Subgroups — C₆₇×#20067 #30699318

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3373	C₂	1	2	yes	maximal, complemented
2	D₄×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1715	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1686	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1681	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1651	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1619	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
10	C₂²×SD₁₆×C₆₇	—	2	2	no	—
11	#559×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
13	D₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#493×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	2	yes	—
17	#489×C₆₇	—	2	2	no	—
18	#486×C₆₇	C₂²	1	2	yes	complemented
19	#476×C₆₇	—	2	2	no	—
20	#475×C₆₇	C₂²	1	2	yes	complemented
21	#465×C₆₇	—	2	2	no	—
22	#463×C₆₇	—	2	2	no	—
23	#461×C₆₇	C₂²	1	2	yes	—
24	#460×C₆₇	C₂²	1	2	yes	—
25	#459×C₆₇	C₂²	1	2	yes	—
26	#450×C₆₇	—	2	2	no	—
27	#448×C₆₇	—	2	2	no	—
28	#446×C₆₇	C₂²	1	2	yes	complemented
29	#442×C₆₇	—	2	2	no	—
30	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
31	C₈×D₄×C₆₇	C₂²	1	2	yes	—
32	#335×C₆₇	C₂²	1	2	yes	complemented
33	#332×C₆₇	C₂²	1	2	yes	complemented
34	#331×C₆₇	C₂²	1	1	yes	characteristic
35	#331×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	complemented
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×SD₁₆×C₆₇	—	2	2	no	—
45	C₂×SD₁₆×C₆₇	—	4	8	no	—
46	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
47	C₂×D₈×C₆₇	—	2	2	no	—
48	C₂×D₈×C₆₇	—	4	1	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
52	#127×C₆₇	C₂³	1	1	yes	characteristic
53	#127×C₆₇	C₂³	1	1	yes	characteristic
54	#124×C₆₇	—	2	2	no	—
55	#123×C₆₇	—	4	1	no	—
56	C₂²⋊Q₈×C₆₇	—	2	2	no	—
57	C₂²⋊Q₈×C₆₇	—	4	2	no	—
58	#121×C₆₇	—	2	2	no	—
59	#120×C₆₇	—	4	1	no	—
60	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	C₂³	1	2	yes	—
64	#106×C₆₇	—	2	2	no	—
65	#106×C₆₇	—	4	1	no	—
66	#106×C₆₇	D₄	1	2	yes	—
67	C₄⋊C₈×C₆₇	—	2	2	no	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	2	no	—
72	#103×C₆₇	D₄	1	2	yes	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	2	no	—
77	#102×C₆₇	D₄	1	2	yes	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	#97×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂⁴×C₆₇	—	4	1	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂×Q₈×C₆₇	—	8	2	no	—
87	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	4	4	no	—
94	C₂×D₄×C₆₇	—	8	2	no	—
95	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	4	1	no	—
98	C₂²×C₄×C₆₇	—	4	2	no	—
99	SD₁₆×C₆₇	—	2	4	no	—
100	SD₁₆×C₆₇	—	4	4	no	—
101	SD₁₆×C₆₇	—	4	4	no	—
102	SD₁₆×C₆₇	—	4	4	no	—
103	SD₁₆×C₆₇	—	8	2	no	—
104	D₈×C₆₇	—	4	4	no	—
105	D₈×C₆₇	—	8	2	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	4	4	no	—
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	4	2	no	—
116	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂²⋊C₄×C₆₇	—	4	2	no	—
120	C₂²⋊C₄×C₆₇	—	4	1	no	—
121	C₂²⋊C₄×C₆₇	—	4	1	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	4	1	no	—
127	C₂³×C₆₇	—	4	1	no	—
128	C₂³×C₆₇	—	8	4	no	—
129	C₂³×C₆₇	—	16	1	no	—
130	Q₈×C₆₇	—	2	2	no	—
131	Q₈×C₆₇	—	2	2	no	—
132	Q₈×C₆₇	—	4	2	no	—
