Subgroups — C₆₇×#19205 #30698456

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3246	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1690	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1688	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1666	C₂	1	4	yes	maximal, complemented
7	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×#135×C₆₇	—	2	1	no	—
10	C₂×#135×C₆₇	—	2	1	no	—
11	#533×C₆₇	C₂²	1	1	yes	characteristic
12	#532×C₆₇	C₂²	1	2	yes	—
13	#495×C₆₇	—	2	1	no	—
14	#495×C₆₇	—	2	1	no	—
15	#494×C₆₇	C₂²	1	1	yes	characteristic
16	#493×C₆₇	C₂²	1	1	yes	characteristic
17	#492×C₆₇	C₂²	1	2	yes	—
18	#476×C₆₇	—	2	2	no	—
19	#475×C₆₇	C₂²	1	4	yes	—
20	#465×C₆₇	—	2	4	no	—
21	#461×C₆₇	—	2	2	no	—
22	#460×C₆₇	C₂²	1	4	yes	—
23	#446×C₆₇	—	2	1	no	—
24	#446×C₆₇	C₂²	1	2	yes	—
25	C₈×Q₈×C₆₇	—	2	1	no	—
26	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
27	#424×C₆₇	—	2	1	no	—
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
30	#405×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#334×C₆₇	C₂²	1	2	yes	—
33	#334×C₆₇	C₂²	1	2	yes	—
34	#332×C₆₇	C₂²	1	4	yes	—
35	#331×C₆₇	C₂²	1	4	yes	—
36	C₂×#41×C₆₇	—	2	2	no	—
37	#135×C₆₇	—	4	4	no	—
38	#135×C₆₇	—	4	4	no	—
39	C₂×Q₁₆×C₆₇	—	2	1	no	—
40	C₂×Q₁₆×C₆₇	—	2	1	no	—
41	C₂×SD₁₆×C₆₇	—	2	4	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	—	2	1	no	—
45	C₂×D₈×C₆₇	—	2	1	no	—
46	C₂×M₁₆×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
53	#127×C₆₇	C₂³	1	2	yes	—
54	#125×C₆₇	—	2	1	no	—
55	#124×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	—	2	1	no	—
57	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
58	#121×C₆₇	—	2	2	no	—
59	#121×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	—	2	1	no	—
61	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	#117×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#106×C₆₇	—	2	4	no	—
67	C₄⋊C₈×C₆₇	—	2	2	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	#103×C₆₇	—	2	4	no	—
74	#102×C₆₇	—	2	4	no	—
75	#97×C₆₇	—	2	2	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#41×C₆₇	—	4	4	no	—
82	#41×C₆₇	—	4	4	no	—
83	C₂×Q₈×C₆₇	—	2	1	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
94	Q₁₆×C₆₇	—	4	2	no	—
95	Q₁₆×C₆₇	—	4	2	no	—
96	SD₁₆×C₆₇	—	4	4	no	—
97	SD₁₆×C₆₇	—	4	4	no	—
98	SD₁₆×C₆₇	—	8	4	no	—
99	D₈×C₆₇	—	4	2	no	—
100	D₈×C₆₇	—	4	2	no	—
101	M₁₆×C₆₇	—	4	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	8	2	no	—
106	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂²⋊C₄×C₆₇	—	4	2	no	—
121	C₂²⋊C₄×C₆₇	—	4	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	#41	1	1	yes	characteristic
126	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	#121	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	4	no	—
130	Q₈×C₆₇	—	4	2	no	—
131	Q₈×C₆₇	—	4	2	no	—
132	Q₈×C₆₇	—	4	2	no	—
133	D₄×C₆₇	—	4	2	no	—
134	D₄×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	8	2	no	—
138	D₄×C₆₇	—	8	4	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	8	4	no	—
145	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
149	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	—	2	1	no	—
155	C₈×C₆₇	—	4	2	no	—
156	C₈×C₆₇	—	4	4	no	—
157	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
158	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	#460	1	2	yes	—
165	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	4	2	no	—
171	C₄×C₆₇	—	8	2	no	—
172	C₄×C₆₇	#460	1	2	yes	—
173	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
174	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₄×C₆₇	#577	1	2	yes	—
176	#19205	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂×C₆₇	—	2	2	no	—
178	C₂×C₆₇	—	16	2	no	—
179	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
180	C₂×C₆₇	#3060	1	1	yes	characteristic
181	C₂×C₆₇	#3067	1	1	yes	characteristic
182	#3246	C₂×C₆₇	1	2	yes	complemented
183	#3147	C₂×C₆₇	1	1	yes	characteristic
184	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#1690	C₂×C₆₇	1	2	yes	complemented
186	#1688	C₂×C₆₇	1	2	yes	complemented
187	#1666	C₂×C₆₇	1	4	yes	complemented
188	C₂×#334	C₂×C₆₇	1	1	yes	characteristic, complemented
189	#1547	C₂×C₆₇	1	2	yes	complemented
