Subgroups — C₂×#498×D₆₇ #31725417

Order 17152.

2398 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	C₂×C₄⋊Q₈×D₆₇	C₂	1	2	yes	maximal
2	C₂×#16863794	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863793	C₂	1	4	yes	maximal, complemented
4	C₂×#16863792	C₂	1	2	yes	maximal, complemented
5	#498×D₆₇	C₂	1	16	yes	maximal, direct product factor
6	C₂×#106×D₆₇	C₂	1	4	yes	maximal
7	C₂×C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₆₇	—	2	2	no	—
10	C₂²×C₈×D₆₇	C₂²	1	2	yes	—
11	C₂×#14524043	C₂²	1	2	yes	—
12	C₂×#14524042	—	2	2	no	—
13	C₂×#14524042	C₂²	1	4	yes	—
14	C₂×#14524041	—	2	2	no	—
15	C₄⋊Q₈×D₆₇	—	2	8	no	—
16	C₄⋊Q₈×D₆₇	C₂²	1	16	yes	—
17	C₂×C₄⋊C₄×D₆₇	—	2	2	no	—
18	C₂×C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
19	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
20	#16863794	C₂²	1	8	yes	complemented
21	#16863793	C₂²	1	32	yes	complemented
22	#16863792	C₂²	1	16	yes	complemented
23	C₂×#14523975	C₂²	1	4	yes	—
24	C₂×#14523974	C₂²	1	8	yes	—
25	#106×D₆₇	C₂²	1	32	yes	—
26	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
28	C₄×C₈×D₆₇	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
30	#498×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#106×C₆₇	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₃₄	—	2	4	no	—
34	C₂²×Dic₁₃₄	—	2	2	no	—
35	C₂×Q₈×D₆₇	—	2	16	no	—
36	C₂×Q₈×D₆₇	—	4	16	no	—
37	C₂²×C₄×D₆₇	—	2	4	no	—
38	C₂²×C₄×D₆₇	D₄	1	2	yes	—
39	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₂×C₈×D₆₇	Q₈	1	16	yes	complemented
42	C₂×C₈×D₆₇	C₂³	1	8	yes	—
43	#14524043	—	2	4	no	—
44	#14524043	C₂³	1	8	yes	—
45	#14524042	—	2	8	no	—
46	#14524042	—	2	16	no	—
47	#14524042	C₂³	1	16	yes	—
48	#14524041	—	2	16	no	—
49	C₂×C₄⋊Dic₆₇	—	2	2	no	—
50	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
51	C₂×Dic₆₇⋊C₄	—	2	4	no	—
52	C₂×Dic₆₇⋊C₄	—	2	4	no	—
53	C₄⋊C₄×D₆₇	—	2	16	no	—
54	C₄⋊C₄×D₆₇	—	2	8	no	—
55	C₄⋊C₄×D₆₇	C₂³	1	16	yes	—
56	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
57	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
58	C₄²×D₆₇	D₄	1	4	yes	—
59	C₄²×D₆₇	C₂³	1	4	yes	—
60	#14523975	C₂³	1	16	yes	—
61	#14523974	C₂³	1	32	yes	—
62	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
63	C₈×Dic₆₇	C₂³	1	8	yes	—
64	C₂²×Q₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₆₇	—	2	4	no	—
67	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	C₂³	1	16	yes	—
72	C₄×C₈×C₆₇	C₂³	1	4	yes	—
73	C₂³×D₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×Dic₁₃₄	—	2	8	no	—
75	C₂×Dic₁₃₄	—	2	16	no	—
76	C₂×Dic₁₃₄	—	4	8	no	—
77	C₂×Dic₁₃₄	—	4	16	no	—
78	Q₈×D₆₇	—	4	64	no	—
79	C₂²×Dic₆₇	—	2	4	no	—
80	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₆₇	—	2	16	no	—
