Subgroups — C₆₇×#11074 #30690325

Order 17152.

866 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic
2	C₂×#419×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2310	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1740	C₂	1	1	yes	maximal, characteristic
5	C₂×H(3,ℤ/4)×C₆₇	C₂	1	2	yes	maximal
6	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂×#117×C₆₇	C₂²	1	2	yes	—
9	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
11	#424×C₆₇	—	2	4	no	—
12	#423×C₆₇	—	2	2	no	—
13	#419×C₆₇	—	2	4	no	—
14	#419×C₆₇	—	2	4	no	—
15	C₂×#103×C₆₇	—	2	4	no	—
16	#391×C₆₇	—	2	2	no	—
17	#391×C₆₇	C₄	1	4	yes	complemented
18	#381×C₆₇	—	2	2	no	—
19	#342×C₆₇	—	2	2	no	—
20	H(3,ℤ/4)×C₆₇	—	2	4	no	—
21	C₂²×Q₈×C₆₇	—	2	1	no	—
22	C₂²×Q₈×C₆₇	—	2	1	no	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×Q₈×C₆₇	—	2	2	no	—
25	C₂²×Q₈×C₆₇	—	2	2	no	—
26	C₂²×Q₈×C₆₇	—	2	2	no	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
30	C₂²×Q₈×C₆₇	Q₈	1	2	yes	—
31	C₂³×C₄×C₆₇	—	2	1	no	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	D₄	1	4	yes	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	2	no	—
42	#117×C₆₇	—	2	2	no	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	—	2	2	no	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	#117×C₆₇	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	—	2	2	no	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	D₄	1	2	yes	—
57	C₄⋊C₈×C₆₇	—	2	8	no	—
58	#103×C₆₇	—	2	16	no	—
59	#95×C₆₇	—	2	8	no	—
60	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	—	2	4	no	—
65	C₂×Q₈×C₆₇	—	2	4	no	—
66	C₂×Q₈×C₆₇	—	2	4	no	—
67	C₂×Q₈×C₆₇	—	2	4	no	—
68	C₂×Q₈×C₆₇	—	2	4	no	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂×Q₈×C₆₇	—	4	1	no	—
71	C₂×Q₈×C₆₇	—	4	2	no	—
72	C₂×Q₈×C₆₇	—	4	1	no	—
73	C₂×Q₈×C₆₇	—	4	2	no	—
74	C₂×Q₈×C₆₇	—	4	1	no	—
75	C₂×Q₈×C₆₇	—	4	2	no	—
76	C₂×Q₈×C₆₇	—	4	2	no	—
77	C₂×Q₈×C₆₇	—	4	1	no	—
78	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	—	4	2	no	—
85	C₂²×C₄×C₆₇	—	4	2	no	—
86	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	2	yes	—
96	C₂²×C₄×C₆₇	#41	1	2	yes	—
97	M₁₆×C₆₇	—	4	2	no	—
98	M₁₆×C₆₇	—	4	2	no	—
99	M₁₆×C₆₇	—	4	2	no	—
100	M₁₆×C₆₇	—	4	2	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₂²⋊C₄×C₆₇	—	2	4	no	—
110	C₂²⋊C₄×C₆₇	—	2	4	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	—	2	2	no	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	4	no	—
116	C₄²×C₆₇	—	2	4	no	—
117	C₄²×C₆₇	—	4	2	no	—
118	C₂³×C₆₇	—	4	2	no	—
119	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₆₇	#121	1	2	yes	—
121	C₂³×C₆₇	#121	1	1	yes	characteristic
122	C₂³×C₆₇	#121	1	2	yes	—
123	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
124	Q₈×C₆₇	—	2	8	no	—
125	Q₈×C₆₇	—	4	4	no	—
126	Q₈×C₆₇	—	4	2	no	—
127	Q₈×C₆₇	—	4	2	no	—
128	Q₈×C₆₇	—	4	2	no	—
129	Q₈×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	4	4	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	8	4	no	—
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₆₇	#120	1	2	yes	—
146	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	2	yes	—
148	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
151	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#123	1	2	yes	—
153	C₂×C₄×C₆₇	#123	1	2	yes	—
154	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#124	1	2	yes	—
156	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
157	C₂×C₄×C₆₇	#126	1	2	yes	—
158	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
159	C₈×C₆₇	—	4	4	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	4	no	—
162	C₂²×C₆₇	—	4	2	no	—
163	C₂²×C₆₇	—	4	1	no	—
164	C₂²×C₆₇	—	4	1	no	—
165	C₂²×C₆₇	#380	1	1	yes	characteristic
166	C₂²×C₆₇	#386	1	1	yes	characteristic
167	C₂²×C₆₇	#390	1	1	yes	characteristic
168	C₂²×C₆₇	#394	1	2	yes	—
169	C₂²×C₆₇	#396	1	2	yes	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	—	8	4	no	—
174	C₄×C₆₇	#385	1	2	yes	—
175	C₄×C₆₇	#388	1	2	yes	—
176	C₄×C₆₇	#392	1	1	yes	characteristic
177	C₄×C₆₇	#393	1	1	yes	characteristic
178	C₄×C₆₇	#397	1	1	yes	characteristic
179	C₄×C₆₇	#399	1	1	yes	characteristic
180	#11074	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₆₇	—	2	1	no	—
182	C₂×C₆₇	—	2	2	no	—
