Subgroups — C₆₇×#21916 #30701167

Order 17152.

818 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	8	yes	maximal, complemented
3	C₆₇×#1817	C₂	1	2	yes	maximal
4	C₆₇×#1797	C₂	1	2	yes	maximal
5	C₆₇×#1749	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×#97×C₆₇	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
17	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
18	#336×C₆₇	C₂²	1	4	yes	—
19	#336×C₆₇	C₂²	1	4	yes	—
20	#336×C₆₇	C₂²	1	8	yes	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
29	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₇	—	2	4	no	—
34	C₂×C₄²×C₆₇	—	4	2	no	—
35	C₂×C₄²×C₆₇	—	4	2	no	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
42	C₄⋊C₈×C₆₇	—	4	4	no	—
43	C₂²⋊C₈×C₆₇	—	2	8	no	—
44	#97×C₆₇	—	4	4	no	—
45	#95×C₆₇	C₂×C₄	1	4	yes	—
46	#95×C₆₇	C₂×C₄	1	4	yes	—
47	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	—	2	2	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	4	1	no	—
57	C₂²×C₄×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	—	4	1	no	—
61	C₂²×C₄×C₆₇	—	4	1	no	—
62	C₂²×C₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	—	4	1	no	—
64	C₂²×C₄×C₆₇	—	4	1	no	—
65	C₂²×C₄×C₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
67	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	#41	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	8	no	—
78	C₂×C₈×C₆₇	—	2	8	no	—
79	C₂×C₈×C₆₇	—	2	8	no	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄²×C₆₇	—	2	4	no	—
83	C₄²×C₆₇	—	2	4	no	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₂³×C₆₇	—	2	1	no	—
89	C₂³×C₆₇	—	2	1	no	—
90	C₂³×C₆₇	—	2	1	no	—
91	C₂³×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	#117	1	2	yes	—
93	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
94	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
95	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	2	8	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	—	2	2	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	#117	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₆₇	—	4	8	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	#432	1	4	yes	—
135	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
136	C₂²×C₆₇	#518	1	1	yes	characteristic
137	C₂²×C₆₇	#519	1	1	yes	characteristic
138	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
139	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
140	C₂²×C₆₇	#568	1	1	yes	characteristic
141	C₂²×C₆₇	#568	1	1	yes	characteristic
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	2	4	no	—
144	C₄×C₆₇	—	2	4	no	—
145	C₄×C₆₇	—	4	2	no	—
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	#539	1	4	yes	—
150	#21916	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	—	2	4	no	—
152	C₂×C₆₇	#2313	1	1	yes	characteristic
153	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
154	C₂×C₆₇	#2921	1	1	yes	characteristic
155	C₂×C₆₇	#2973	1	1	yes	characteristic
156	C₂×C₆₇	#2975	1	1	yes	characteristic
157	C₂×C₆₇	#2983	1	1	yes	characteristic
158	C₂×C₆₇	#2983	1	1	yes	characteristic
159	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
160	#1917	C₂×C₆₇	1	8	yes	complemented
161	#1817	C₂×C₆₇	1	2	yes	—
162	#1797	C₂×C₆₇	1	2	yes	—
163	#1749	C₂×C₆₇	1	2	yes	—
164	C₆₇	#21916	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
166	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
167	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
168	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
171	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
172	C₂×#97	C₂²×C₆₇	1	4	yes	—
173	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	—
174	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
175	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
176	C₂×#95	C₄×C₆₇	1	1	yes	characteristic
177	#336	C₂²×C₆₇	1	4	yes	—
178	#336	C₂²×C₆₇	1	8	yes	—
179	#336	C₂²×C₆₇	1	4	yes	—
180	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
181	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
192	C₂×C₄²	—	2	4	no	—
193	C₂×C₄²	—	4	2	no	—
194	C₂×C₄²	—	4	2	no	—
195	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
198	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
199	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
200	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
201	C₄⋊C₈	—	4	4	no	—
202	C₂²⋊C₈	—	2	8	no	—
203	#97	—	4	4	no	—
204	#95	C₂×C₄×C₆₇	1	4	yes	—
205	#95	C₂×C₄×C₆₇	1	4	yes	—
206	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
226	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
234	C₂²×C₄	#41×C₆₇	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	#117×C₆₇	1	2	yes	—
252	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
253	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
254	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	#117×C₆₇	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
282	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
283	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
284	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
285	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
286	C₈	—	4	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#432×C₆₇	1	4	yes	—
294	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
295	C₂²	#518×C₆₇	1	1	yes	characteristic
296	C₂²	#519×C₆₇	1	1	yes	characteristic
297	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
298	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
299	C₂²	#568×C₆₇	1	1	yes	characteristic
300	C₂²	#568×C₆₇	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	#539×C₆₇	1	4	yes	—
309	C₂	—	2	4	no	—
310	C₂	C₆₇×#2313	1	1	yes	characteristic
311	C₂	C₂×#432×C₆₇	1	1	yes	characteristic
312	C₂	C₆₇×#2921	1	1	yes	characteristic
313	C₂	C₆₇×#2973	1	1	yes	characteristic
314	C₂	C₆₇×#2975	1	1	yes	characteristic
315	C₂	C₆₇×#2983	1	1	yes	characteristic
316	C₂	C₆₇×#2983	1	1	yes	characteristic

	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
150. #21916	C₆₇	direct (N × H)
160. #1917	C₂×C₆₇	semidirect
164. C₆₇	#21916	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×#1817	2
C₆₇×#1797	2
C₆₇×#1749	2
#21916	67

Subgroups — C67×#21916 #30701167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21916 #30701167