Subgroups — C₆₇×#10862 #30690113

Order 17152.

982 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	C₂²×#107×C₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#340×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#107×C₆₇	—	2	4	no	—
10	C₂×#107×C₆₇	—	2	4	no	—
11	C₂×#107×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₆₇	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	complemented
14	C₂×#95×C₆₇	—	2	4	no	—
15	#340×C₆₇	—	2	4	no	—
16	C₂³×C₄×C₆₇	—	2	4	no	—
17	C₂³×C₄×C₆₇	D₄	1	2	yes	—
18	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	—	2	2	no	—
20	C₂²×C₈×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	D₄	1	2	yes	—
25	C₂²×C₈×C₆₇	D₄	1	4	yes	—
26	C₂²×C₈×C₆₇	D₄	1	2	yes	—
27	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
28	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₆₇	—	2	2	no	—
36	C₂×C₄²×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
39	#107×C₆₇	—	2	16	no	—
40	C₄×C₈×C₆₇	—	2	8	no	—
41	#95×C₆₇	—	4	8	no	—
42	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	8	no	—
46	C₂²×C₄×C₆₇	—	4	4	no	—
47	C₂²×C₄×C₆₇	—	4	4	no	—
48	C₂²×C₄×C₆₇	—	4	8	no	—
49	C₂²×C₄×C₆₇	—	4	8	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₆₇	D₈	1	4	yes	—
52	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	#41	1	2	yes	—
62	C₂²×C₄×C₆₇	#41	1	2	yes	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	C₄²	1	4	yes	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
73	C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₄⋊C₄×C₆₇	—	4	4	no	—
76	C₄⋊C₄×C₆₇	—	4	4	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₆₇	C₄×D₄	1	2	yes	—
82	C₂³×C₆₇	#121	1	2	yes	—
83	C₂³×C₆₇	#127	1	1	yes	characteristic
84	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×D₈	1	2	yes	—
86	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
87	C₂³×C₆₇	#135	1	4	yes	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	8	no	—
91	C₂×C₄×C₆₇	—	2	8	no	—
92	C₂×C₄×C₆₇	—	4	8	no	—
93	C₂×C₄×C₆₇	—	4	16	no	—
94	C₂×C₄×C₆₇	—	8	8	no	—
95	C₂×C₄×C₆₇	—	8	4	no	—
96	C₂×C₄×C₆₇	—	8	4	no	—
97	C₂×C₄×C₆₇	#107	1	8	yes	complemented
98	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₆₇	#121	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₆₇	#124	1	2	yes	—
113	C₂×C₄×C₆₇	#125	1	2	yes	—
114	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
116	C₈×C₆₇	—	2	4	no	—
117	C₈×C₆₇	#95	1	8	yes	complemented
118	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₆₇	#380	1	1	yes	characteristic
120	C₂²×C₆₇	#384	1	1	yes	characteristic
121	C₂²×C₆₇	#389	1	2	yes	—
122	C₂²×C₆₇	#390	1	2	yes	—
123	C₂²×C₆₇	C₂×#107	1	2	yes	—
124	C₂²×C₆₇	#427	1	2	yes	—
125	C₂²×C₆₇	C₄×D₈	1	4	yes	—
126	C₂²×C₆₇	C₄×Q₁₆	1	4	yes	—
127	C₂²×C₆₇	#466	1	4	yes	—
128	C₂²×C₆₇	#467	1	4	yes	—
129	C₂²×C₆₇	#493	1	1	yes	characteristic
130	C₂²×C₆₇	#494	1	1	yes	characteristic
131	C₂²×C₆₇	#495	1	2	yes	—
132	C₂²×C₆₇	#499	1	2	yes	—
133	C₂²×C₆₇	#500	1	2	yes	—
134	C₄×C₆₇	—	2	8	no	—
135	C₄×C₆₇	—	8	8	no	—
136	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
137	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₆₇	#380	1	1	yes	characteristic
139	C₄×C₆₇	#382	1	1	yes	characteristic
140	C₄×C₆₇	#389	1	2	yes	—
141	C₄×C₆₇	#391	1	2	yes	—
142	#10862	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
144	C₂×C₆₇	C₄×#107	1	2	yes	—
145	C₂×C₆₇	#1808	1	2	yes	—
146	C₂×C₆₇	#1845	1	1	yes	characteristic
147	C₂×C₆₇	#1846	1	1	yes	characteristic
148	C₂×C₆₇	#1942	1	4	yes	—
149	C₂×C₆₇	#1969	1	2	yes	—
150	C₂×C₆₇	#1970	1	2	yes	—
151	C₂²×#107	C₂×C₆₇	1	2	yes	—
152	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
153	C₂×#340	C₂×C₆₇	1	2	yes	—
154	C₄⋊#95	C₂×C₆₇	1	2	yes	—
155	C₆₇	#10862	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
159	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×#107	—	2	4	no	—
161	C₂×#107	—	2	4	no	—
162	C₂×#107	C₄×C₆₇	1	8	yes	complemented
163	C₂×C₄×C₈	—	2	4	no	—
164	C₂×C₄×C₈	C₄×C₆₇	1	4	yes	complemented
165	C₂×#95	—	2	4	no	—
166	#340	—	2	4	no	—
