Subgroups — C₆₇×#33068 #30712319

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1948	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1920	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1864	C₂	1	4	yes	maximal
6	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1802	C₂	1	2	yes	maximal
8	C₂×#125×C₆₇	—	2	1	no	—
9	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	—	2	1	no	—
12	C₂×#106×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#102×C₆₇	C₂²	1	4	yes	—
15	#389×C₆₇	—	2	2	no	—
16	#389×C₆₇	C₂²	1	4	yes	—
17	#384×C₆₇	C₂²	1	2	yes	—
18	#382×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
22	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	#340×C₆₇	C₂²	1	4	yes	—
25	#340×C₆₇	C₂²	1	4	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	—	2	2	no	—
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	#125×C₆₇	—	4	4	no	—
32	C₄×D₄×C₆₇	—	2	4	no	—
33	C₄×D₄×C₆₇	—	2	4	no	—
34	C₄×D₄×C₆₇	—	2	2	no	—
35	C₄×D₄×C₆₇	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	—	4	4	no	—
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#106×C₆₇	—	2	8	no	—
53	C₄⋊C₈×C₆₇	—	2	4	no	—
54	#102×C₆₇	—	2	8	no	—
55	#95×C₆₇	—	2	2	no	—
56	#95×C₆₇	—	2	1	no	—
57	#95×C₆₇	—	2	4	no	—
58	C₂⁴×C₆₇	—	2	1	no	—
59	C₂×D₄×C₆₇	—	2	1	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂×D₄×C₆₇	#41	1	4	yes	—
63	C₂²×C₄×C₆₇	—	2	8	no	—
64	C₂²×C₄×C₆₇	—	2	1	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₄⋊C₄×C₆₇	—	4	8	no	—
86	C₄⋊C₄×C₆₇	—	4	4	no	—
87	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
88	C₄⋊C₄×C₆₇	#41	1	4	yes	—
89	C₂²⋊C₄×C₆₇	—	2	8	no	—
90	C₂²⋊C₄×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	4	2	no	—
95	C₄²×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	—	4	2	no	—
97	C₄²×C₆₇	SD₁₆	1	4	yes	—
98	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
99	C₄²×C₆₇	#41	1	2	yes	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	2	4	no	—
102	C₂³×C₆₇	—	4	1	no	—
103	C₂³×C₆₇	—	4	1	no	—
104	C₂³×C₆₇	—	4	2	no	—
105	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₆₇	—	2	4	no	—
107	D₄×C₆₇	—	2	2	no	—
108	D₄×C₆₇	#125	1	4	yes	complemented
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	4	16	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	8	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
135	C₈×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	—	2	4	no	—
137	C₂²×C₆₇	—	4	2	no	—
138	C₂²×C₆₇	—	4	4	no	—
139	C₂²×C₆₇	—	4	4	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₆₇	#525	1	2	yes	—
143	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂²×C₆₇	#575	1	2	yes	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	2	4	no	—
148	C₄×C₆₇	—	4	4	no	—
149	C₄×C₆₇	—	4	2	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	—	8	4	no	—
152	C₄×C₆₇	#476	1	4	yes	—
153	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
154	C₄×C₆₇	#529	1	2	yes	—
155	C₄×C₆₇	#533	1	1	yes	characteristic
156	#33068	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	—	4	4	no	—
158	C₂×C₆₇	#2851	1	1	yes	characteristic
159	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
160	C₂×C₆₇	#3080	1	1	yes	characteristic
161	C₂×C₆₇	#3101	1	2	yes	—
162	C₂×C₆₇	#3115	1	2	yes	—
163	#2859	C₂×C₆₇	1	1	yes	characteristic, complemented
164	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
165	#1948	C₂×C₆₇	1	2	yes	complemented
166	#1920	C₂×C₆₇	1	4	yes	complemented
167	#1864	C₂×C₆₇	1	4	yes	—
168	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#1802	C₂×C₆₇	1	2	yes	—
170	C₆₇	#33068	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#125	—	2	1	no	—
172	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
173	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
174	C₂²×C₄²	—	2	1	no	—
175	C₂×#106	C₂²×C₆₇	1	4	yes	—
176	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
177	C₂×#102	C₂²×C₆₇	1	4	yes	—
178	#389	—	2	2	no	—
179	#389	C₂²×C₆₇	1	4	yes	—
180	#384	C₂²×C₆₇	1	2	yes	—
181	#382	—	2	1	no	—
182	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
183	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
184	C₄×C₂²⋊C₄	—	2	1	no	—
185	C₄×C₂²⋊C₄	—	2	2	no	—
186	C₄³	C₂²×C₆₇	1	1	yes	characteristic
187	#340	C₂²×C₆₇	1	4	yes	—
188	#340	C₂²×C₆₇	1	4	yes	—
189	#336	C₂²×C₆₇	1	4	yes	—
190	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	—	2	1	no	—
193	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
194	#125	—	4	4	no	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	Q₈×C₆₇	1	4	yes	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
203	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	4	4	no	—
210	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₆₇	1	2	yes	—
212	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
214	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
215	#106	—	2	8	no	—
216	C₄⋊C₈	—	2	4	no	—
217	#102	—	2	8	no	—
218	#95	—	2	2	no	—
219	#95	—	2	1	no	—
220	#95	—	2	4	no	—
221	C₂⁴	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
225	C₂×D₄	#41×C₆₇	1	4	yes	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	8	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
251	C₄⋊C₄	#41×C₆₇	1	4	yes	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	4	no	—
259	C₄²	—	4	2	no	—
260	C₄²	SD₁₆×C₆₇	1	4	yes	—
261	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
262	C₄²	#41×C₆₇	1	2	yes	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	2	no	—
268	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	D₄	#125×C₆₇	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	16	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
290	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
291	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
292	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
293	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
294	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
296	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
305	C₂²	#525×C₆₇	1	2	yes	—
306	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
307	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
308	C₂²	#575×C₆₇	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	8	4	no	—
315	C₄	#476×C₆₇	1	4	yes	—
316	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
317	C₄	#529×C₆₇	1	2	yes	—
318	C₄	#533×C₆₇	1	1	yes	characteristic
319	C₂	—	4	4	no	—
320	C₂	C₆₇×#2851	1	1	yes	characteristic
321	C₂	C₂×#476×C₆₇	1	1	yes	characteristic
322	C₂	C₆₇×#3080	1	1	yes	characteristic
323	C₂	C₆₇×#3101	1	2	yes	—
324	C₂	C₆₇×#3115	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2859	C₂	semidirect
3. C₆₇×#1948	C₂	semidirect
4. C₆₇×#1920	C₂	semidirect
6. C₆₇×#1831	C₂	semidirect
108. D₄×C₆₇	#125	semidirect
156. #33068	C₆₇	direct (N × H)
163. #2859	C₂×C₆₇	semidirect
165. #1948	C₂×C₆₇	semidirect
166. #1920	C₂×C₆₇	semidirect
168. #1831	C₂×C₆₇	semidirect
170. C₆₇	#33068	direct (N × H)
271. D₄	#125×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2859	2
C₄²×D₄×C₆₇	2
C₆₇×#1948	2
C₆₇×#1920	2
C₆₇×#1864	2
C₆₇×#1831	2
C₆₇×#1802	2
#33068	67

Subgroups — C67×#33068 #30712319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33068 #30712319