Subgroups — C₆₇×#20437 #30699688

Order 17152.

1166 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1770	C₂	1	4	yes	maximal, complemented
6	C₈×C₄⋊C₄×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1729	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#423×C₆₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
15	#405×C₆₇	—	2	8	no	—
16	C₂×#97×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	#384×C₆₇	C₄	1	2	yes	—
22	#382×C₆₇	C₄	1	2	yes	—
23	#380×C₆₇	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
25	#336×C₆₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	4	no	—
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₇	D₄	1	4	yes	—
36	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#117×C₆₇	—	2	4	no	—
42	#117×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	—
53	#97×C₆₇	—	2	8	no	—
54	C₄×C₈×C₆₇	—	2	8	no	—
55	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
57	#95×C₆₇	C₂×C₄	1	8	yes	—
58	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	—	2	8	no	—
60	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₆₇	#41	1	2	yes	—
70	C₂²×C₄×C₆₇	#41	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₆₇	—	2	16	no	—
73	C₂×C₈×C₆₇	—	2	8	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	4	no	—
78	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	8	no	—
88	C₄⋊C₄×C₆₇	—	2	16	no	—
89	C₂²⋊C₄×C₆₇	—	2	8	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	C₄²	1	4	yes	—
93	C₄²×C₆₇	C₄²	1	4	yes	—
94	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
95	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
100	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	16	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	16	no	—
105	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₆₇	#117	1	4	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	#131	1	8	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₈×C₆₇	—	2	4	no	—
122	C₈×C₆₇	—	2	8	no	—
123	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
124	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×#131	1	4	yes	—
133	C₄×C₆₇	—	2	4	no	—
134	C₄×C₆₇	—	4	8	no	—
135	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
136	C₄×C₆₇	#405	1	8	yes	complemented
137	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
139	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	#20437	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₆₇	—	2	4	no	—
143	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#405	1	2	yes	—
145	C₂×C₆₇	#2931	1	2	yes	—
146	C₂×C₆₇	#2960	1	2	yes	—
147	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
148	C₂×#405	C₂×C₆₇	1	2	yes	—
149	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
150	#2285	C₂×C₆₇	1	1	yes	characteristic
151	#1770	C₂×C₆₇	1	4	yes	complemented
152	C₈×C₄⋊C₄	C₂×C₆₇	1	4	yes	complemented
153	#1729	C₂×C₆₇	1	2	yes	—
154	C₆₇	#20437	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
156	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
158	C₂×#117	C₂²×C₆₇	1	2	yes	—
159	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
160	#423	C₄×C₆₇	1	8	yes	complemented
161	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
162	#405	—	2	8	no	—
163	C₂×#97	C₂²×C₆₇	1	4	yes	—
164	C₂×C₄×C₈	C₄×C₆₇	1	8	yes	complemented
165	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
166	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
168	#384	C₄×C₆₇	1	2	yes	—
169	#382	C₄×C₆₇	1	2	yes	—
170	#380	C₄×C₆₇	1	4	yes	—
171	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
172	#336	C₂²×C₆₇	1	8	yes	—
173	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
174	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
182	C₂²×C₈	D₄×C₆₇	1	4	yes	—
183	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
184	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
185	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
186	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
188	#117	—	2	4	no	—
189	#117	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
197	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
198	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
199	C₄⋊C₈	C₂×C₄×C₆₇	1	16	yes	—
200	#97	—	2	8	no	—
201	C₄×C₈	—	2	8	no	—
202	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
203	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
204	#95	C₂×C₄×C₆₇	1	8	yes	—
205	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	C₄²×C₆₇	1	8	yes	—
208	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
209	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
216	C₂²×C₄	#41×C₆₇	1	2	yes	—
217	C₂²×C₄	#41×C₆₇	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
219	M₁₆	—	2	16	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₄⋊C₄×C₆₇	1	16	yes	complemented
226	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
227	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
228	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
230	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
232	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
233	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	16	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₄²×C₆₇	1	4	yes	—
240	C₄²	C₄²×C₆₇	1	4	yes	—
241	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
242	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
245	C₂³	C₄×D₄×C₆₇	1	2	yes	—
246	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
247	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	16	no	—
252	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
253	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
254	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
256	C₂×C₄	#117×C₆₇	1	4	yes	—
257	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
258	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
259	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
260	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
261	C₂×C₄	#131×C₆₇	1	8	yes	—
262	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	2	8	no	—
270	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
271	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
276	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
277	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
278	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
279	C₂²	C₂×#131×C₆₇	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	4	8	no	—
282	C₄	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
283	C₄	#405×C₆₇	1	8	yes	complemented
284	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
285	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
286	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
287	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
290	C₂	C₂×#405×C₆₇	1	2	yes	—
291	C₂	C₆₇×#2931	1	2	yes	—
292	C₂	C₆₇×#2960	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1770	C₂	semidirect
6. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
13. #423×C₆₇	C₄	semidirect
17. C₂×C₄×C₈×C₆₇	C₄	semidirect
78. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
136. C₄×C₆₇	#405	semidirect
141. #20437	C₆₇	direct (N × H)
151. #1770	C₂×C₆₇	semidirect
152. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
154. C₆₇	#20437	direct (N × H)
160. #423	C₄×C₆₇	semidirect
164. C₂×C₄×C₈	C₄×C₆₇	semidirect
225. C₂×C₈	C₄⋊C₄×C₆₇	semidirect
283. C₄	#405×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₆₇	2
C₂×#405×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2285	2
C₆₇×#1770	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1729	2
#20437	67

Subgroups — C67×#20437 #30699688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20437 #30699688