Subgroups — C₆₇×#20503 #30699754

Order 17152.

1206 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#442×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2275	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1776	C₂	1	2	yes	maximal, complemented
5	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1762	C₂	1	2	yes	maximal, complemented
7	C₄×#102×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
12	#442×C₆₇	C₄	1	16	yes	complemented
13	C₄×D₈×C₆₇	C₄	1	8	yes	complemented
14	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₆₇	C₂²	1	2	yes	—
16	C₂×#102×C₆₇	C₂²	1	2	yes	—
17	C₂×#102×C₆₇	C₂²	1	4	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
25	C₄×C₂²⋊C₄×C₆₇	—	2	4	no	—
26	#376×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#95×C₆₇	—	2	2	no	—
28	#340×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	4	yes	—
30	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	—	2	4	no	—
32	C₂³×C₄×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₆₇	—	2	8	no	—
39	C₄×D₄×C₆₇	—	2	4	no	—
40	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
41	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	—	2	4	no	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	#107×C₆₇	C₂×C₄	1	4	yes	complemented
51	#106×C₆₇	C₂×C₄	1	8	yes	complemented
52	#102×C₆₇	C₂×C₄	1	8	yes	complemented
53	#102×C₆₇	C₂×C₄	1	16	yes	complemented
54	#97×C₆₇	—	2	2	no	—
55	#97×C₆₇	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₆₇	—	2	4	no	—
57	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
58	#95×C₆₇	—	2	4	no	—
59	#95×C₆₇	—	2	2	no	—
60	#95×C₆₇	D₄	1	2	yes	—
61	#95×C₆₇	D₄	1	2	yes	—
62	C₂⁴×C₆₇	—	2	2	no	—
63	C₂×D₄×C₆₇	—	2	4	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	16	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	8	4	no	—
72	C₂²×C₄×C₆₇	—	8	4	no	—
73	C₂²×C₄×C₆₇	—	8	4	no	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	D₈×C₆₇	C₄²	1	16	yes	complemented
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₆₇	—	2	8	no	—
91	C₂²⋊C₄×C₆₇	—	2	16	no	—
92	C₄²×C₆₇	—	8	4	no	—
93	C₄²×C₆₇	—	8	2	no	—
94	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
96	C₄²×C₆₇	#41	1	4	yes	—
97	C₄²×C₆₇	#41	1	2	yes	—
98	C₂³×C₆₇	—	2	8	no	—
99	C₂³×C₆₇	—	2	4	no	—
100	C₂³×C₆₇	—	4	4	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₆₇	—	2	4	no	—
105	D₄×C₆₇	—	2	8	no	—
106	D₄×C₆₇	C₂×C₄²	1	8	yes	—
107	C₂×C₄×C₆₇	—	2	4	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	8	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	8	16	no	—
117	C₂×C₄×C₆₇	—	8	8	no	—
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
120	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	—	2	4	no	—
127	C₈×C₆₇	C₂×C₄²	1	4	yes	—
128	C₂²×C₆₇	—	2	8	no	—
129	C₂²×C₆₇	—	4	8	no	—
130	C₂²×C₆₇	—	4	8	no	—
131	C₂²×C₆₇	—	4	4	no	—
132	C₂²×C₆₇	—	4	4	no	—
133	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
134	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₆₇	#576	1	2	yes	—
136	C₂²×C₆₇	#577	1	2	yes	—
137	C₄×C₆₇	—	2	8	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	4	8	no	—
141	C₄×C₆₇	—	4	4	no	—
142	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
144	C₄×C₆₇	C₄×#41	1	2	yes	—
145	#20503	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₆₇	—	4	8	no	—
147	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
148	C₂×C₆₇	#2946	1	4	yes	—
149	C₂×C₆₇	#2948	1	1	yes	characteristic
150	C₂×C₆₇	#2952	1	1	yes	characteristic
151	C₂×#442	C₂×C₆₇	1	2	yes	—
152	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
153	#2275	C₂×C₆₇	1	2	yes	complemented
154	#1776	C₂×C₆₇	1	2	yes	complemented
155	C₄×#106	C₂×C₆₇	1	1	yes	characteristic, complemented
156	#1762	C₂×C₆₇	1	2	yes	complemented
157	C₄×#102	C₂×C₆₇	1	4	yes	complemented
158	#1726	C₂×C₆₇	1	1	yes	characteristic, complemented
159	C₆₇	#20503	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
161	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
162	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
163	#442	C₄×C₆₇	1	16	yes	complemented
164	C₄×D₈	C₄×C₆₇	1	8	yes	complemented
165	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×#106	C₂²×C₆₇	1	2	yes	—
167	C₂×#102	C₂²×C₆₇	1	2	yes	—
168	C₂×#102	C₂²×C₆₇	1	4	yes	—
169	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