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	8	4	no	—
135	D₄×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	4	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	8	1	no	—
141	D₄×C₆₇	—	8	1	no	—
142	D₄×C₆₇	—	8	4	no	—
143	D₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	8	4	no	—
151	C₂×C₄×C₆₇	—	16	1	no	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₈×C₆₇	—	2	2	no	—
155	C₈×C₆₇	—	2	2	no	—
156	C₈×C₆₇	—	2	2	no	—
157	C₈×C₆₇	—	4	2	no	—
158	C₈×C₆₇	—	4	2	no	—
159	C₂²×C₆₇	—	2	4	no	—
160	C₂²×C₆₇	—	4	2	no	—
161	C₂²×C₆₇	—	4	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	—	8	1	no	—
165	C₂²×C₆₇	—	8	1	no	—
166	C₂²×C₆₇	—	16	4	no	—
167	C₂²×C₆₇	D₄²	1	1	yes	characteristic
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	4	2	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	C₄×C₆₇	#577	1	2	yes	—
175	#20067	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	4	4	no	—
177	C₂×C₆₇	—	16	1	no	—
178	C₂×C₆₇	—	16	1	no	—
179	C₂×C₆₇	#3265	1	2	yes	—
180	C₂×C₆₇	#3291	1	1	yes	characteristic
181	#3373	C₂×C₆₇	1	2	yes	complemented
182	D₄×SD₁₆	C₂×C₆₇	1	2	yes	complemented
183	#1715	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#1686	C₂×C₆₇	1	2	yes	complemented
185	#1681	C₂×C₆₇	1	2	yes	complemented
186	#1651	C₂×C₆₇	1	2	yes	complemented
187	#1620	C₂×C₆₇	1	1	yes	characteristic, complemented
188	#1619	C₂×C₆₇	1	1	yes	characteristic
189	#1585	C₂×C₆₇	1	2	yes	complemented
190	C₆₇	#20067	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×SD₁₆	—	2	2	no	—
192	#559	C₂²×C₆₇	1	1	yes	characteristic, complemented
193	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
194	D₄²	C₂²×C₆₇	1	1	yes	characteristic
195	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
196	#493	C₂²×C₆₇	1	1	yes	characteristic
197	#492	C₂²×C₆₇	1	2	yes	—
198	#489	—	2	2	no	—
199	#486	C₂²×C₆₇	1	2	yes	complemented
200	#476	—	2	2	no	—
201	#475	C₂²×C₆₇	1	2	yes	complemented
202	#465	—	2	2	no	—
203	#463	—	2	2	no	—
204	#461	C₂²×C₆₇	1	2	yes	—
205	#460	C₂²×C₆₇	1	2	yes	—
206	#459	C₂²×C₆₇	1	2	yes	—
207	#450	—	2	2	no	—
208	#448	—	2	2	no	—
209	#446	C₂²×C₆₇	1	2	yes	complemented
210	#442	—	2	2	no	—
211	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
212	C₈×D₄	C₂²×C₆₇	1	2	yes	—
213	#335	C₂²×C₆₇	1	2	yes	complemented
214	#332	C₂²×C₆₇	1	2	yes	complemented
215	#331	C₂²×C₆₇	1	1	yes	characteristic
216	#331	C₂²×C₆₇	1	1	yes	characteristic, complemented
217	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	complemented
218	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
219	#322	C₂²×C₆₇	1	1	yes	characteristic, complemented
220	C₂²×Q₈	—	2	2	no	—
221	C₂²×D₄	—	2	2	no	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	4	8	no	—
227	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
228	C₂×D₈	—	2	2	no	—
229	C₂×D₈	—	4	1	no	—
230	C₂²×C₈	—	2	2	no	—
231	C₄⋊Q₈	—	2	2	no	—
232	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
233	#127	C₂³×C₆₇	1	1	yes	characteristic
234	#127	C₂³×C₆₇	1	1	yes	characteristic
235	#124	—	2	2	no	—
236	#123	—	4	1	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₂²⋊Q₈	—	4	2	no	—
239	#121	—	2	2	no	—
240	#120	—	4	1	no	—
241	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	C₂³×C₆₇	1	2	yes	—