83	C₂×C₄×D₆₇	—	4	16	no	—
84	C₂×C₄×D₆₇	SD₁₆	1	16	yes	—
85	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×D₆₇	C₂×D₄	1	8	yes	—
87	C₂×C₄×D₆₇	C₂×Q₈	1	8	yes	—
88	C₂×C₄×D₆₇	C₂⁴	1	2	yes	—
89	C₂×C₆₇⋊C₈	C₂×Q₈	1	8	yes	complemented
90	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
91	C₈×D₆₇	C₂×Q₈	1	32	yes	complemented
92	C₄⋊Dic₆₇	—	2	8	no	—
93	C₄⋊Dic₆₇	—	2	4	no	—
94	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
95	Dic₆₇⋊C₄	—	2	16	no	—
96	Dic₆₇⋊C₄	—	2	16	no	—
97	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
98	C₄×Dic₆₇	C₂×D₄	1	8	yes	—
99	C₄×Dic₆₇	C₂⁴	1	2	yes	—
100	C₂×Q₈×C₆₇	—	2	8	no	—
101	C₂×Q₈×C₆₇	—	4	8	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	complemented
106	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
107	C₄⋊C₄×C₆₇	—	2	8	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
110	C₄²×C₆₇	C₂×D₄	1	2	yes	—
111	C₄²×C₆₇	C₂⁴	1	2	yes	—
112	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂²×D₆₇	C₄⋊Q₈	1	4	yes	—
114	C₂²×D₆₇	C₂×SD₁₆	1	8	yes	—
115	C₂²×D₆₇	C₂²×D₄	1	2	yes	—
116	Q₈×C₆₇	—	4	16	no	—
117	C₂×C₄×C₆₇	—	2	8	no	—
118	C₂×C₄×C₆₇	—	4	8	no	—
119	C₂×C₄×C₆₇	C₂×SD₁₆	1	8	yes	—
120	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
122	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
124	C₂×Dic₆₇	—	2	8	no	—
125	C₂×Dic₆₇	—	4	8	no	—
126	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
127	C₂×Dic₆₇	C₂×SD₁₆	1	8	yes	—
128	C₂×Dic₆₇	C₂²×D₄	1	2	yes	—
129	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
130	C₄×D₆₇	—	4	32	no	—
131	C₄×D₆₇	C₄⋊Q₈	1	8	yes	—
132	C₄×D₆₇	C₂×SD₁₆	1	32	yes	—
133	C₄×D₆₇	C₂²×Q₈	1	8	yes	—
134	Dic₁₃₄	—	4	16	no	—
135	Dic₁₃₄	—	4	32	no	—
136	C₈×C₆₇	C₂²×Q₈	1	8	yes	complemented
137	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	complemented
138	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
139	C₂²×C₆₇	C₂²×SD₁₆	1	4	yes	—
140	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
141	C₂×D₆₇	#498	1	16	yes	direct product factor
142	C₂×D₆₇	C₂×C₄⋊Q₈	1	4	yes	—
143	C₂×D₆₇	C₂²×SD₁₆	1	8	yes	—
144	C₄×C₆₇	—	4	8	no	—
145	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
146	C₄×C₆₇	C₂²×SD₁₆	1	8	yes	—
147	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
148	Dic₆₇	—	4	8	no	—
149	Dic₆₇	C₂×C₄⋊Q₈	1	4	yes	—
150	Dic₆₇	C₂²×SD₁₆	1	8	yes	—
151	C₂²×#498	—	67	1	no	maximal, Sylow
152	C₂×C₆₇	C₂×#498	1	4	yes	complemented
153	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₆₇	C₂³×SD₁₆	1	2	yes	—
155	D₆₇	C₂×#498	1	8	yes	direct product factor
156	C₂²×C₄⋊Q₈	—	67	2	no	—
157	C₂×#498	—	67	16	no	—
158	C₂×#498	—	67	4	no	—
159	C₂×#498	—	67	1	no	—
160	C₂×#498	—	67	2	no	—
161	C₂×#498	D₆₇	1	1	yes	characteristic, direct product factor