170	C₂×#97	C₂²×C₆₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
174	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
175	C₄×C₂²⋊C₄	—	2	2	no	—
176	C₄×C₂²⋊C₄	—	2	4	no	—
177	#376	C₂²×C₆₇	1	1	yes	characteristic
178	C₂×#95	—	2	2	no	—
179	#340	C₂²×C₆₇	1	2	yes	—
180	#340	C₂²×C₆₇	1	4	yes	—
181	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
185	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	—
186	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
189	C₄×D₄	—	2	8	no	—
190	C₄×D₄	—	2	4	no	—
191	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
192	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
193	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	4	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
201	#107	C₂×C₄×C₆₇	1	4	yes	complemented
202	#106	C₂×C₄×C₆₇	1	8	yes	complemented
203	#102	C₂×C₄×C₆₇	1	8	yes	complemented
204	#102	C₂×C₄×C₆₇	1	16	yes	complemented
205	#97	—	2	2	no	—
206	#97	C₂×C₄×C₆₇	1	8	yes	complemented
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	D₄×C₆₇	1	2	yes	—
212	#95	D₄×C₆₇	1	2	yes	—
213	C₂⁴	—	2	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
217	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	16	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	—	8	4	no	—
224	C₂²×C₄	—	8	4	no	—
225	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
227	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
229	D₈	C₄²×C₆₇	1	16	yes	complemented
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
240	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₄²	—	8	4	no	—
244	C₄²	—	8	2	no	—
245	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
246	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
247	C₄²	#41×C₆₇	1	4	yes	—
248	C₄²	#41×C₆₇	1	2	yes	—
249	C₂³	—	2	8	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	8	no	—
257	D₄	C₂×C₄²×C₆₇	1	8	yes	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	16	no	—
268	C₂×C₄	—	8	8	no	—
269	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
270	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
271	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
272	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
273	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
275	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	4	no	—
278	C₈	C₂×C₄²×C₆₇	1	4	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
285	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
286	C₂²	#576×C₆₇	1	2	yes	—
287	C₂²	#577×C₆₇	1	2	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	8	no	—
292	C₄	—	4	4	no	—
293	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
294	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
295	C₄	C₄×#41×C₆₇	1	2	yes	—
296	C₂	—	4	8	no	—
297	C₂	C₄²×D₄×C₆₇	1	1	yes	characteristic
298	C₂	C₆₇×#2946	1	4	yes	—
299	C₂	C₆₇×#2948	1	1	yes	characteristic
300	C₂	C₆₇×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₆₇×#2275	C₂	semidirect
4. C₆₇×#1776	C₂	semidirect
5. C₄×#106×C₆₇	C₂	semidirect
6. C₆₇×#1762	C₂	semidirect
7. C₄×#102×C₆₇	C₂	semidirect
8. C₆₇×#1726	C₂	semidirect
12. #442×C₆₇	C₄	semidirect
13. C₄×D₈×C₆₇	C₄	semidirect
40. C₄×D₄×C₆₇	C₂×C₄	semidirect
41. C₄×D₄×C₆₇	C₂×C₄	semidirect
50. #107×C₆₇	C₂×C₄	semidirect
51. #106×C₆₇	C₂×C₄	semidirect
52. #102×C₆₇	C₂×C₄	semidirect
53. #102×C₆₇	C₂×C₄	semidirect
55. #97×C₆₇	C₂×C₄	semidirect
57. C₄×C₈×C₆₇	C₂×C₄	semidirect
78. D₈×C₆₇	C₄²	semidirect
145. #20503	C₆₇	direct (N × H)
153. #2275	C₂×C₆₇	semidirect
154. #1776	C₂×C₆₇	semidirect
155. C₄×#106	C₂×C₆₇	semidirect
156. #1762	C₂×C₆₇	semidirect
157. C₄×#102	C₂×C₆₇	semidirect
158. #1726	C₂×C₆₇	semidirect
159. C₆₇	#20503	direct (N × H)
163. #442	C₄×C₆₇	semidirect
164. C₄×D₈	C₄×C₆₇	semidirect
191. C₄×D₄	C₂×C₄×C₆₇	semidirect
192. C₄×D₄	C₂×C₄×C₆₇	semidirect
201. #107	C₂×C₄×C₆₇	semidirect
202. #106	C₂×C₄×C₆₇	semidirect
203. #102	C₂×C₄×C₆₇	semidirect
204. #102	C₂×C₄×C₆₇	semidirect
206. #97	C₂×C₄×C₆₇	semidirect
208. C₄×C₈	C₂×C₄×C₆₇	semidirect
229. D₈	C₄²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#442×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2275	2
C₆₇×#1776	2
C₄×#106×C₆₇	2
C₆₇×#1762	2
C₄×#102×C₆₇	2
C₆₇×#1726	2
#20503	67

Subgroups — C67×#20503 #30699754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20503 #30699754