245	#106	—	2	2	no	—
246	#106	—	4	1	no	—
247	#106	D₄×C₆₇	1	2	yes	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
250	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
251	#103	—	2	2	no	—
252	#103	—	2	2	no	—
253	#103	D₄×C₆₇	1	2	yes	—
254	#102	—	2	2	no	—
255	#102	—	2	2	no	—
256	#102	—	2	2	no	—
257	#102	—	2	2	no	—
258	#102	D₄×C₆₇	1	2	yes	—
259	C₂²⋊C₈	—	2	2	no	—
260	#97	—	2	2	no	—
261	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₆₇	1	2	yes	—
263	C₂⁴	—	4	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×Q₈	—	8	2	no	—
268	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	4	no	—
275	C₂×D₄	—	8	2	no	—
276	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	—	4	2	no	—
280	SD₁₆	—	2	4	no	—
281	SD₁₆	—	4	4	no	—
282	SD₁₆	—	4	4	no	—
283	SD₁₆	—	4	4	no	—
284	SD₁₆	—	8	2	no	—
285	D₈	—	4	4	no	—
286	D₈	—	8	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	4	no	—
291	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
298	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
306	C₂³	—	2	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	1	no	—
309	C₂³	—	8	4	no	—
310	C₂³	—	16	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	8	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	4	no	—
324	D₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	8	4	no	—
332	C₂×C₄	—	16	1	no	—
333	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
334	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	2	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	1	no	—
347	C₂²	—	16	4	no	—
348	C₂²	D₄²×C₆₇	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
355	C₄	#577×C₆₇	1	2	yes	—
356	C₂	—	4	4	no	—
357	C₂	—	16	1	no	—
358	C₂	—	16	1	no	—
359	C₂	C₆₇×#3265	1	2	yes	—
360	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#1619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3373	C₂	semidirect
2. D₄×SD₁₆×C₆₇	C₂	semidirect
3. C₆₇×#1715	C₂	semidirect
4. C₆₇×#1686	C₂	semidirect
5. C₆₇×#1681	C₂	semidirect
6. C₆₇×#1651	C₂	semidirect
7. C₆₇×#1620	C₂	semidirect
9. C₆₇×#1585	C₂	semidirect
11. #559×C₆₇	C₂²	semidirect
14. #498×C₆₇	C₂²	semidirect
18. #486×C₆₇	C₂²	semidirect
20. #475×C₆₇	C₂²	semidirect
28. #446×C₆₇	C₂²	semidirect
32. #335×C₆₇	C₂²	semidirect
33. #332×C₆₇	C₂²	semidirect
35. #331×C₆₇	C₂²	semidirect
36. Q₈⋊C₈×C₆₇	C₂²	semidirect
38. #322×C₆₇	C₂²	semidirect
175. #20067	C₆₇	direct (N × H)
181. #3373	C₂×C₆₇	semidirect
182. D₄×SD₁₆	C₂×C₆₇	semidirect
183. #1715	C₂×C₆₇	semidirect
184. #1686	C₂×C₆₇	semidirect
185. #1681	C₂×C₆₇	semidirect
186. #1651	C₂×C₆₇	semidirect
187. #1620	C₂×C₆₇	semidirect
189. #1585	C₂×C₆₇	semidirect
190. C₆₇	#20067	direct (N × H)
192. #559	C₂²×C₆₇	semidirect
195. #498	C₂²×C₆₇	semidirect
199. #486	C₂²×C₆₇	semidirect
201. #475	C₂²×C₆₇	semidirect
209. #446	C₂²×C₆₇	semidirect
213. #335	C₂²×C₆₇	semidirect
214. #332	C₂²×C₆₇	semidirect
216. #331	C₂²×C₆₇	semidirect
217. Q₈⋊C₈	C₂²×C₆₇	semidirect
219. #322	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3373	2
D₄×SD₁₆×C₆₇	2
C₆₇×#1715	2
C₆₇×#1686	2
C₆₇×#1681	2
C₆₇×#1651	2
C₆₇×#1620	2
C₆₇×#1619	2
C₆₇×#1585	2
#20067	67

Subgroups — C67×#20067 #30699318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20067 #30699318