Subgroups — C₆₇×#21953 #30701204

Order 17152.

942 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1940	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1939	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1797	C₂	1	4	yes	maximal
7	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#432×C₆₇	—	2	4	no	—
11	#431×C₆₇	—	2	4	no	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂×#95×C₆₇	C₄	1	2	yes	—
25	#336×C₆₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	—	4	4	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	4	yes	—
42	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	—	2	4	no	—
47	C₄⋊C₈×C₆₇	—	2	4	no	—
48	C₄⋊C₈×C₆₇	—	4	8	no	—
49	C₂²⋊C₈×C₆₇	—	2	4	no	—
50	C₂²⋊C₈×C₆₇	—	2	4	no	—
51	#97×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	—	2	4	no	—
53	#95×C₆₇	C₂×C₄	1	2	yes	—
54	#95×C₆₇	C₂×C₄	1	4	yes	—
55	#95×C₆₇	C₂×C₄	1	2	yes	—
56	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	8	no	—
62	C₂²×C₄×C₆₇	—	2	8	no	—
63	C₂²×C₄×C₆₇	—	4	4	no	—
64	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	8	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	M₁₆	1	8	yes	—
85	C₄²×C₆₇	—	2	8	no	—
86	C₄²×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
89	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₆₇	C₂×D₄	1	2	yes	—
91	C₄²×C₆₇	C₂×D₄	1	2	yes	—
92	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
93	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂³×C₆₇	—	2	1	no	—
95	C₂³×C₆₇	—	2	1	no	—
96	C₂³×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₆₇	#131	1	4	yes	—
99	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	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	4	no	—
107	C₂×C₄×C₆₇	—	2	8	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	8	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	4	16	no	—
115	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
121	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
125	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₆₇	—	4	4	no	—
127	C₈×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	#424	1	4	yes	—
133	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₇	#518	1	1	yes	characteristic
135	C₂²×C₆₇	#519	1	1	yes	characteristic
136	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
137	C₂²×C₆₇	C₂×#131	1	2	yes	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	8	no	—
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	#423	1	4	yes	—
145	C₄×C₆₇	#537	1	2	yes	—
146	C₄×C₆₇	#544	1	2	yes	—
147	#21953	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₆₇	—	2	4	no	—
149	C₂×C₆₇	#2316	1	1	yes	characteristic
150	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
152	C₂×C₆₇	#2974	1	2	yes	—
153	C₂×C₆₇	#2984	1	2	yes	—
154	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
155	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
156	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
157	#1940	C₂×C₆₇	1	4	yes	complemented
158	#1939	C₂×C₆₇	1	4	yes	complemented
159	#1797	C₂×C₆₇	1	4	yes	—
160	C₆₇	#21953	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
162	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
163	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
164	#432	—	2	4	no	—
165	#431	—	2	4	no	—
166	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
170	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
171	C₂×#97	C₂²×C₆₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
173	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
174	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
176	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
178	C₂×#95	C₄×C₆₇	1	2	yes	—
179	#336	C₂²×C₆₇	1	8	yes	—
180	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
181	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
183	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
185	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
190	C₂×C₄²	—	4	4	no	—
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	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
195	C₂×C₄²	D₄×C₆₇	1	4	yes	—
196	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
197	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
198	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	4	8	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	C₂²⋊C₈	—	2	4	no	—
205	#97	—	2	4	no	—
206	C₄×C₈	—	2	4	no	—
207	#95	C₂×C₄×C₆₇	1	2	yes	—
208	#95	C₂×C₄×C₆₇	1	4	yes	—
209	#95	C₂×C₄×C₆₇	1	2	yes	—
210	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
219	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	M₁₆×C₆₇	1	8	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
243	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
244	C₄²	C₂×D₄×C₆₇	1	2	yes	—
245	C₄²	C₂×D₄×C₆₇	1	2	yes	—
246	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
247	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
252	C₂³	#131×C₆₇	1	4	yes	—
253	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	16	no	—
269	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
270	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
274	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
275	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
276	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
278	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
279	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
280	C₈	—	4	4	no	—
281	C₈	—	4	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	#424×C₆₇	1	4	yes	—
287	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
288	C₂²	#518×C₆₇	1	1	yes	characteristic
289	C₂²	#519×C₆₇	1	1	yes	characteristic
290	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
291	C₂²	C₂×#131×C₆₇	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	4	no	—
298	C₄	#423×C₆₇	1	4	yes	—
299	C₄	#537×C₆₇	1	2	yes	—
300	C₄	#544×C₆₇	1	2	yes	—
301	C₂	—	2	4	no	—
302	C₂	C₆₇×#2316	1	1	yes	characteristic
303	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
304	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
305	C₂	C₆₇×#2974	1	2	yes	—
306	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
4. C₆₇×#1940	C₂	semidirect
5. C₆₇×#1939	C₂	semidirect
147. #21953	C₆₇	direct (N × H)
157. #1940	C₂×C₆₇	semidirect
158. #1939	C₂×C₆₇	semidirect
160. C₆₇	#21953	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×#432×C₆₇	2
C₂×#431×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1797	2
#21953	67

Subgroups — C67×#21953 #30701204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21953 #30701204