Subgroups — C₆₇×#33609 #30712860

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3058	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2220	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2217	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
13	#536×C₆₇	—	2	1	no	—
14	#535×C₆₇	—	2	1	no	—
15	C₂×SD₃₂×C₆₇	—	2	4	no	—
16	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
17	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
18	#504×C₆₇	—	2	4	no	—
19	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₆₇	C₂²	1	2	yes	complemented
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	complemented
23	#470×C₆₇	—	2	2	no	—
24	#469×C₆₇	—	2	2	no	—
25	#467×C₆₇	C₂²	1	4	yes	complemented
26	#466×C₆₇	C₂²	1	4	yes	—
27	#427×C₆₇	C₂²	1	1	yes	characteristic
28	#427×C₆₇	C₂²	1	1	yes	characteristic
29	#419×C₆₇	—	2	1	no	—
30	#418×C₆₇	—	2	1	no	—
31	#367×C₆₇	C₂²	1	2	yes	complemented
32	#366×C₆₇	C₂²	1	2	yes	complemented
33	#358×C₆₇	C₂²	1	4	yes	complemented
34	#357×C₆₇	C₂²	1	4	yes	—
35	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂×Q₁₆×C₆₇	—	2	2	no	—
39	C₂×Q₁₆×C₆₇	—	2	1	no	—
40	C₂×Q₁₆×C₆₇	—	2	2	no	—
41	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×D₈×C₆₇	—	2	2	no	—
43	C₂×D₈×C₆₇	—	2	1	no	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	C₂³	1	2	yes	—
46	#131×C₆₇	—	2	2	no	—
47	#131×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
49	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	4	1	no	—
53	#127×C₆₇	—	4	1	no	—
54	#124×C₆₇	—	4	1	no	—
55	#124×C₆₇	—	4	1	no	—
56	#123×C₆₇	—	4	2	no	—
57	C₂²⋊Q₈×C₆₇	—	2	4	no	—
58	#121×C₆₇	—	2	4	no	—
59	#120×C₆₇	—	4	2	no	—
60	#117×C₆₇	—	2	1	no	—
61	#117×C₆₇	—	2	1	no	—
62	SD₃₂×C₆₇	—	2	4	no	—
63	SD₃₂×C₆₇	—	2	4	no	—
64	M₃₂×C₆₇	—	2	2	no	—
65	M₃₂×C₆₇	—	2	4	no	—
66	C₂×C₁₆×C₆₇	—	2	4	no	—
67	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
68	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
71	#107×C₆₇	C₂³	1	2	yes	—
72	#107×C₆₇	C₂³	1	2	yes	—
73	#106×C₆₇	—	2	1	no	—
74	#106×C₆₇	—	2	1	no	—
75	#103×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	4	no	—
77	C₂⁴×C₆₇	—	4	1	no	—
78	#41×C₆₇	—	2	2	no	—
79	#41×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	4	1	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	4	4	no	—
87	C₂×D₄×C₆₇	—	4	2	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	D₈	1	2	yes	—
91	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₆₇	—	2	2	no	—
96	Q₁₆×C₆₇	—	2	2	no	—
97	Q₁₆×C₆₇	—	4	2	no	—
98	D₈×C₆₇	—	2	2	no	—
99	D₈×C₆₇	—	2	2	no	—
100	D₈×C₆₇	—	4	2	no	—
101	M₁₆×C₆₇	—	2	2	no	—
102	M₁₆×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₇	#41	1	2	yes	—
109	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	4	4	no	—
113	C₂²⋊C₄×C₆₇	—	4	1	no	—
114	C₂²⋊C₄×C₆₇	—	4	1	no	—
115	C₂²⋊C₄×C₆₇	—	4	4	no	—
116	C₂²⋊C₄×C₆₇	—	4	4	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	—	4	1	no	—
119	C₁₆×C₆₇	—	2	2	no	—
120	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
121	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
122	C₂³×C₆₇	—	4	2	no	—
123	C₂³×C₆₇	—	8	2	no	—
124	C₂³×C₆₇	—	16	2	no	—
125	C₂³×C₆₇	C₂×D₈	1	2	yes	—
126	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₆₇	—	2	2	no	—
128	Q₈×C₆₇	—	4	2	no	—
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	8	2	no	—
131	D₄×C₆₇	—	2	4	no	—
132	D₄×C₆₇	—	2	2	no	—
133	D₄×C₆₇	—	4	2	no	—
134	D₄×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	8	2	no	—
136	D₄×C₆₇	—	16	4	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	8	2	no	—
143	C₂×C₄×C₆₇	#121	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#135	1	2	yes	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	#121	1	2	yes	—
150	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	4	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	8	4	no	—
156	C₂²×C₆₇	—	8	2	no	—
157	C₂²×C₆₇	—	16	4	no	—
158	C₂²×C₆₇	#466	1	2	yes	—
159	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	8	2	no	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	#466	1	2	yes	—
165	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
166	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
167	#33609	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₆₇	—	2	2	no	—
169	C₂×C₆₇	—	4	2	no	—
170	C₂×C₆₇	—	16	2	no	—
171	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
172	C₂×C₆₇	#3416	1	2	yes	—
173	C₂²×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
174	C₂×#504	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#3059	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#3058	C₂×C₆₇	1	1	yes	characteristic
