Subgroups — C₆₇×#21914 #30701165

Order 17152.

838 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×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1917	C₂	1	4	yes	maximal, complemented
3	C₆₇×#1916	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1817	C₂	1	2	yes	maximal
5	C₆₇×#1797	C₂	1	2	yes	maximal
6	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#95×C₆₇	C₄	1	2	yes	—
22	#336×C₆₇	C₂²	1	2	yes	—
23	#336×C₆₇	C₂²	1	2	yes	—
24	#336×C₆₇	C₂²	1	4	yes	—
25	#336×C₆₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
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	2	yes	—
38	C₂×C₄²×C₆₇	—	2	4	no	—
39	C₂×C₄²×C₆₇	—	4	2	no	—
40	C₂×C₄²×C₆₇	—	4	2	no	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
49	C₄⋊C₈×C₆₇	—	4	4	no	—
50	C₂²⋊C₈×C₆₇	—	2	4	no	—
51	C₂²⋊C₈×C₆₇	—	2	4	no	—
52	#97×C₆₇	—	4	2	no	—
53	C₄×C₈×C₆₇	—	4	2	no	—
54	#95×C₆₇	C₂×C₄	1	2	yes	—
55	#95×C₆₇	C₂×C₄	1	2	yes	—
56	#95×C₆₇	C₂×C₄	1	2	yes	—
57	#95×C₆₇	C₂×C₄	1	2	yes	—
58	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	#41	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	M₁₆	1	8	yes	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	4	1	no	—
96	C₄²×C₆₇	—	4	2	no	—
97	C₄²×C₆₇	—	4	1	no	—
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	#117	1	2	yes	—
102	C₂³×C₆₇	#131	1	4	yes	—
103	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	—	2	8	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	1	no	—
109	C₂×C₄×C₆₇	—	2	1	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	8	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	4	8	no	—
123	C₂×C₄×C₆₇	—	4	8	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	#117	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
128	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
132	C₈×C₆₇	—	4	4	no	—
133	C₈×C₆₇	—	4	4	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	4	no	—
137	C₂²×C₆₇	—	2	4	no	—
138	C₂²×C₆₇	#433	1	4	yes	—
139	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
140	C₂²×C₆₇	#518	1	1	yes	characteristic
141	C₂²×C₆₇	#519	1	1	yes	characteristic
142	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
143	C₂²×C₆₇	C₂×#131	1	2	yes	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	2	4	no	—
146	C₄×C₆₇	—	2	4	no	—
147	C₄×C₆₇	—	2	4	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	4	2	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	#539	1	4	yes	—
152	#21914	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₆₇	—	2	4	no	—
154	C₂×C₆₇	#2313	1	1	yes	characteristic
155	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
156	C₂×C₆₇	#2920	1	1	yes	characteristic
157	C₂×C₆₇	#2974	1	2	yes	—
158	C₂×C₆₇	#2984	1	2	yes	—
159	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
160	#1917	C₂×C₆₇	1	4	yes	complemented
161	#1916	C₂×C₆₇	1	4	yes	complemented
162	#1817	C₂×C₆₇	1	2	yes	—
163	#1797	C₂×C₆₇	1	2	yes	—
164	#1749	C₂×C₆₇	1	1	yes	characteristic
165	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
166	C₆₇	#21914	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
168	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
170	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
172	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
174	C₂×#97	C₂²×C₆₇	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
176	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
177	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
179	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
180	C₂×#95	C₄×C₆₇	1	2	yes	—
181	#336	C₂²×C₆₇	1	4	yes	—
182	#336	C₂²×C₆₇	1	8	yes	—
183	#336	C₂²×C₆₇	1	2	yes	—
184	#336	C₂²×C₆₇	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
187	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
188	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
189	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
191	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
192	C₂×C₄⋊C₄	—	2	8	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	—	4	2	no	—
199	C₂×C₄²	—	4	2	no	—
200	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
201	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
203	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
208	C₄⋊C₈	—	4	4	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	#97	—	4	2	no	—
212	C₄×C₈	—	4	2	no	—
213	#95	C₂×C₄×C₆₇	1	2	yes	—
214	#95	C₂×C₄×C₆₇	1	2	yes	—
215	#95	C₂×C₄×C₆₇	1	2	yes	—
216	#95	C₂×C₄×C₆₇	1	2	yes	—
217	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
237	C₂²×C₄	#41×C₆₇	1	4	yes	—
238	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	M₁₆×C₆₇	1	8	yes	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	#117×C₆₇	1	2	yes	—
261	C₂³	#131×C₆₇	1	4	yes	—
262	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	#117×C₆₇	1	2	yes	—
285	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
286	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
287	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
288	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
289	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
290	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
291	C₈	—	4	4	no	—
292	C₈	—	4	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	#433×C₆₇	1	4	yes	—
298	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
299	C₂²	#518×C₆₇	1	1	yes	characteristic
300	C₂²	#519×C₆₇	1	1	yes	characteristic
301	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
302	C₂²	C₂×#131×C₆₇	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	#539×C₆₇	1	4	yes	—
311	C₂	—	2	4	no	—
312	C₂	C₆₇×#2313	1	1	yes	characteristic
313	C₂	C₂×#433×C₆₇	1	1	yes	characteristic
314	C₂	C₆₇×#2920	1	1	yes	characteristic
315	C₂	C₆₇×#2974	1	2	yes	—
316	C₂	C₆₇×#2984	1	2	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
2. C₆₇×#1917	C₂	semidirect
3. C₆₇×#1916	C₂	semidirect
152. #21914	C₆₇	direct (N × H)
160. #1917	C₂×C₆₇	semidirect
161. #1916	C₂×C₆₇	semidirect
166. C₆₇	#21914	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×C₄⋊C₄×C₆₇	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1817	2
C₆₇×#1797	2
C₆₇×#1749	2
C₄×C₂²⋊C₈×C₆₇	2
#21914	67

Subgroups — C67×#21914 #30701165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21914 #30701165