162	C₂²×#106	—	67	4	no	—
163	C₂²×C₄×C₈	—	67	1	no	—
164	C₆₇	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
165	C₂³×Q₈	—	134	2	no	—
166	C₂³×C₈	—	67	2	no	—
167	C₂×C₄⋊Q₈	—	67	16	no	—
168	C₂×C₄⋊Q₈	—	67	4	no	—
169	C₂×C₄⋊Q₈	—	67	2	no	—
170	C₂×C₄⋊Q₈	—	134	2	no	—
171	C₂×C₄⋊Q₈	—	134	8	no	—
172	C₂×C₄⋊Q₈	—	134	2	no	—
173	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
174	C₂²×C₄⋊C₄	—	67	4	no	—
175	C₂²×C₄⋊C₄	—	134	2	no	—
176	C₂²×C₄²	—	67	1	no	—
177	#498	—	67	16	no	—
178	#498	—	67	32	no	—
179	#498	—	67	8	no	—
180	#498	C₂×D₆₇	1	8	yes	direct product factor
181	C₂×#106	—	67	4	no	—
182	C₂×#106	—	67	8	no	—
183	C₂×#106	—	67	32	no	—
184	C₂×#106	C₂×D₆₇	1	4	yes	—
185	C₂×C₄×C₈	—	67	2	no	—
186	C₂×C₄×C₈	—	67	1	no	—
187	C₂×C₄×C₈	—	67	8	no	—
188	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	268	16	no	—
191	C₂²×Q₈	—	134	4	no	—
192	C₂²×Q₈	—	134	16	no	—
193	C₂²×Q₈	—	134	2	no	—
194	C₂³×C₄	—	67	2	no	—
195	C₂³×C₄	—	67	1	no	—
196	C₂³×C₄	—	134	4	no	—
197	C₂²×C₈	—	67	2	no	—
198	C₂²×C₈	—	67	16	no	—
199	C₂²×C₈	—	67	8	no	—
200	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	—	67	16	no	—
203	C₄⋊Q₈	—	67	8	no	—
204	C₄⋊Q₈	—	134	16	no	—
205	C₄⋊Q₈	—	134	8	no	—
206	C₄⋊Q₈	—	134	16	no	—
207	C₄⋊Q₈	—	134	4	no	—
208	C₄⋊Q₈	C₂²×D₆₇	1	8	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	67	4	no	—
211	C₂×C₄⋊C₄	—	67	16	no	—
212	C₂×C₄⋊C₄	—	134	8	no	—
213	C₂×C₄⋊C₄	—	134	16	no	—
214	C₂×C₄⋊C₄	—	134	4	no	—
215	C₂×C₄⋊C₄	—	134	4	no	—
216	C₂×C₄⋊C₄	—	134	2	no	—
217	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
218	C₂×C₄²	—	67	2	no	—
219	C₂×C₄²	—	67	1	no	—
220	C₂×C₄²	—	67	4	no	—
221	C₂×C₄²	—	67	4	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	#106	—	67	16	no	—
224	#106	—	67	32	no	—
225	#106	C₂²×D₆₇	1	16	yes	—
226	C₄×C₈	—	67	8	no	—
227	C₄×C₈	—	67	4	no	—
228	C₄×C₈	C₂²×D₆₇	1	4	yes	—
229	C₂⁴	—	67	1	no	—
230	C₂×Q₈	—	2	8	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×Q₈	—	268	16	no	—
233	C₂×Q₈	—	268	64	no	—
234	C₂×Q₈	—	268	8	no	—
235	C₂×Q₈	—	134	16	no	—
236	C₂×Q₈	—	134	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	268	16	no	—
239	C₂²×C₄	—	67	1	no	—
240	C₂²×C₄	—	67	2	no	—
241	C₂²×C₄	—	67	16	no	—
242	C₂²×C₄	—	67	2	no	—
243	C₂²×C₄	—	67	8	no	—
244	C₂²×C₄	—	67	8	no	—
245	C₂²×C₄	—	67	2	no	—
246	C₂²×C₄	—	134	16	no	—
247	C₂²×C₄	—	134	4	no	—
248	C₂²×C₄	D₄×D₆₇	1	2	yes	—
249	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
250	C₂×C₈	—	67	8	no	—
251	C₂×C₈	—	67	4	no	—
252	C₂×C₈	—	67	32	no	—
253	C₂×C₈	Q₈×D₆₇	1	8	yes	complemented
254	C₂×C₈	C₂³×D₆₇	1	4	yes	—
255	C₄⋊C₄	—	2	8	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	67	8	no	—
258	C₄⋊C₄	—	134	16	no	—
