Subgroups — C₆₇×#9878 #30689129

Order 17152.

680 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1280	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
9	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	—	2	1	no	—
12	#391×C₆₇	—	2	1	no	—
13	#391×C₆₇	—	2	1	no	—
14	#391×C₆₇	—	2	1	no	—
15	#391×C₆₇	—	2	1	no	—
16	C₂×#95×C₆₇	C₄	1	2	yes	—
17	C₂×#95×C₆₇	C₂²	1	2	yes	—
18	#336×C₆₇	—	2	1	no	—
19	#336×C₆₇	—	2	1	no	—
20	C₂²×Q₈×C₆₇	—	2	1	no	—
21	C₂²×Q₈×C₆₇	—	2	1	no	—
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂²×Q₈×C₆₇	—	4	2	no	—
24	C₂²×Q₈×C₆₇	—	4	2	no	—
25	C₂²×Q₈×C₆₇	C₈	1	4	yes	complemented
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
36	C₂×C₄²×C₆₇	—	4	2	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	4	2	no	—
42	C₂²⋊C₈×C₆₇	—	2	2	no	—
43	C₂²⋊C₈×C₆₇	—	2	4	no	—
44	C₂²⋊C₈×C₆₇	Q₈	1	4	yes	complemented
45	C₄×C₈×C₆₇	—	4	2	no	—
46	#95×C₆₇	—	2	2	no	—
47	#95×C₆₇	—	2	2	no	—
48	#95×C₆₇	—	2	2	no	—
49	#95×C₆₇	—	2	2	no	—
50	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	—	4	8	no	—
54	C₂×Q₈×C₆₇	—	4	2	no	—
55	C₂×Q₈×C₆₇	—	4	2	no	—
56	C₂×Q₈×C₆₇	—	4	8	no	—
57	C₂×Q₈×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	4	1	no	—
68	C₂²×C₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	4	1	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
80	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₂×C₈×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	4	2	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	4	2	no	—
91	C₄⋊C₄×C₆₇	—	4	2	no	—
92	C₄⋊C₄×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	4	1	no	—
96	C₄²×C₆₇	—	4	1	no	—
97	C₂³×C₆₇	—	2	1	no	—
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
103	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂³×C₆₇	#131	1	2	yes	—
106	Q₈×C₆₇	—	4	8	no	—
107	Q₈×C₆₇	—	4	8	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	4	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	8	4	no	—
133	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
135	C₂×C₄×C₆₇	#99	1	2	yes	—
136	C₂×C₄×C₆₇	#100	1	2	yes	—
137	C₂×C₄×C₆₇	#101	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
141	C₈×C₆₇	—	4	4	no	—
142	C₈×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	2	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	#391	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂²×C₆₇	#408	1	1	yes	characteristic
154	C₂²×C₆₇	C₂×#99	1	1	yes	characteristic
155	C₂²×C₆₇	#410	1	1	yes	characteristic
156	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
158	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
159	C₂²×C₆₇	#446	1	2	yes	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	2	2	no	—
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	2	1	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	4	4	no	—
166	C₄×C₆₇	—	4	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	#323	1	4	yes	—
169	#9878	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	—	2	2	no	—
172	C₂×C₆₇	C₂×#323	1	1	yes	characteristic
173	C₂×C₆₇	#1461	1	1	yes	characteristic
174	C₂×C₆₇	#1469	1	1	yes	characteristic
175	C₂×C₆₇	#1472	1	1	yes	characteristic
176	C₂×C₆₇	#1984	1	1	yes	characteristic
177	C₂×C₆₇	#1996	1	1	yes	characteristic
178	C₂×C₆₇	#1997	1	1	yes	characteristic
179	C₂×#391	C₂×C₆₇	1	1	yes	characteristic
180	#1797	C₂×C₆₇	1	1	yes	characteristic
181	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
182	#1280	C₂×C₆₇	1	4	yes	—
183	C₆₇	#9878	1	1	yes	characteristic, direct product factor, Sylow
184	C₂³×Q₈	C₄×C₆₇	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
186	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	2	1	no	—
188	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
189	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
190	C₂×C₄×C₈	—	2	1	no	—
191	#391	—	2	1	no	—
192	#391	—	2	1	no	—
193	#391	—	2	1	no	—
194	#391	—	2	1	no	—
195	C₂×#95	C₄×C₆₇	1	2	yes	—
196	C₂×#95	C₂²×C₆₇	1	2	yes	—
197	#336	—	2	1	no	—
198	#336	—	2	1	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	4	2	no	—
203	C₂²×Q₈	—	4	2	no	—
204	C₂²×Q₈	C₈×C₆₇	1	4	yes	complemented
205	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
206	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
208	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
209	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	4	2	no	—
215	C₂×C₄²	—	4	2	no	—
216	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
220	C₄⋊C₈	—	4	2	no	—
221	C₂²⋊C₈	—	2	2	no	—
222	C₂²⋊C₈	—	2	4	no	—
223	C₂²⋊C₈	Q₈×C₆₇	1	4	yes	complemented
224	C₄×C₈	—	4	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
258	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
259	C₂²×C₄	M₁₆×C₆₇	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
263	C₂²×C₄	#41×C₆₇	1	2	yes	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
281	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
282	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
283	C₂³	C₂×M₁₆×C₆₇	1	1	yes	characteristic
284	C₂³	#131×C₆₇	1	2	yes	—
285	Q₈	—	4	8	no	—
286	Q₈	—	4	8	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
313	C₂×C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
314	C₂×C₄	#99×C₆₇	1	2	yes	—
315	C₂×C₄	#100×C₆₇	1	2	yes	—
316	C₂×C₄	#101×C₆₇	1	2	yes	—
317	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
318	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
319	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
320	C₈	—	4	4	no	—
321	C₈	—	8	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	#391×C₆₇	1	1	yes	characteristic
331	C₂²	C₂×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
332	C₂²	#408×C₆₇	1	1	yes	characteristic
333	C₂²	C₂×#99×C₆₇	1	1	yes	characteristic
334	C₂²	#410×C₆₇	1	1	yes	characteristic
335	C₂²	C₂×#100×C₆₇	1	1	yes	characteristic
336	C₂²	C₂×#101×C₆₇	1	1	yes	characteristic
337	C₂²	C₈×Q₈×C₆₇	1	2	yes	—
338	C₂²	#446×C₆₇	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	4	no	—
345	C₄	—	4	1	no	—
346	C₄	—	8	2	no	—
347	C₄	#323×C₆₇	1	4	yes	—
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	C₂×#323×C₆₇	1	1	yes	characteristic
351	C₂	C₆₇×#1461	1	1	yes	characteristic
352	C₂	C₆₇×#1469	1	1	yes	characteristic
353	C₂	C₆₇×#1472	1	1	yes	characteristic
354	C₂	C₆₇×#1984	1	1	yes	characteristic
355	C₂	C₆₇×#1996	1	1	yes	characteristic
356	C₂	C₆₇×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₂²×Q₈×C₆₇	C₈	semidirect
44. C₂²⋊C₈×C₆₇	Q₈	semidirect
169. #9878	C₆₇	direct (N × H)
183. C₆₇	#9878	direct (N × H)
204. C₂²×Q₈	C₈×C₆₇	semidirect
223. C₂²⋊C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#391×C₆₇	2
C₆₇×#1797	2
C₄×C₂²⋊C₈×C₆₇	2
C₆₇×#1280	2
#9878	67

Subgroups — C67×#9878 #30689129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9878 #30689129