Subgroups — C₆₇×#10888 #30690139

Order 17152.

762 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2317	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic
3	C₂×#343×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1733	C₂	1	2	yes	maximal
5	C₂×H(3,ℤ/4)×C₆₇	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
7	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
11	#389×C₆₇	—	2	2	no	—
12	#385×C₆₇	—	2	2	no	—
13	#384×C₆₇	C₄	1	2	yes	complemented
14	#382×C₆₇	C₄	1	2	yes	complemented
15	C₄×C₄⋊C₄×C₆₇	—	2	4	no	—
16	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
17	#343×C₆₇	—	2	2	no	—
18	#339×C₆₇	—	2	4	no	—
19	H(3,ℤ/4)×C₆₇	C₄	1	8	yes	complemented
20	C₂³×C₄×C₆₇	D₄	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	#117×C₆₇	—	2	2	no	—
30	#117×C₆₇	—	2	2	no	—
31	#117×C₆₇	—	2	2	no	—
32	#117×C₆₇	—	2	2	no	—
33	#117×C₆₇	—	2	2	no	—
34	#117×C₆₇	C₂×C₄	1	4	yes	—
35	#117×C₆₇	C₂×C₄	1	4	yes	—
36	#117×C₆₇	Q₈	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	—	2	4	no	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	4	yes	—
53	#95×C₆₇	—	2	4	no	—
54	#95×C₆₇	—	2	4	no	—
55	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	8	2	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₆₇	#41	1	2	yes	—
70	C₂²×C₄×C₆₇	#41	1	2	yes	—
71	C₂²×C₄×C₆₇	#41	1	2	yes	—
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	2	4	no	—
74	M₁₆×C₆₇	—	4	2	no	—
75	M₁₆×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	8	4	no	—
87	C₂²⋊C₄×C₆₇	—	2	4	no	—
88	C₂²⋊C₄×C₆₇	—	2	4	no	—
89	C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₂²⋊C₄×C₆₇	—	4	1	no	—
91	C₂²⋊C₄×C₆₇	—	4	1	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	4	2	no	—
95	C₄²×C₆₇	—	8	4	no	—
96	C₄²×C₆₇	C₄²	1	4	yes	complemented
97	C₄²×C₆₇	C₄⋊C₄	1	8	yes	complemented
98	C₂³×C₆₇	—	4	1	no	—
99	C₂³×C₆₇	—	4	1	no	—
100	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₆₇	C₄×D₄	1	2	yes	—
102	C₂³×C₆₇	#120	1	1	yes	characteristic
103	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₆₇	#123	1	1	yes	characteristic
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	8	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	1	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	—	8	2	no	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₇	#117	1	2	yes	—
122	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	#121	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₆₇	#123	1	2	yes	—
131	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#125	1	2	yes	—
133	C₂×C₄×C₆₇	#126	1	2	yes	—
134	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
135	C₈×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₆₇	#385	1	2	yes	—
146	C₂²×C₆₇	#386	1	1	yes	characteristic
147	C₂²×C₆₇	#387	1	2	yes	—
148	C₂²×C₆₇	#391	1	1	yes	characteristic
149	C₄×C₆₇	—	4	4	no	—
150	C₄×C₆₇	—	4	8	no	—
151	C₄×C₆₇	—	4	4	no	—
152	C₄×C₆₇	—	8	1	no	—
153	C₄×C₆₇	—	8	1	no	—
154	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
155	C₄×C₆₇	#387	1	2	yes	—
156	C₄×C₆₇	#388	1	2	yes	—
157	C₄×C₆₇	#389	1	2	yes	—
158	#10888	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	—	2	2	no	—
161	C₂×C₆₇	—	2	2	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	#1436	1	1	yes	characteristic
164	C₂×C₆₇	#1911	1	1	yes	characteristic
165	C₂×C₆₇	#1912	1	1	yes	characteristic
166	#2317	C₂×C₆₇	1	1	yes	characteristic
167	#2274	C₂×C₆₇	1	1	yes	characteristic
168	C₂×#343	C₂×C₆₇	1	1	yes	characteristic
169	#1733	C₂×C₆₇	1	2	yes	—
