Subgroups — C₆₇×#12734 #30691985

Order 17152.

932 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#359×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#41×C₆₇	—	2	1	no	—
6	C₂×#135×C₆₇	—	2	8	no	—
7	C₂×#135×C₆₇	—	2	1	no	—
8	C₂×#135×C₆₇	—	2	1	no	—
9	C₂×#135×C₆₇	C₄	1	2	yes	—
10	C₂×#135×C₆₇	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₂²×SD₁₆×C₆₇	—	2	1	no	—
13	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	complemented
16	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
17	#407×C₆₇	—	2	1	no	—
18	#359×C₆₇	—	2	8	no	—
19	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
20	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
21	#341×C₆₇	—	2	1	no	—
22	C₂×#41×C₆₇	—	2	8	no	—
23	C₂×#41×C₆₇	—	2	2	no	—
24	C₂×#41×C₆₇	—	2	2	no	—
25	C₂×#41×C₆₇	—	4	4	no	—
26	C₂×#41×C₆₇	—	4	1	no	—
27	C₂×#41×C₆₇	—	4	1	no	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₂²×D₄×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	4	1	no	—
31	C₂³×C₄×C₆₇	—	4	1	no	—
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	#135×C₆₇	—	2	8	no	—
34	#135×C₆₇	—	2	8	no	—
35	#135×C₆₇	—	2	4	no	—
36	#135×C₆₇	—	2	4	no	—
37	#135×C₆₇	—	2	4	no	—
38	#135×C₆₇	D₄	1	8	yes	—
39	C₂×Q₁₆×C₆₇	—	2	4	no	—
40	C₂×Q₁₆×C₆₇	—	2	1	no	—
41	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×SD₁₆×C₆₇	—	2	8	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×SD₁₆×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	4	no	—
46	C₂×D₈×C₆₇	—	2	1	no	—
47	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	4	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₆₇	—	2	2	no	—
57	C₂×C₁₆×C₆₇	—	2	4	no	—
58	#108×C₆₇	—	2	4	no	—
59	C₂²⋊C₈×C₆₇	—	4	1	no	—
60	C₂²⋊C₈×C₆₇	—	4	1	no	—
61	C₂⁴×C₆₇	—	4	1	no	—
62	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
63	#41×C₆₇	—	2	8	no	—
64	#41×C₆₇	—	2	8	no	—
65	#41×C₆₇	—	4	8	no	—
66	#41×C₆₇	—	4	4	no	—
67	#41×C₆₇	—	4	8	no	—
68	#41×C₆₇	—	4	4	no	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂×Q₈×C₆₇	—	2	2	no	—
71	C₂×Q₈×C₆₇	—	4	2	no	—
72	C₂×Q₈×C₆₇	—	4	1	no	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	2	2	no	—
75	C₂×D₄×C₆₇	—	4	2	no	—
76	C₂×D₄×C₆₇	—	4	8	no	—
77	C₂×D₄×C₆₇	—	4	2	no	—
78	C₂×D₄×C₆₇	—	4	2	no	—
79	C₂×D₄×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	4	8	no	—
83	C₂²×C₄×C₆₇	—	4	1	no	—
84	C₂²×C₄×C₆₇	—	4	2	no	—
85	C₂²×C₄×C₆₇	—	4	2	no	—
86	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₆₇	—	2	4	no	—
93	Q₁₆×C₆₇	—	2	2	no	—
94	Q₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
95	SD₁₆×C₆₇	—	2	8	no	—
96	SD₁₆×C₆₇	—	2	8	no	—
97	D₈×C₆₇	—	2	4	no	—
98	D₈×C₆₇	—	2	2	no	—
99	D₈×C₆₇	C₂²⋊C₄	1	4	yes	—
100	M₁₆×C₆₇	—	4	2	no	—
101	M₁₆×C₆₇	—	4	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	2	no	—
108	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	2	2	no	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	—	4	1	no	—
117	C₂³×C₆₇	—	4	4	no	—
118	C₂³×C₆₇	—	4	2	no	—
119	C₂³×C₆₇	—	8	2	no	—
120	C₂³×C₆₇	—	8	2	no	—
121	C₂³×C₆₇	#102	1	1	yes	characteristic
122	C₂³×C₆₇	#102	1	1	yes	characteristic
123	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
124	Q₈×C₆₇	—	2	4	no	—
125	Q₈×C₆₇	—	4	4	no	—
126	Q₈×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	2	4	no	—
128	D₄×C₆₇	—	4	4	no	—
129	D₄×C₆₇	—	4	8	no	—
130	D₄×C₆₇	—	4	8	no	—
131	D₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	4	8	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	8	no	—
143	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#120	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
149	C₈×C₆₇	—	2	1	no	—
150	C₈×C₆₇	—	2	1	no	—
151	C₈×C₆₇	—	8	2	no	—
152	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₈×C₆₇	#120	1	2	yes	—
154	C₂²×C₆₇	—	2	1	no	—
155	C₂²×C₆₇	—	2	1	no	—
156	C₂²×C₆₇	—	2	1	no	—
157	C₂²×C₆₇	—	2	1	no	—
158	C₂²×C₆₇	—	2	1	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	4	4	no	—
162	C₂²×C₆₇	—	4	4	no	—
163	C₂²×C₆₇	—	8	8	no	—
164	C₂²×C₆₇	—	8	2	no	—
165	C₂²×C₆₇	—	8	2	no	—
166	C₂²×C₆₇	#359	1	2	yes	complemented
167	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
168	C₂²×C₆₇	#450	1	2	yes	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	#379	1	1	yes	characteristic
173	C₄×C₆₇	#418	1	1	yes	characteristic
174	C₄×C₆₇	#447	1	2	yes	—
175	#12734	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	2	2	no	—
177	C₂×C₆₇	—	2	2	no	—
178	C₂×C₆₇	—	4	1	no	—
179	C₂×C₆₇	—	8	4	no	—
180	C₂×C₆₇	#1786	1	1	yes	characteristic
181	C₂×C₆₇	C₂×#359	1	1	yes	characteristic
182	C₂×C₆₇	#2143	1	1	yes	characteristic
183	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
184	C₂×#359	C₂×C₆₇	1	4	yes	complemented
185	C₂×C₂²⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