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	—	2	2	no	—
185	C₂×C₆₇	#1439	1	1	yes	characteristic
186	C₂×C₆₇	#1909	1	2	yes	—
187	#2905	C₂×C₆₇	1	1	yes	characteristic
188	C₂×#419	C₂×C₆₇	1	2	yes	—
189	#2310	C₂×C₆₇	1	1	yes	characteristic
190	#1740	C₂×C₆₇	1	1	yes	characteristic
191	C₂×H(3,ℤ/4)	C₂×C₆₇	1	2	yes	—
192	C₆₇	#11074	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
195	C₂×#117	C₂²×C₆₇	1	2	yes	—
196	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
198	#424	—	2	4	no	—
199	#423	—	2	2	no	—
200	#419	—	2	4	no	—
201	#419	—	2	4	no	—
202	C₂×#103	—	2	4	no	—
203	#391	—	2	2	no	—
204	#391	C₄×C₆₇	1	4	yes	complemented
205	#381	—	2	2	no	—
206	#342	—	2	2	no	—
207	H(3,ℤ/4)	—	2	4	no	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×Q₈	—	2	1	no	—
211	C₂²×Q₈	—	2	2	no	—
212	C₂²×Q₈	—	2	2	no	—
213	C₂²×Q₈	—	2	2	no	—
214	C₂²×Q₈	—	2	2	no	—
215	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
216	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
217	C₂²×Q₈	Q₈×C₆₇	1	2	yes	—
218	C₂³×C₄	—	2	1	no	—
219	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂²×C₈	D₄×C₆₇	1	4	yes	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	—	2	2	no	—
231	#117	—	2	2	no	—
232	#117	—	2	2	no	—
233	#117	—	2	2	no	—
234	#117	D₄×C₆₇	1	4	yes	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₂²⋊C₄	—	2	1	no	—
240	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
243	C₂×C₄²	D₄×C₆₇	1	2	yes	—
244	C₄⋊C₈	—	2	8	no	—
245	#103	—	2	16	no	—
246	#95	—	2	8	no	—
247	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
274	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
277	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	#41×C₆₇	1	2	yes	—
283	C₂²×C₄	#41×C₆₇	1	2	yes	—
284	M₁₆	—	4	2	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	—	4	2	no	—
287	M₁₆	—	4	2	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	4	no	—
290	C₂×C₈	—	2	4	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	2	4	no	—
297	C₂²⋊C₄	—	2	4	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	4	no	—
303	C₄²	—	2	4	no	—
304	C₄²	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
307	C₂³	#121×C₆₇	1	2	yes	—
308	C₂³	#121×C₆₇	1	1	yes	characteristic
309	C₂³	#121×C₆₇	1	2	yes	—
310	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
311	Q₈	—	2	8	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	4	2	no	—
316	Q₈	—	4	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
331	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
332	C₂×C₄	#120×C₆₇	1	2	yes	—
333	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
334	C₂×C₄	#121×C₆₇	1	2	yes	—
335	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
336	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
337	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
338	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
339	C₂×C₄	#123×C₆₇	1	2	yes	—
340	C₂×C₄	#123×C₆₇	1	2	yes	—
341	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
342	C₂×C₄	#124×C₆₇	1	2	yes	—
343	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
344	C₂×C₄	#126×C₆₇	1	2	yes	—
345	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
346	C₈	—	4	4	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	#380×C₆₇	1	1	yes	characteristic
353	C₂²	#386×C₆₇	1	1	yes	characteristic
354	C₂²	#390×C₆₇	1	1	yes	characteristic
355	C₂²	#394×C₆₇	1	2	yes	—
356	C₂²	#396×C₆₇	1	2	yes	—
357	C₄	—	4	4	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	4	no	—
361	C₄	#385×C₆₇	1	2	yes	—
362	C₄	#388×C₆₇	1	2	yes	—
363	C₄	#392×C₆₇	1	1	yes	characteristic
364	C₄	#393×C₆₇	1	1	yes	characteristic
365	C₄	#397×C₆₇	1	1	yes	characteristic
366	C₄	#399×C₆₇	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	2	no	—
369	C₂	—	2	1	no	—
370	C₂	—	2	2	no	—
371	C₂	C₆₇×#1439	1	1	yes	characteristic
372	C₂	C₆₇×#1909	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. #391×C₆₇	C₄	semidirect
180. #11074	C₆₇	direct (N × H)
192. C₆₇	#11074	direct (N × H)
204. #391	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2905	2
C₂×#419×C₆₇	2
C₆₇×#2310	2
C₆₇×#1740	2
C₂×H(3,ℤ/4)×C₆₇	2
#11074	67

Subgroups — C67×#11074 #30690325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11074 #30690325