Subgroups — C₆₇×#21976 #30701227

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1940	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1850	C₂	1	4	yes	maximal
6	C₆₇×#1842	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1831	C₂	1	2	yes	maximal, complemented
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×#117×C₆₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#432×C₆₇	—	2	4	no	—
12	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#97×C₆₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
25	#376×C₆₇	C₄	1	2	yes	—
26	C₄³×C₆₇	C₂²	1	2	yes	—
27	#336×C₆₇	C₂²	1	8	yes	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	—	2	4	no	—
32	C₂²×C₈×C₆₇	—	4	4	no	—
33	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	#117×C₆₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	D₄	1	4	yes	—
46	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₆₇	—	2	4	no	—
51	C₄⋊C₈×C₆₇	—	2	4	no	—
52	C₄⋊C₈×C₆₇	—	2	2	no	—
53	C₄⋊C₈×C₆₇	—	2	2	no	—
54	C₄⋊C₈×C₆₇	—	4	8	no	—
55	C₂²⋊C₈×C₆₇	—	2	4	no	—
56	#97×C₆₇	—	2	4	no	—
57	#95×C₆₇	C₂×C₄	1	2	yes	—
58	#95×C₆₇	C₂×C₄	1	2	yes	—
59	#95×C₆₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	4	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	8	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	4	8	no	—
84	C₄⋊C₄×C₆₇	—	2	8	no	—
85	C₂²⋊C₄×C₆₇	—	2	4	no	—
86	C₂²⋊C₄×C₆₇	M₁₆	1	8	yes	—
87	C₄²×C₆₇	—	2	8	no	—
88	C₄²×C₆₇	—	2	4	no	—
89	C₄²×C₆₇	—	2	4	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
93	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₄²×C₆₇	C₂×D₄	1	2	yes	—
96	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
97	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
102	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	8	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	8	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
123	C₂×C₄×C₆₇	#131	1	4	yes	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
128	C₈×C₆₇	—	4	2	no	—
129	C₈×C₆₇	—	4	4	no	—
130	C₈×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	8	no	—
134	C₂²×C₆₇	#423	1	4	yes	—
135	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₆₇	#520	1	2	yes	—
137	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
138	C₂²×C₆₇	C₂×#131	1	2	yes	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	2	8	no	—
144	C₄×C₆₇	—	4	4	no	—
145	C₄×C₆₇	#424	1	4	yes	—
146	C₄×C₆₇	#535	1	1	yes	characteristic
147	C₄×C₆₇	#536	1	1	yes	characteristic
148	C₄×C₆₇	#544	1	2	yes	—
149	#21976	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₆₇	—	2	4	no	—
151	C₂×C₆₇	#2317	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
153	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
154	C₂×C₆₇	#2990	1	4	yes	—
155	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
156	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
157	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
158	#1940	C₂×C₆₇	1	4	yes	complemented
159	#1850	C₂×C₆₇	1	4	yes	—
160	#1842	C₂×C₆₇	1	2	yes	complemented
161	#1831	C₂×C₆₇	1	2	yes	complemented
162	C₆₇	#21976	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
164	C₂×#117	C₄×C₆₇	1	2	yes	—
165	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
166	#432	—	2	4	no	—
167	C₂×C₄⋊C₈	—	2	4	no	—
168	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
174	C₂×#97	C₂²×C₆₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
177	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
178	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
179	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
180	#376	C₄×C₆₇	1	2	yes	—
181	C₄³	C₂²×C₆₇	1	2	yes	—
182	#336	C₂²×C₆₇	1	8	yes	—
183	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	4	4	no	—
188	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
189	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
190	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
191	#117	—	2	8	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₆₇	1	4	yes	—
201	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
202	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
203	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	4	8	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	#97	—	2	4	no	—
212	#95	C₂×C₄×C₆₇	1	2	yes	—
213	#95	C₂×C₄×C₆₇	1	2	yes	—
214	#95	C₂×C₄×C₆₇	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
221	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
228	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	M₁₆×C₆₇	1	8	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
248	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
249	C₄²	C₂×D₄×C₆₇	1	2	yes	—
250	C₄²	C₂×D₄×C₆₇	1	2	yes	—
251	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
252	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
256	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
257	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
277	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
278	C₂×C₄	#131×C₆₇	1	4	yes	—
279	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
281	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
282	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
283	C₈	—	4	4	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	8	no	—
289	C₂²	#423×C₆₇	1	4	yes	—
290	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
291	C₂²	#520×C₆₇	1	2	yes	—
292	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
293	C₂²	C₂×#131×C₆₇	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	8	no	—
299	C₄	—	4	4	no	—
300	C₄	#424×C₆₇	1	4	yes	—
301	C₄	#535×C₆₇	1	1	yes	characteristic
302	C₄	#536×C₆₇	1	1	yes	characteristic
303	C₄	#544×C₆₇	1	2	yes	—
304	C₂	—	2	4	no	—
305	C₂	C₆₇×#2317	1	1	yes	characteristic
306	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
307	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
308	C₂	C₆₇×#2990	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1940	C₂	semidirect
6. C₆₇×#1842	C₂	semidirect
7. C₆₇×#1831	C₂	semidirect
149. #21976	C₆₇	direct (N × H)
158. #1940	C₂×C₆₇	semidirect
160. #1842	C₂×C₆₇	semidirect
161. #1831	C₂×C₆₇	semidirect
162. C₆₇	#21976	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×#432×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₄×#117×C₆₇	2
C₆₇×#1940	2
C₆₇×#1850	2
C₆₇×#1842	2
C₆₇×#1831	2
#21976	67

Subgroups — C67×#21976 #30701227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21976 #30701227