Subgroups — C₆₇×#22014 #30701265

Order 17152.

846 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1955	C₂	1	2	yes	maximal
3	C₆₇×#1940	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1924	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1817	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1797	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
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×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₄×D₄×C₆₇	—	2	8	no	—
37	C₄×D₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	—	4	4	no	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
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	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₆₇	—	2	4	no	—
53	C₄⋊C₈×C₆₇	—	4	4	no	—
54	C₂²⋊C₈×C₆₇	—	4	4	no	—
55	C₂²⋊C₈×C₆₇	—	4	4	no	—
56	#97×C₆₇	—	2	4	no	—
57	#97×C₆₇	—	4	2	no	—
58	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
59	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
60	#95×C₆₇	C₂×C₄	1	2	yes	—
61	#95×C₆₇	C₂×C₄	1	2	yes	—
62	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×D₄×C₆₇	—	2	4	no	—
64	C₂×D₄×C₆₇	—	2	2	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	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
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	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₂²⋊C₄×C₆₇	—	2	4	no	—
95	C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₂²⋊C₄×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₄²×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	—	4	2	no	—
104	C₄²×C₆₇	—	4	2	no	—
105	C₄²×C₆₇	#41	1	4	yes	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	2	2	no	—
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	—	2	2	no	—
112	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
113	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	D₄×C₆₇	—	2	8	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	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	4	8	no	—
131	C₂×C₄×C₆₇	—	4	8	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	#117	1	2	yes	—
135	C₂×C₄×C₆₇	#117	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
141	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₆₇	—	4	4	no	—
143	C₈×C₆₇	—	4	2	no	—
144	C₈×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	—	2	4	no	—
146	C₂²×C₆₇	—	2	4	no	—
147	C₂²×C₆₇	—	2	4	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	—	4	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
153	C₂²×C₆₇	#518	1	1	yes	characteristic
154	C₂²×C₆₇	#520	1	1	yes	characteristic
155	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
156	C₂²×C₆₇	#568	1	2	yes	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	4	no	—
160	C₄×C₆₇	—	2	4	no	—
161	C₄×C₆₇	—	4	2	no	—
162	C₄×C₆₇	—	4	2	no	—
163	C₄×C₆₇	—	4	2	no	—
164	C₄×C₆₇	#540	1	2	yes	—
165	C₄×C₆₇	#542	1	2	yes	—
166	#22014	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₆₇	—	4	4	no	—
168	C₂×C₆₇	#2319	1	1	yes	characteristic
169	C₂×C₆₇	#2920	1	1	yes	characteristic
170	C₂×C₆₇	#2923	1	1	yes	characteristic
171	C₂×C₆₇	#2973	1	2	yes	—
172	C₂×C₆₇	#2989	1	2	yes	—
173	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
174	#1955	C₂×C₆₇	1	2	yes	—
175	#1940	C₂×C₆₇	1	2	yes	complemented
176	#1924	C₂×C₆₇	1	4	yes	complemented
177	#1817	C₂×C₆₇	1	2	yes	complemented
178	#1797	C₂×C₆₇	1	2	yes	complemented
179	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#1745	C₂×C₆₇	1	1	yes	characteristic, complemented
181	C₆₇	#22014	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
183	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
184	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
185	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
186	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
187	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
188	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
189	C₂×#97	C₂²×C₆₇	1	2	yes	—
190	C₂×#97	C₂²×C₆₇	1	2	yes	—
191	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
192	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
193	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
194	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
195	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
196	C₄³	C₂²×C₆₇	1	1	yes	characteristic
197	#336	C₂²×C₆₇	1	1	yes	characteristic
198	#336	C₂²×C₆₇	1	2	yes	—
199	#336	C₂²×C₆₇	1	1	yes	characteristic
200	#336	C₂²×C₆₇	1	4	yes	—
201	#336	C₂²×C₆₇	1	2	yes	—
202	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
203	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
204	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
205	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
206	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
207	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
208	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
209	C₄×D₄	—	2	8	no	—
210	C₄×D₄	—	2	4	no	—
211	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
214	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
215	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
216	C₂×C₄²	—	4	4	no	—
217	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
218	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
219	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
220	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
221	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
224	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	4	4	no	—
227	C₂²⋊C₈	—	4	4	no	—
228	C₂²⋊C₈	—	4	4	no	—
229	#97	—	2	4	no	—
230	#97	—	4	2	no	—
231	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
232	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
233	#95	C₂×C₄×C₆₇	1	2	yes	—
234	#95	C₂×C₄×C₆₇	1	2	yes	—
235	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	M₁₆×C₆₇	1	8	yes	—
248	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	#41×C₆₇	1	4	yes	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
286	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
287	D₄	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	#117×C₆₇	1	2	yes	—
308	C₂×C₄	#117×C₆₇	1	2	yes	—
309	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
310	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
311	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
312	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
313	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
314	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
315	C₈	—	4	2	no	—
316	C₈	—	4	4	no	—
317	C₈	—	4	2	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
326	C₂²	#518×C₆₇	1	1	yes	characteristic
327	C₂²	#520×C₆₇	1	1	yes	characteristic
328	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
329	C₂²	#568×C₆₇	1	2	yes	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	#540×C₆₇	1	2	yes	—
338	C₄	#542×C₆₇	1	2	yes	—
339	C₂	—	4	4	no	—
340	C₂	C₆₇×#2319	1	1	yes	characteristic
341	C₂	C₆₇×#2920	1	1	yes	characteristic
342	C₂	C₆₇×#2923	1	1	yes	characteristic
343	C₂	C₆₇×#2973	1	2	yes	—
344	C₂	C₆₇×#2989	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	C₂²×D₄×C₆₇	C₂×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
3. C₆₇×#1940	C₂	semidirect
4. C₆₇×#1924	C₂	semidirect
5. C₆₇×#1817	C₂	semidirect
6. C₆₇×#1797	C₂	semidirect
7. C₆₇×#1767	C₂	semidirect
8. C₆₇×#1745	C₂	semidirect
166. #22014	C₆₇	direct (N × H)
175. #1940	C₂×C₆₇	semidirect
176. #1924	C₂×C₆₇	semidirect
177. #1817	C₂×C₆₇	semidirect
178. #1797	C₂×C₆₇	semidirect
179. #1767	C₂×C₆₇	semidirect
180. #1745	C₂×C₆₇	semidirect
181. C₆₇	#22014	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×D₄×C₆₇	2
C₆₇×#1955	2
C₆₇×#1940	2
C₆₇×#1924	2
C₆₇×#1817	2
C₆₇×#1797	2
C₆₇×#1767	2
C₆₇×#1745	2
#22014	67

Subgroups — C67×#22014 #30701265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22014 #30701265