Subgroups — C₆₇×#33484 #30712735

Order 17152.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2183	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2163	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#346×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
10	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	—
12	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₈×D₄×C₆₇	C₄	1	2	yes	—
15	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
16	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₆₇	C₂²	1	2	yes	—
18	#431×C₆₇	C₄	1	1	yes	characteristic
19	#431×C₆₇	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
23	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
24	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
25	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
26	#346×C₆₇	C₂²	1	1	yes	characteristic
27	#346×C₆₇	C₂²	1	2	yes	—
28	#346×C₆₇	C₂²	1	4	yes	—
29	#346×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
31	C₂×#41×C₆₇	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
37	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₆₇	C₈	1	2	yes	—
39	C₄×D₄×C₆₇	C₈	1	1	yes	characteristic
40	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	#117×C₆₇	C₈	1	1	yes	characteristic
43	#117×C₆₇	C₈	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₆₇	—	2	4	no	—
47	C₂×C₁₆×C₆₇	—	2	8	no	—
48	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
49	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
51	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	2	yes	—
65	#41×C₆₇	—	2	4	no	—
66	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂²×C₄	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	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
86	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
89	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
92	C₄²×C₆₇	C₂×C₈	1	2	yes	—
93	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₆₇	C₂×C₈	1	2	yes	—
95	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₆₇	—	2	4	no	—
100	C₁₆×C₆₇	—	2	2	no	—
101	C₁₆×C₆₇	#41	1	4	yes	complemented
102	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
104	Q₈×C₆₇	—	2	2	no	—
105	D₄×C₆₇	—	2	4	no	—
106	D₄×C₆₇	M₃₂	1	4	yes	complemented
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₆₇	—	2	2	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	#131	1	2	yes	—
129	C₈×C₆₇	C₂×#41	1	2	yes	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	1	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	C₂×M₃₂	1	4	yes	—
136	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
145	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
147	#33484	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₆₇	—	2	4	no	—
149	C₂×C₆₇	—	4	1	no	—
150	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
151	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₆₇	#3407	1	1	yes	characteristic
153	C₈×#41	C₂×C₆₇	1	1	yes	characteristic
154	#2183	C₂×C₆₇	1	2	yes	complemented
155	#2168	C₂×C₆₇	1	4	yes	complemented
156	D₄×C₁₆	C₂×C₆₇	1	2	yes	complemented
157	#2163	C₂×C₆₇	1	1	yes	characteristic, complemented
158	#2163	C₂×C₆₇	1	2	yes	complemented
159	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
160	C₂×#346	C₂×C₆₇	1	2	yes	complemented
161	C₆₇	#33484	1	1	yes	characteristic, direct product factor, Sylow
162	C₄×#41	C₄×C₆₇	1	1	yes	characteristic
163	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
164	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	—
165	C₈×Q₈	C₄×C₆₇	1	1	yes	characteristic
166	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
167	C₈×D₄	C₄×C₆₇	1	2	yes	—
168	C₈×D₄	C₄×C₆₇	1	1	yes	characteristic
169	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
170	C₈×D₄	C₂²×C₆₇	1	2	yes	—
171	#431	C₄×C₆₇	1	1	yes	characteristic
172	#431	C₄×C₆₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
175	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
176	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
177	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
178	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
179	#346	C₂²×C₆₇	1	1	yes	characteristic
180	#346	C₂²×C₆₇	1	2	yes	—
181	#346	C₂²×C₆₇	1	4	yes	—
182	#346	C₂²×C₆₇	1	1	yes	characteristic
183	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
184	C₂×#41	C₈×C₆₇	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
186	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
189	C₄×Q₈	C₈×C₆₇	1	1	yes	characteristic
190	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
191	C₄×D₄	C₈×C₆₇	1	2	yes	—
192	C₄×D₄	C₈×C₆₇	1	1	yes	characteristic
193	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
194	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
195	#117	C₈×C₆₇	1	1	yes	characteristic
196	#117	C₈×C₆₇	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
199	M₃₂	—	2	4	no	—
200	C₂×C₁₆	—	2	8	no	—
201	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
202	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
203	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
204	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
205	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
206	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
208	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
209	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
210	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
211	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
212	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
214	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₆₇	1	2	yes	—
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×C₈×C₆₇	1	1	yes	characteristic
220	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
221	C₂×D₄	C₂×C₈×C₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
223	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
239	C₄⋊C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
240	C₄⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
241	C₄⋊C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
243	C₂²⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
244	C₂²⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
245	C₄²	C₂×C₈×C₆₇	1	2	yes	—
246	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
247	C₄²	C₂×C₈×C₆₇	1	2	yes	—
248	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
249	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
250	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
251	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
252	C₁₆	—	2	4	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	#41×C₆₇	1	4	yes	complemented
255	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
256	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	M₃₂×C₆₇	1	4	yes	complemented
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
271	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
273	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
275	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	#131×C₆₇	1	2	yes	—
282	C₈	C₂×#41×C₆₇	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×M₃₂×C₆₇	1	4	yes	—
289	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	C₂×M₃₂×C₆₇	1	2	yes	—
298	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
299	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	4	1	no	—
302	C₂	C₈×#41×C₆₇	1	1	yes	characteristic
303	C₂	C₂²×M₃₂×C₆₇	1	1	yes	characteristic
304	C₂	C₆₇×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂×#41×C₆₇	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2183	C₂	semidirect
3. C₆₇×#2168	C₂	semidirect
4. D₄×C₁₆×C₆₇	C₂	semidirect
5. C₆₇×#2163	C₂	semidirect
6. C₆₇×#2163	C₂	semidirect
7. C₄×M₃₂×C₆₇	C₂	semidirect
8. C₂×#346×C₆₇	C₂	semidirect
101. C₁₆×C₆₇	#41	semidirect
106. D₄×C₆₇	M₃₂	semidirect
147. #33484	C₆₇	direct (N × H)
154. #2183	C₂×C₆₇	semidirect
155. #2168	C₂×C₆₇	semidirect
156. D₄×C₁₆	C₂×C₆₇	semidirect
157. #2163	C₂×C₆₇	semidirect
158. #2163	C₂×C₆₇	semidirect
159. C₄×M₃₂	C₂×C₆₇	semidirect
160. C₂×#346	C₂×C₆₇	semidirect
161. C₆₇	#33484	direct (N × H)
254. C₁₆	#41×C₆₇	semidirect
259. D₄	M₃₂×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈×#41×C₆₇	2
C₆₇×#2183	2
C₆₇×#2168	2
D₄×C₁₆×C₆₇	2
C₆₇×#2163	2
C₄×M₃₂×C₆₇	2
C₂×#346×C₆₇	2
#33484	67

Subgroups — C67×#33484 #30712735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33484 #30712735