177	#2220	C₂×C₆₇	1	4	yes	complemented
178	#2217	C₂×C₆₇	1	4	yes	complemented
179	#2158	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#2141	C₂×C₆₇	1	1	yes	characteristic, complemented
181	#2140	C₂×C₆₇	1	1	yes	characteristic, complemented
182	C₆₇	#33609	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
184	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
186	#536	—	2	1	no	—
187	#535	—	2	1	no	—
188	C₂×SD₃₂	—	2	4	no	—
189	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
190	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
191	#504	—	2	4	no	—
192	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×M₃₂	C₂²×C₆₇	1	2	yes	complemented
194	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
195	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	complemented
196	#470	—	2	2	no	—
197	#469	—	2	2	no	—
198	#467	C₂²×C₆₇	1	4	yes	complemented
199	#466	C₂²×C₆₇	1	4	yes	—
200	#427	C₂²×C₆₇	1	1	yes	characteristic
201	#427	C₂²×C₆₇	1	1	yes	characteristic
202	#419	—	2	1	no	—
203	#418	—	2	1	no	—
204	#367	C₂²×C₆₇	1	2	yes	complemented
205	#366	C₂²×C₆₇	1	2	yes	complemented
206	#358	C₂²×C₆₇	1	4	yes	complemented
207	#357	C₂²×C₆₇	1	4	yes	—
208	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	2	2	no	—
214	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	2	no	—
218	C₂×D₈	C₂³×C₆₇	1	2	yes	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
222	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
223	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
224	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	4	1	no	—
226	#127	—	4	1	no	—
227	#124	—	4	1	no	—
228	#124	—	4	1	no	—
229	#123	—	4	2	no	—
230	C₂²⋊Q₈	—	2	4	no	—
231	#121	—	2	4	no	—
232	#120	—	4	2	no	—
233	#117	—	2	1	no	—
234	#117	—	2	1	no	—
235	SD₃₂	—	2	4	no	—
236	SD₃₂	—	2	4	no	—
237	M₃₂	—	2	2	no	—
238	M₃₂	—	2	4	no	—
239	C₂×C₁₆	—	2	4	no	—
240	C₂×C₁₆	D₄×C₆₇	1	4	yes	complemented
241	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
242	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
244	#107	C₂³×C₆₇	1	2	yes	—
245	#107	C₂³×C₆₇	1	2	yes	—
246	#106	—	2	1	no	—
247	#106	—	2	1	no	—
248	#103	—	2	4	no	—
249	#102	—	2	4	no	—
250	C₂⁴	—	4	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	D₈×C₆₇	1	2	yes	—
264	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	4	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	4	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
281	C₂×C₈	#41×C₆₇	1	2	yes	—
282	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
294	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	16	2	no	—
298	C₂³	C₂×D₈×C₆₇	1	2	yes	—
299	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	#121×C₆₇	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
319	C₂×C₄	#135×C₆₇	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#121×C₆₇	1	2	yes	—
323	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
324	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	—	16	4	no	—
331	C₂²	#466×C₆₇	1	2	yes	—
332	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	#466×C₆₇	1	2	yes	—
338	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
339	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
344	C₂	C₆₇×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#41×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₃₂×C₆₇	C₂	semidirect
2. C₂×#504×C₆₇	C₂	semidirect
3. C₆₇×#3059	C₂	semidirect
5. C₆₇×#2220	C₂	semidirect
6. C₆₇×#2217	C₂	semidirect
7. C₆₇×#2158	C₂	semidirect
8. C₆₇×#2141	C₂	semidirect
9. C₆₇×#2140	C₂	semidirect
20. C₂×M₃₂×C₆₇	C₂²	semidirect
22. C₂²×C₁₆×C₆₇	C₂²	semidirect
25. #467×C₆₇	C₂²	semidirect
31. #367×C₆₇	C₂²	semidirect
32. #366×C₆₇	C₂²	semidirect
33. #358×C₆₇	C₂²	semidirect
67. C₂×C₁₆×C₆₇	D₄	semidirect
167. #33609	C₆₇	direct (N × H)
173. C₂²×SD₃₂	C₂×C₆₇	semidirect
174. C₂×#504	C₂×C₆₇	semidirect
175. #3059	C₂×C₆₇	semidirect
177. #2220	C₂×C₆₇	semidirect
178. #2217	C₂×C₆₇	semidirect
179. #2158	C₂×C₆₇	semidirect
180. #2141	C₂×C₆₇	semidirect
181. #2140	C₂×C₆₇	semidirect
182. C₆₇	#33609	direct (N × H)
193. C₂×M₃₂	C₂²×C₆₇	semidirect
195. C₂²×C₁₆	C₂²×C₆₇	semidirect
198. #467	C₂²×C₆₇	semidirect
204. #367	C₂²×C₆₇	semidirect
205. #366	C₂²×C₆₇	semidirect
206. #358	C₂²×C₆₇	semidirect
240. C₂×C₁₆	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₆₇	2
C₂×#504×C₆₇	2
C₆₇×#3059	2
C₆₇×#3058	2
C₆₇×#2220	2
C₆₇×#2217	2
C₆₇×#2158	2
C₆₇×#2141	2
C₆₇×#2140	2
#33609	67

Subgroups — C67×#33609 #30712860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33609 #30712860