Subgroups — C₆₇×#33138 #30712389

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2624	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1942	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1936	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₆₇	C₂²	1	2	yes	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	#466×C₆₇	—	2	4	no	—
15	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#102×C₆₇	C₂²	1	2	yes	—
18	C₂×#102×C₆₇	C₂²	1	4	yes	—
19	#405×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#399×C₆₇	—	2	2	no	—
23	#397×C₆₇	—	2	2	no	—
24	#394×C₆₇	—	2	4	no	—
25	#390×C₆₇	C₂²	1	4	yes	—
26	#389×C₆₇	C₂²	1	4	yes	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	#340×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	4	yes	—
30	C₂²×D₄×C₆₇	—	2	2	no	—
31	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×D₈×C₆₇	—	2	2	no	—
34	C₂×D₈×C₆₇	—	2	4	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	4	no	—
37	C₂²×C₈×C₆₇	—	2	4	no	—
38	C₂²×C₈×C₆₇	D₄	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	#121×C₆₇	—	2	4	no	—
43	#121×C₆₇	—	2	4	no	—
44	#117×C₆₇	—	2	2	no	—
45	#117×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
55	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
57	#107×C₆₇	—	2	2	no	—
58	#107×C₆₇	—	2	4	no	—
59	#102×C₆₇	—	2	4	no	—
60	#102×C₆₇	—	2	8	no	—
61	#97×C₆₇	—	2	4	no	—
62	C₄×C₈×C₆₇	—	2	4	no	—
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	4	no	—
65	C₂⁴×C₆₇	—	2	2	no	—
66	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	2	4	no	—
69	C₂×D₄×C₆₇	—	2	2	no	—
70	C₂×D₄×C₆₇	—	2	4	no	—
71	C₂×D₄×C₆₇	—	4	4	no	—
72	C₂×D₄×C₆₇	—	8	4	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	D₈×C₆₇	—	4	4	no	—
84	M₁₆×C₆₇	—	2	8	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	8	no	—
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₆₇	#41	1	4	yes	—
92	C₂×C₈×C₆₇	#41	1	4	yes	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	8	4	no	—
100	C₂²⋊C₄×C₆₇	—	2	4	no	—
101	C₂²⋊C₄×C₆₇	—	4	4	no	—
102	C₂²⋊C₄×C₆₇	—	8	4	no	—
103	C₂²⋊C₄×C₆₇	—	8	4	no	—
104	C₂²⋊C₄×C₆₇	—	8	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	2	1	no	—
111	C₂³×C₆₇	—	4	2	no	—
112	C₂³×C₆₇	—	4	4	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	#121	1	2	yes	—
117	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	D₄×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	4	2	no	—
120	D₄×C₆₇	—	4	4	no	—
121	D₄×C₆₇	—	8	8	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	4	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	#121	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	—	2	4	no	—
144	C₈×C₆₇	#123	1	4	yes	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	—	2	2	no	—
147	C₂²×C₆₇	—	2	2	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	4	no	—
152	C₂²×C₆₇	—	8	4	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
156	C₂²×C₆₇	#532	1	2	yes	—
157	C₂²×C₆₇	#576	1	2	yes	—
158	C₂²×C₆₇	#576	1	1	yes	characteristic
159	C₂²×C₆₇	#578	1	1	yes	characteristic
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	4	2	no	—
162	C₄×C₆₇	—	4	2	no	—
163	C₄×C₆₇	—	8	4	no	—
164	C₄×C₆₇	—	8	2	no	—
165	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
166	C₄×C₆₇	#525	1	1	yes	characteristic
167	C₄×C₆₇	#529	1	2	yes	—
168	#33138	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂×C₆₇	—	4	2	no	—
170	C₂×C₆₇	—	16	2	no	—
171	C₂×C₆₇	#2854	1	1	yes	characteristic
172	C₂×C₆₇	#3058	1	1	yes	characteristic
173	C₂×C₆₇	#3061	1	1	yes	characteristic
174	C₂×C₆₇	#3175	1	2	yes	—
175	C₂×C₆₇	#3176	1	2	yes	—
176	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
177	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
178	#2624	C₂×C₆₇	1	2	yes	complemented
179	#1970	C₂×C₆₇	1	2	yes	complemented
180	#1942	C₂×C₆₇	1	2	yes	complemented
181	#1936	C₂×C₆₇	1	4	yes	complemented
182	#1895	C₂×C₆₇	1	2	yes	complemented
183	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
184	C₆₇	#33138	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
186	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