170	C₂×H(3,ℤ/4)	C₂×C₆₇	1	2	yes	—
171	C₆₇	#10888	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
173	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#117	C₂²×C₆₇	1	2	yes	—
176	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
177	#389	—	2	2	no	—
178	#385	—	2	2	no	—
179	#384	C₄×C₆₇	1	2	yes	complemented
180	#382	C₄×C₆₇	1	2	yes	complemented
181	C₄×C₄⋊C₄	—	2	4	no	—
182	C₄×C₂²⋊C₄	—	2	2	no	—
183	#343	—	2	2	no	—
184	#339	—	2	4	no	—
185	H(3,ℤ/4)	C₄×C₆₇	1	8	yes	complemented
186	C₂³×C₄	D₄×C₆₇	1	2	yes	—
187	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	C₂×C₄×C₆₇	1	4	yes	—
201	#117	C₂×C₄×C₆₇	1	4	yes	—
202	#117	Q₈×C₆₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₂²⋊C₄	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
218	C₂×C₄²	D₄×C₆₇	1	4	yes	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	8	2	no	—
229	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
235	C₂²×C₄	#41×C₆₇	1	2	yes	—
236	C₂²×C₄	#41×C₆₇	1	2	yes	—
237	C₂²×C₄	#41×C₆₇	1	2	yes	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	8	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	8	4	no	—
262	C₄²	C₄²×C₆₇	1	4	yes	complemented
263	C₄²	C₄⋊C₄×C₆₇	1	8	yes	complemented
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
267	C₂³	C₄×D₄×C₆₇	1	2	yes	—
268	C₂³	#120×C₆₇	1	1	yes	characteristic
269	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
270	C₂³	#123×C₆₇	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
287	C₂×C₄	#117×C₆₇	1	2	yes	—
288	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
289	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
290	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
291	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
292	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
293	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
294	C₂×C₄	#121×C₆₇	1	2	yes	—
295	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
296	C₂×C₄	#123×C₆₇	1	2	yes	—
297	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
298	C₂×C₄	#125×C₆₇	1	2	yes	—
299	C₂×C₄	#126×C₆₇	1	2	yes	—
300	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
301	C₈	—	4	4	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
311	C₂²	#385×C₆₇	1	2	yes	—
312	C₂²	#386×C₆₇	1	1	yes	characteristic
313	C₂²	#387×C₆₇	1	2	yes	—
314	C₂²	#391×C₆₇	1	1	yes	characteristic
315	C₄	—	4	4	no	—
316	C₄	—	4	8	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
321	C₄	#387×C₆₇	1	2	yes	—
322	C₄	#388×C₆₇	1	2	yes	—
323	C₄	#389×C₆₇	1	2	yes	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	2	2	no	—
327	C₂	—	2	1	no	—
328	C₂	C₆₇×#1436	1	1	yes	characteristic
329	C₂	C₆₇×#1911	1	1	yes	characteristic
330	C₂	C₆₇×#1912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#386
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×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
13. #384×C₆₇	C₄	semidirect
14. #382×C₆₇	C₄	semidirect
19. H(3,ℤ/4)×C₆₇	C₄	semidirect
96. C₄²×C₆₇	C₄²	semidirect
97. C₄²×C₆₇	C₄⋊C₄	semidirect
158. #10888	C₆₇	direct (N × H)
171. C₆₇	#10888	direct (N × H)
179. #384	C₄×C₆₇	semidirect
180. #382	C₄×C₆₇	semidirect
185. H(3,ℤ/4)	C₄×C₆₇	semidirect
262. C₄²	C₄²×C₆₇	semidirect
263. C₄²	C₄⋊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×#2317	2
C₆₇×#2274	2
C₂×#343×C₆₇	2
C₆₇×#1733	2
C₂×H(3,ℤ/4)×C₆₇	2
#10888	67

Subgroups — C67×#10888 #30690139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10888 #30690139