Subgroups — C₆₇×#22020 #30701271

Order 17152.

706 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1985	C₂	1	2	yes	maximal
3	C₆₇×#1940	C₂	1	2	yes	maximal
4	C₆₇×#1916	C₂	1	4	yes	maximal
5	C₆₇×#1839	C₂	1	2	yes	maximal
6	C₆₇×#1831	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×#97×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	C₄³×C₆₇	C₂²	1	2	yes	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	#336×C₆₇	C₂²	1	4	yes	—
24	#336×C₆₇	C₂²	1	1	yes	characteristic
25	#336×C₆₇	C₂²	1	1	yes	characteristic
26	#336×C₆₇	C₂²	1	2	yes	—
27	#336×C₆₇	C₂²	1	2	yes	—
28	#336×C₆₇	C₂²	1	4	yes	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₄×Q₈×C₆₇	—	2	8	no	—
34	C₄×Q₈×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	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	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	—	2	4	no	—
52	C₄⋊C₈×C₆₇	—	4	4	no	—
53	#97×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	—	4	2	no	—
55	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
56	#95×C₆₇	C₂×C₄	1	2	yes	—
57	#95×C₆₇	C₂×C₄	1	2	yes	—
58	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×Q₈×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	M₁₆	1	8	yes	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	—	2	4	no	—
88	C₄²×C₆₇	—	2	4	no	—
89	C₄²×C₆₇	—	2	4	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	#41	1	4	yes	—
97	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	Q₈×C₆₇	—	2	8	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	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	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	#117	1	2	yes	—
116	C₂×C₄×C₆₇	#117	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₆₇	#131	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₆₇	—	4	2	no	—
126	C₈×C₆₇	—	4	4	no	—
127	C₈×C₆₇	—	4	2	no	—
128	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
129	C₂²×C₆₇	#519	1	1	yes	characteristic
130	C₂²×C₆₇	#520	1	1	yes	characteristic
131	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
132	C₂²×C₆₇	C₂×#131	1	2	yes	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	4	no	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	4	2	no	—
140	C₄×C₆₇	—	4	2	no	—
141	C₄×C₆₇	#541	1	2	yes	—
142	C₄×C₆₇	#542	1	2	yes	—
143	#22020	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₆₇	#2320	1	1	yes	characteristic
145	C₂×C₆₇	#2920	1	1	yes	characteristic
146	C₂×C₆₇	#2922	1	1	yes	characteristic
147	C₂×C₆₇	#2984	1	2	yes	—
148	C₂×C₆₇	#2990	1	2	yes	—
149	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
150	#1985	C₂×C₆₇	1	2	yes	—
151	#1940	C₂×C₆₇	1	2	yes	—
152	#1916	C₂×C₆₇	1	4	yes	—
153	#1839	C₂×C₆₇	1	2	yes	—
154	#1831	C₂×C₆₇	1	2	yes	—
155	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
156	#1745	C₂×C₆₇	1	1	yes	characteristic
157	C₆₇	#22020	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	—
159	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
162	C₂×#97	C₂²×C₆₇	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
164	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
165	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
166	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
167	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
169	C₄³	C₂²×C₆₇	1	2	yes	—
170	C₄³	C₂²×C₆₇	1	1	yes	characteristic
171	#336	C₂²×C₆₇	1	1	yes	characteristic
172	#336	C₂²×C₆₇	1	2	yes	—
173	#336	C₂²×C₆₇	1	4	yes	—
174	#336	C₂²×C₆₇	1	1	yes	characteristic
175	#336	C₂²×C₆₇	1	4	yes	—
176	#336	C₂²×C₆₇	1	4	yes	—
177	#336	C₂²×C₆₇	1	2	yes	—
178	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
181	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
189	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
193	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
195	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
199	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	4	4	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	4	2	no	—
204	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
205	#95	C₂×C₄×C₆₇	1	2	yes	—
206	#95	C₂×C₄×C₆₇	1	2	yes	—
207	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	M₁₆×C₆₇	1	8	yes	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	2	no	—
245	C₄²	#41×C₆₇	1	4	yes	—
246	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	#117×C₆₇	1	2	yes	—
265	C₂×C₄	#117×C₆₇	1	2	yes	—
266	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
267	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
268	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
269	C₂×C₄	#131×C₆₇	1	4	yes	—
270	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
271	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
272	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
278	C₂²	#519×C₆₇	1	1	yes	characteristic
279	C₂²	#520×C₆₇	1	1	yes	characteristic
280	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
281	C₂²	C₂×#131×C₆₇	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	#541×C₆₇	1	2	yes	—
291	C₄	#542×C₆₇	1	2	yes	—
292	C₂	C₆₇×#2320	1	1	yes	characteristic
293	C₂	C₆₇×#2920	1	1	yes	characteristic
294	C₂	C₆₇×#2922	1	1	yes	characteristic
295	C₂	C₆₇×#2984	1	2	yes	—
296	C₂	C₆₇×#2990	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂³×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
143. #22020	C₆₇	direct (N × H)
157. C₆₇	#22020	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₆₇	2
C₆₇×#1985	2
C₆₇×#1940	2
C₆₇×#1916	2
C₆₇×#1839	2
C₆₇×#1831	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1745	2
#22020	67

Subgroups — C67×#22020 #30701271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22020 #30701271