190	C₆₇	#19205	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×#135	—	2	1	no	—
192	C₂×#135	—	2	1	no	—
193	#533	C₂²×C₆₇	1	1	yes	characteristic
194	#532	C₂²×C₆₇	1	2	yes	—
195	#495	—	2	1	no	—
196	#495	—	2	1	no	—
197	#494	C₂²×C₆₇	1	1	yes	characteristic
198	#493	C₂²×C₆₇	1	1	yes	characteristic
199	#492	C₂²×C₆₇	1	2	yes	—
200	#476	—	2	2	no	—
201	#475	C₂²×C₆₇	1	4	yes	—
202	#465	—	2	4	no	—
203	#461	—	2	2	no	—
204	#460	C₂²×C₆₇	1	4	yes	—
205	#446	—	2	1	no	—
206	#446	C₂²×C₆₇	1	2	yes	—
207	C₈×Q₈	—	2	1	no	—
208	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
209	#424	—	2	1	no	—
210	#423	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
212	#405	—	2	1	no	—
213	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
214	#334	C₂²×C₆₇	1	2	yes	—
215	#334	C₂²×C₆₇	1	2	yes	—
216	#332	C₂²×C₆₇	1	4	yes	—
217	#331	C₂²×C₆₇	1	4	yes	—
218	C₂×#41	—	2	2	no	—
219	#135	—	4	4	no	—
220	#135	—	4	4	no	—
221	C₂×Q₁₆	—	2	1	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×SD₁₆	—	2	4	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	—	2	1	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
232	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
233	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
234	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
235	#127	C₂³×C₆₇	1	2	yes	—
236	#125	—	2	1	no	—
237	#124	—	2	2	no	—
238	C₂²⋊Q₈	—	2	1	no	—
239	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
240	#121	—	2	2	no	—
241	#121	—	2	2	no	—
242	C₄×Q₈	—	2	1	no	—
243	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
244	C₄×D₄	—	2	2	no	—
245	C₄×D₄	—	2	2	no	—
246	#117	—	2	1	no	—
247	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
248	#106	—	2	4	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
254	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
255	#103	—	2	4	no	—
256	#102	—	2	4	no	—
257	#97	—	2	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
260	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
263	#41	—	4	4	no	—
264	#41	—	4	4	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
276	Q₁₆	—	4	2	no	—
277	Q₁₆	—	4	2	no	—
278	SD₁₆	—	4	4	no	—
279	SD₁₆	—	4	4	no	—
280	SD₁₆	—	8	4	no	—
281	D₈	—	4	2	no	—
282	D₈	—	4	2	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	8	2	no	—
288	C₂×C₈	SD₁₆×C₆₇	1	4	yes	complemented
289	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₄²	#41×C₆₇	1	1	yes	characteristic
308	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
309	C₂³	—	4	2	no	—
310	C₂³	#121×C₆₇	1	1	yes	characteristic
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
328	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
329	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
331	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	4	no	—
339	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
340	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#460×C₆₇	1	2	yes	—
347	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	#460×C₆₇	1	2	yes	—
355	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
356	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
357	C₄	#577×C₆₇	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₂×#460×C₆₇	1	1	yes	characteristic
361	C₂	C₆₇×#3060	1	1	yes	characteristic
362	C₂	C₆₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3246	C₂	semidirect
3. C₆₇×#2960	C₂	semidirect
4. C₆₇×#1690	C₂	semidirect
5. C₆₇×#1688	C₂	semidirect
6. C₆₇×#1666	C₂	semidirect
7. C₂×#334×C₆₇	C₂	semidirect
8. C₆₇×#1547	C₂	semidirect
106. C₂×C₈×C₆₇	SD₁₆	semidirect
176. #19205	C₆₇	direct (N × H)
182. #3246	C₂×C₆₇	semidirect
184. #2960	C₂×C₆₇	semidirect
185. #1690	C₂×C₆₇	semidirect
186. #1688	C₂×C₆₇	semidirect
187. #1666	C₂×C₆₇	semidirect
188. C₂×#334	C₂×C₆₇	semidirect
189. #1547	C₂×C₆₇	semidirect
190. C₆₇	#19205	direct (N × H)
288. C₂×C₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3246	2
C₆₇×#3147	2
C₆₇×#2960	2
C₆₇×#1690	2
C₆₇×#1688	2
C₆₇×#1666	2
C₂×#334×C₆₇	2
C₆₇×#1547	2
#19205	67

Subgroups — C67×#19205 #30698456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19205 #30698456