
24. #347×C₆₇	C₂²	semidirect
42. #14523983	C₂×C₄	semidirect
44. #14523982	C₂×C₄	semidirect
45. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
46. C₁₆×D₆₇	C₂×C₄	semidirect
54. M₃₂×C₆₇	C₂×C₄	semidirect
56. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
87. C₆₇⋊C₁₆	C₂²×C₄	semidirect
99. C₁₆×C₆₇	C₂²×C₄	semidirect
141. C₂×D₆₇	#347	direct (N × H)
155. C₂×C₆₇	C₂×#347	semidirect
158. D₆₇	C₂×#347	direct (N × H)
164. C₂×#347	D₆₇	direct (N × H)
165. C₆₇	C₂²×#347	semidirect
183. #347	C₂×D₆₇	direct (N × H)
207. M₃₂	C₄×D₆₇	semidirect
211. C₂×C₁₆	C₄×D₆₇	semidirect
258. C₁₆	C₂×C₄×D₆₇	semidirect
315. C₂	#347×D₆₇	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×M₃₂×D₆₇	2
C₂×#97×D₆₇	2
C₂×#16863174	2
C₂×#16863173	2
#347×D₆₇	2
C₂×#347×C₆₇	2
C₂²×#347	67

Subgroups — C2×#347×D67 #31008483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#347×D₆₇ #31008483