186	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
187	C₆₇	#12734	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×#41	—	2	1	no	—
189	C₂×#135	—	2	8	no	—
190	C₂×#135	—	2	1	no	—
191	C₂×#135	—	2	1	no	—
192	C₂×#135	C₄×C₆₇	1	2	yes	—
193	C₂×#135	C₂²×C₆₇	1	2	yes	—
194	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
195	C₂²×SD₁₆	—	2	1	no	—
196	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
197	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
198	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	complemented
199	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
200	#407	—	2	1	no	—
201	#359	—	2	8	no	—
202	C₂²⋊C₁₆	—	2	1	no	—
203	C₂²⋊C₁₆	—	2	1	no	—
204	#341	—	2	1	no	—
205	C₂×#41	—	2	8	no	—
206	C₂×#41	—	2	2	no	—
207	C₂×#41	—	2	2	no	—
208	C₂×#41	—	4	4	no	—
209	C₂×#41	—	4	1	no	—
210	C₂×#41	—	4	1	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	4	1	no	—
214	C₂³×C₄	—	4	1	no	—
215	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
216	#135	—	2	8	no	—
217	#135	—	2	8	no	—
218	#135	—	2	4	no	—
219	#135	—	2	4	no	—
220	#135	—	2	4	no	—
221	#135	D₄×C₆₇	1	8	yes	—
222	C₂×Q₁₆	—	2	4	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
225	C₂×SD₁₆	—	2	8	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×D₈	—	2	4	no	—
229	C₂×D₈	—	2	1	no	—
230	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂²×C₈	—	2	4	no	—
233	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
236	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
237	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
238	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
239	C₂×C₁₆	—	2	2	no	—
240	C₂×C₁₆	—	2	4	no	—
241	#108	—	2	4	no	—
242	C₂²⋊C₈	—	4	1	no	—
243	C₂²⋊C₈	—	4	1	no	—
244	C₂⁴	—	4	1	no	—
245	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
246	#41	—	2	8	no	—
247	#41	—	2	8	no	—
248	#41	—	4	8	no	—
249	#41	—	4	4	no	—
250	#41	—	4	8	no	—
251	#41	—	4	4	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	8	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	4	8	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
272	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
273	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
275	Q₁₆	—	2	4	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
278	SD₁₆	—	2	8	no	—
279	SD₁₆	—	2	8	no	—
280	D₈	—	2	4	no	—
281	D₈	—	2	2	no	—
282	D₈	C₂²⋊C₄×C₆₇	1	4	yes	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
292	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
293	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
295	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₁₆	—	4	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	—	8	2	no	—
304	C₂³	#102×C₆₇	1	1	yes	characteristic
305	C₂³	#102×C₆₇	1	1	yes	characteristic
306	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	4	2	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	8	no	—
313	D₄	—	4	8	no	—
314	D₄	—	4	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	8	no	—
326	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
327	C₂×C₄	#102×C₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄	#120×C₆₇	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
331	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	8	2	no	—
335	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
336	C₈	#120×C₆₇	1	2	yes	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	8	8	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	#359×C₆₇	1	2	yes	complemented
350	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
351	C₂²	#450×C₆₇	1	2	yes	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	4	no	—
355	C₄	#379×C₆₇	1	1	yes	characteristic
356	C₄	#418×C₆₇	1	1	yes	characteristic
357	C₄	#447×C₆₇	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	2	2	no	—
360	C₂	—	4	1	no	—
361	C₂	—	8	4	no	—
362	C₂	C₆₇×#1786	1	1	yes	characteristic
363	C₂	C₂×#359×C₆₇	1	1	yes	characteristic
364	C₂	C₆₇×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#359×C₆₇	C₂	semidirect
3. C₂×C₂²⋊C₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1810	C₂	semidirect
15. C₂²×C₁₆×C₆₇	C₂²	semidirect
16. C₂×#108×C₆₇	C₂²	semidirect
166. C₂²×C₆₇	#359	semidirect
175. #12734	C₆₇	direct (N × H)
184. C₂×#359	C₂×C₆₇	semidirect
185. C₂×C₂²⋊C₁₆	C₂×C₆₇	semidirect
186. #1810	C₂×C₆₇	semidirect
187. C₆₇	#12734	direct (N × H)
198. C₂²×C₁₆	C₂²×C₆₇	semidirect
199. C₂×#108	C₂²×C₆₇	semidirect
349. C₂²	#359×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂×#359×C₆₇	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₆₇×#1810	2
#12734	67

Subgroups — C67×#12734 #30691985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#12734 #30691985