259	C₄⋊C₄	—	134	16	no	—
260	C₄⋊C₄	—	134	4	no	—
261	C₄⋊C₄	—	134	8	no	—
262	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
263	C₄²	—	67	8	no	—
264	C₄²	—	67	2	no	—
265	C₄²	—	67	2	no	—
266	C₄²	D₄×D₆₇	1	2	yes	—
267	C₄²	C₂³×D₆₇	1	2	yes	—
268	C₂³	—	67	8	no	—
269	C₂³	—	67	4	no	—
270	C₂³	—	67	2	no	—
271	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
272	Q₈	—	4	16	no	—
273	Q₈	—	268	32	no	—
274	Q₈	—	268	16	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	268	32	no	—
278	C₂×C₄	—	268	8	no	—
279	C₂×C₄	—	67	8	no	—
280	C₂×C₄	—	67	4	no	—
281	C₂×C₄	—	67	2	no	—
282	C₂×C₄	—	67	4	no	—
283	C₂×C₄	—	67	32	no	—
284	C₂×C₄	—	67	8	no	—
285	C₂×C₄	—	67	8	no	—
286	C₂×C₄	—	134	8	no	—
287	C₂×C₄	SD₁₆×D₆₇	1	8	yes	—
288	C₂×C₄	C₂×D₄×D₆₇	1	4	yes	—
289	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂×Q₈×D₆₇	1	4	yes	—
291	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
292	C₈	—	67	8	no	—
293	C₈	C₂×Q₈×D₆₇	1	8	yes	complemented
294	C₂²	—	67	8	no	—
295	C₂²	—	67	4	no	—
296	C₂²	—	67	16	no	—
297	C₂²	C₄⋊Q₈×D₆₇	1	2	yes	—
298	C₂²	C₂×SD₁₆×D₆₇	1	4	yes	—
299	C₂²	C₂²×D₄×D₆₇	1	1	yes	characteristic
300	C₄	—	4	8	no	—
301	C₄	—	268	8	no	—
302	C₄	—	67	8	no	—
303	C₄	—	67	4	no	—
304	C₄	C₄⋊Q₈×D₆₇	1	2	yes	—
305	C₄	C₂×SD₁₆×D₆₇	1	8	yes	—
306	C₄	C₂²×Q₈×D₆₇	1	2	yes	—
307	C₂	—	67	8	no	—
308	C₂	#498×D₆₇	1	4	yes	direct product factor
309	C₂	C₂×C₄⋊Q₈×D₆₇	1	1	yes	characteristic
310	C₂	C₂²×SD₁₆×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#498×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂³×D₆₇	C₂×D₄
hypercenter	C₂×#498	D₆₇
hypocenter	C₆₇	C₂²×#498
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. C₂×#16863794	C₂	semidirect
3. C₂×#16863793	C₂	semidirect
4. C₂×#16863792	C₂	semidirect
5. #498×D₆₇	C₂	direct (N × H)
8. C₂×#498×C₆₇	C₂	semidirect
20. #16863794	C₂²	semidirect
21. #16863793	C₂²	semidirect
22. #16863792	C₂²	semidirect
30. #498×C₆₇	C₂²	semidirect
41. C₂×C₈×D₆₇	Q₈	semidirect
89. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
91. C₈×D₆₇	C₂×Q₈	semidirect
105. C₂×C₈×C₆₇	C₂×Q₈	semidirect
136. C₈×C₆₇	C₂²×Q₈	semidirect
137. C₆₇⋊C₈	C₂²×Q₈	semidirect
141. C₂×D₆₇	#498	direct (N × H)
152. C₂×C₆₇	C₂×#498	semidirect
155. D₆₇	C₂×#498	direct (N × H)
161. C₂×#498	D₆₇	direct (N × H)
164. C₆₇	C₂²×#498	semidirect
180. #498	C₂×D₆₇	direct (N × H)
253. C₂×C₈	Q₈×D₆₇	semidirect
293. C₈	C₂×Q₈×D₆₇	semidirect
308. C₂	#498×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×C₄⋊Q₈×D₆₇	2
C₂×#16863794	2
C₂×#16863793	2
C₂×#16863792	2
#498×D₆₇	2
C₂×#106×D₆₇	2
C₂×C₄×C₈×D₆₇	2
C₂×#498×C₆₇	2
C₂²×#498	67

Subgroups — C2×#498×D67 #31725417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#498×D₆₇ #31725417