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	D₄×C₆₇	1	2	yes	—
169	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
174	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
175	C₂²×C₈	D₄×C₆₇	1	2	yes	—
176	C₂²×C₈	D₄×C₆₇	1	4	yes	—
177	C₂²×C₈	D₄×C₆₇	1	2	yes	—
178	C₂²×C₈	Q₈×C₆₇	1	2	yes	—
179	C₂²×C₈	Q₈×C₆₇	1	2	yes	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
189	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
190	#107	—	2	16	no	—
191	C₄×C₈	—	2	8	no	—
192	#95	—	4	8	no	—
193	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	4	8	no	—
201	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
202	C₂²×C₄	D₈×C₆₇	1	4	yes	—
203	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
204	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
212	C₂²×C₄	#41×C₆₇	1	2	yes	—
213	C₂²×C₄	#41×C₆₇	1	2	yes	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₄²×C₆₇	1	4	yes	—
219	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
220	C₂×C₈	C₂²⋊C₄×C₆₇	1	8	yes	—
221	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
222	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
223	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	4	no	—
231	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
232	C₂³	C₄×D₄×C₆₇	1	2	yes	—
233	C₂³	#121×C₆₇	1	2	yes	—
234	C₂³	#127×C₆₇	1	1	yes	characteristic
235	C₂³	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
236	C₂³	C₂×D₈×C₆₇	1	2	yes	—
237	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
238	C₂³	#135×C₆₇	1	4	yes	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	4	16	no	—
245	C₂×C₄	—	8	8	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	#107×C₆₇	1	8	yes	complemented
249	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
256	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
258	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
259	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
260	C₂×C₄	#121×C₆₇	1	2	yes	—
261	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
262	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
263	C₂×C₄	#124×C₆₇	1	2	yes	—
264	C₂×C₄	#125×C₆₇	1	2	yes	—
265	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
266	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	#95×C₆₇	1	8	yes	complemented
269	C₂²	C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
270	C₂²	#380×C₆₇	1	1	yes	characteristic
271	C₂²	#384×C₆₇	1	1	yes	characteristic
272	C₂²	#389×C₆₇	1	2	yes	—
273	C₂²	#390×C₆₇	1	2	yes	—
274	C₂²	C₂×#107×C₆₇	1	2	yes	—
275	C₂²	#427×C₆₇	1	2	yes	—
276	C₂²	C₄×D₈×C₆₇	1	4	yes	—
277	C₂²	C₄×Q₁₆×C₆₇	1	4	yes	—
278	C₂²	#466×C₆₇	1	4	yes	—
279	C₂²	#467×C₆₇	1	4	yes	—
280	C₂²	#493×C₆₇	1	1	yes	characteristic
281	C₂²	#494×C₆₇	1	1	yes	characteristic
282	C₂²	#495×C₆₇	1	2	yes	—
283	C₂²	#499×C₆₇	1	2	yes	—
284	C₂²	#500×C₆₇	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	8	8	no	—
287	C₄	C₂×#95×C₆₇	1	1	yes	characteristic
288	C₄	C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
289	C₄	#380×C₆₇	1	1	yes	characteristic
290	C₄	#382×C₆₇	1	1	yes	characteristic
291	C₄	#389×C₆₇	1	2	yes	—
292	C₄	#391×C₆₇	1	2	yes	—
293	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
294	C₂	C₄×#107×C₆₇	1	2	yes	—
295	C₂	C₆₇×#1808	1	2	yes	—
296	C₂	C₆₇×#1845	1	1	yes	characteristic
297	C₂	C₆₇×#1846	1	1	yes	characteristic
298	C₂	C₆₇×#1942	1	4	yes	—
299	C₂	C₆₇×#1969	1	2	yes	—
300	C₂	C₆₇×#1970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂×#107×C₆₇	C₄	semidirect
13. C₂×C₄×C₈×C₆₇	C₄	semidirect
97. C₂×C₄×C₆₇	#107	semidirect
117. C₈×C₆₇	#95	semidirect
142. #10862	C₆₇	direct (N × H)
155. C₆₇	#10862	direct (N × H)
162. C₂×#107	C₄×C₆₇	semidirect
164. C₂×C₄×C₈	C₄×C₆₇	semidirect
248. C₂×C₄	#107×C₆₇	semidirect
268. C₈	#95×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#107×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×#340×C₆₇	2
C₆₇×C₄⋊#95	2
#10862	67

Subgroups — C67×#10862 #30690113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10862 #30690113