187	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
188	C₂×#121	C₂²×C₆₇	1	2	yes	—
189	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
190	#466	—	2	4	no	—
191	C₂×#107	C₂²×C₆₇	1	2	yes	—
192	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×#102	C₂²×C₆₇	1	4	yes	—
194	C₂×#102	C₂²×C₆₇	1	2	yes	—
195	#405	—	2	4	no	—
196	C₂×#97	C₂²×C₆₇	1	2	yes	—
197	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
198	#399	—	2	2	no	—
199	#397	—	2	2	no	—
200	#394	—	2	4	no	—
201	#390	C₂²×C₆₇	1	4	yes	—
202	#389	C₂²×C₆₇	1	4	yes	—
203	#380	C₂²×C₆₇	1	2	yes	—
204	#340	C₂²×C₆₇	1	2	yes	—
205	#340	C₂²×C₆₇	1	4	yes	—
206	C₂²×D₄	—	2	2	no	—
207	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
208	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	4	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	D₄×C₆₇	1	4	yes	—
215	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
217	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
218	#121	—	2	4	no	—
219	#121	—	2	4	no	—
220	#117	—	2	2	no	—
221	#117	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
224	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
232	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
233	#107	—	2	2	no	—
234	#107	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	8	no	—
237	#97	—	2	4	no	—
238	C₄×C₈	—	2	4	no	—
239	#95	—	2	2	no	—
240	#95	—	2	4	no	—
241	C₂⁴	—	2	2	no	—
242	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	4	no	—
248	C₂×D₄	—	8	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
257	C₂²×C₄	#41×C₆₇	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
259	D₈	—	4	4	no	—
260	M₁₆	—	2	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	8	no	—
265	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
267	C₂×C₈	#41×C₆₇	1	4	yes	—
268	C₂×C₈	#41×C₆₇	1	4	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	8	4	no	—
276	C₂²⋊C₄	—	2	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	8	4	no	—
279	C₂²⋊C₄	—	8	4	no	—
280	C₂²⋊C₄	—	8	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	4	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	#121×C₆₇	1	2	yes	—
293	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	4	no	—
297	D₄	—	8	8	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	#121×C₆₇	1	2	yes	—
314	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
316	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
317	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	4	no	—
320	C₈	#123×C₆₇	1	4	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	—	8	4	no	—
331	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
332	C₂²	#532×C₆₇	1	2	yes	—
333	C₂²	#576×C₆₇	1	2	yes	—
334	C₂²	#576×C₆₇	1	1	yes	characteristic
335	C₂²	#578×C₆₇	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	4	no	—
341	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
342	C₄	#525×C₆₇	1	1	yes	characteristic
343	C₄	#529×C₆₇	1	2	yes	—
344	C₂	—	4	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₆₇×#2854	1	1	yes	characteristic
347	C₂	C₆₇×#3058	1	1	yes	characteristic
348	C₂	C₆₇×#3061	1	1	yes	characteristic
349	C₂	C₆₇×#3175	1	2	yes	—
350	C₂	C₆₇×#3176	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#466×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#405×C₆₇	C₂	semidirect
3. C₆₇×#2624	C₂	semidirect
4. C₆₇×#1970	C₂	semidirect
5. C₆₇×#1942	C₂	semidirect
6. C₆₇×#1936	C₂	semidirect
7. C₆₇×#1895	C₂	semidirect
8. C₆₇×#1808	C₂	semidirect
168. #33138	C₆₇	direct (N × H)
177. C₂×#405	C₂×C₆₇	semidirect
178. #2624	C₂×C₆₇	semidirect
179. #1970	C₂×C₆₇	semidirect
180. #1942	C₂×C₆₇	semidirect
181. #1936	C₂×C₆₇	semidirect
182. #1895	C₂×C₆₇	semidirect
183. #1808	C₂×C₆₇	semidirect
184. C₆₇	#33138	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#466×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2624	2
C₆₇×#1970	2
C₆₇×#1942	2
C₆₇×#1936	2
C₆₇×#1895	2
C₆₇×#1808	2
#33138	67

Subgroups — C67×#33138 #30712389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33138 #30712389