Subgroups — C₆₇×#22015 #30701266

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2293	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×#1818	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1798	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	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	#390×C₆₇	C₄	1	2	yes	—
18	#389×C₆₇	C₄	1	2	yes	—
19	#388×C₆₇	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	#336×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	C₂²	1	1	yes	characteristic
25	#336×C₆₇	C₂²	1	2	yes	—
26	#336×C₆₇	C₂²	1	2	yes	—
27	#336×C₆₇	C₂²	1	4	yes	—
28	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
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×D₄×C₆₇	—	2	4	no	—
34	#117×C₆₇	—	4	4	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×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	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	—	2	4	no	—
48	C₄⋊C₈×C₆₇	—	4	4	no	—
49	C₂²⋊C₈×C₆₇	—	4	4	no	—
50	C₂²⋊C₈×C₆₇	—	4	4	no	—
51	#97×C₆₇	—	2	4	no	—
52	#97×C₆₇	—	4	2	no	—
53	#95×C₆₇	C₂×C₄	1	4	yes	—
54	#95×C₆₇	C₂×C₄	1	2	yes	—
55	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	4	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	2	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	4	4	no	—
81	C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₂²⋊C₄×C₆₇	—	4	4	no	—
83	C₂²⋊C₄×C₆₇	—	4	4	no	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₄²×C₆₇	—	4	2	no	—
89	C₄²×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	#41	1	4	yes	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
97	D₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	2	2	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	4	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	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	#117	1	2	yes	—
115	C₂×C₄×C₆₇	#117	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
120	C₈×C₆₇	—	4	4	no	—
121	C₈×C₆₇	—	4	2	no	—
122	C₈×C₆₇	—	4	2	no	—
123	C₂²×C₆₇	—	4	2	no	—
124	C₂²×C₆₇	—	4	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	—	4	4	no	—
127	C₂²×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
129	C₂²×C₆₇	#518	1	1	yes	characteristic
130	C₂²×C₆₇	#520	1	1	yes	characteristic
131	C₂²×C₆₇	#568	1	2	yes	—
132	C₂²×C₆₇	#568	1	2	yes	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	4	2	no	—
136	C₄×C₆₇	—	4	2	no	—
137	C₄×C₆₇	—	4	4	no	—
138	C₄×C₆₇	—	4	2	no	—
139	C₄×C₆₇	#540	1	2	yes	—
140	C₄×C₆₇	#542	1	2	yes	—
141	#22015	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₆₇	—	8	2	no	—
143	C₂×C₆₇	#2319	1	1	yes	characteristic
144	C₂×C₆₇	#2921	1	1	yes	characteristic
145	C₂×C₆₇	#2923	1	1	yes	characteristic
146	C₂×C₆₇	#2980	1	2	yes	—
147	C₂×C₆₇	#2995	1	2	yes	—
148	#2293	C₂×C₆₇	1	1	yes	characteristic
149	#1955	C₂×C₆₇	1	2	yes	—
150	#1940	C₂×C₆₇	1	2	yes	complemented
151	#1924	C₂×C₆₇	1	4	yes	complemented
152	#1818	C₂×C₆₇	1	2	yes	complemented
153	#1798	C₂×C₆₇	1	2	yes	complemented
154	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
155	#1745	C₂×C₆₇	1	1	yes	characteristic, complemented
156	C₆₇	#22015	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
158	C₂×#117	C₂²×C₆₇	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
161	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
163	C₂×#97	C₂²×C₆₇	1	2	yes	—
164	C₂×#97	C₂²×C₆₇	1	2	yes	—
165	#390	C₄×C₆₇	1	2	yes	—
166	#389	C₄×C₆₇	1	2	yes	—
167	#388	C₄×C₆₇	1	4	yes	—
168	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
169	C₄×C₂²⋊C₄	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	1	yes	characteristic
174	#336	C₂²×C₆₇	1	4	yes	—
175	#336	C₂²×C₆₇	1	2	yes	—
176	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
178	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
179	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
181	C₄×D₄	—	2	4	no	—
182	#117	—	4	4	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
191	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
193	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	—	4	4	no	—
197	C₂²⋊C₈	—	4	4	no	—
198	C₂²⋊C₈	—	4	4	no	—
199	#97	—	2	4	no	—
200	#97	—	4	2	no	—
201	#95	C₂×C₄×C₆₇	1	4	yes	—
202	#95	C₂×C₄×C₆₇	1	2	yes	—
203	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	#41×C₆₇	1	4	yes	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
245	D₄	—	4	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	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	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	#117×C₆₇	1	2	yes	—
263	C₂×C₄	#117×C₆₇	1	2	yes	—
264	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
265	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
266	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
267	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
277	C₂²	#518×C₆₇	1	1	yes	characteristic
278	C₂²	#520×C₆₇	1	1	yes	characteristic
279	C₂²	#568×C₆₇	1	2	yes	—
280	C₂²	#568×C₆₇	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	4	2	no	—
287	C₄	#540×C₆₇	1	2	yes	—
288	C₄	#542×C₆₇	1	2	yes	—
289	C₂	—	8	2	no	—
290	C₂	C₆₇×#2319	1	1	yes	characteristic
291	C₂	C₆₇×#2921	1	1	yes	characteristic
292	C₂	C₆₇×#2923	1	1	yes	characteristic
293	C₂	C₆₇×#2980	1	2	yes	—
294	C₂	C₆₇×#2995	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×#1818	C₂	semidirect
6. C₆₇×#1798	C₂	semidirect
7. C₆₇×#1767	C₂	semidirect
8. C₆₇×#1745	C₂	semidirect
141. #22015	C₆₇	direct (N × H)
150. #1940	C₂×C₆₇	semidirect
151. #1924	C₂×C₆₇	semidirect
152. #1818	C₂×C₆₇	semidirect
153. #1798	C₂×C₆₇	semidirect
154. #1767	C₂×C₆₇	semidirect
155. #1745	C₂×C₆₇	semidirect
156. C₆₇	#22015	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2293	2
C₆₇×#1955	2
C₆₇×#1940	2
C₆₇×#1924	2
C₆₇×#1818	2
C₆₇×#1798	2
C₆₇×#1767	2
C₆₇×#1745	2
#22015	67

Subgroups — C67×#22015 #30701266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22015 #30701266