Subgroups — C₆₇×#33132 #30712383

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 340 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×#1972	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1944	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1934	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1807	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×#106×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	#106×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	2	no	—
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	4	2	no	—
111	C₂³×C₆₇	—	4	4	no	—
112	C₂³×C₆₇	—	4	4	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	#121	1	2	yes	—
116	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	4	4	no	—
120	D₄×C₆₇	—	8	8	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	#121	1	2	yes	—
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	2	4	no	—
141	C₈×C₆₇	#123	1	4	yes	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	4	4	no	—
146	C₂²×C₆₇	—	8	2	no	—
147	C₂²×C₆₇	—	8	4	no	—
148	C₂²×C₆₇	—	8	4	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
152	C₂²×C₆₇	#532	1	2	yes	—
153	C₂²×C₆₇	#576	1	2	yes	—
154	C₂²×C₆₇	#577	1	2	yes	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	8	4	no	—
159	C₄×C₆₇	—	8	2	no	—
160	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
161	C₄×C₆₇	#525	1	1	yes	characteristic
162	C₄×C₆₇	#529	1	2	yes	—
163	#33132	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	4	2	no	—
165	C₂×C₆₇	—	16	2	no	—
166	C₂×C₆₇	#2854	1	1	yes	characteristic
167	C₂×C₆₇	#3058	1	1	yes	characteristic
168	C₂×C₆₇	#3060	1	1	yes	characteristic
169	C₂×C₆₇	#3179	1	4	yes	—
170	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
171	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#2624	C₂×C₆₇	1	2	yes	complemented
173	#1972	C₂×C₆₇	1	2	yes	complemented
174	#1944	C₂×C₆₇	1	2	yes	complemented
175	#1934	C₂×C₆₇	1	4	yes	complemented
176	#1895	C₂×C₆₇	1	2	yes	complemented
177	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
178	C₆₇	#33132	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
180	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
181	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×#121	C₂²×C₆₇	1	2	yes	—
183	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
184	#466	—	2	4	no	—
185	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×#106	C₂²×C₆₇	1	2	yes	—
187	C₂×#102	C₂²×C₆₇	1	4	yes	—
188	C₂×#102	C₂²×C₆₇	1	2	yes	—
189	#405	—	2	4	no	—
190	C₂×#97	C₂²×C₆₇	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
192	#399	—	2	2	no	—
193	#397	—	2	2	no	—
194	#394	—	2	4	no	—
195	#390	C₂²×C₆₇	1	4	yes	—
196	#389	C₂²×C₆₇	1	4	yes	—
197	#380	C₂²×C₆₇	1	2	yes	—
198	#340	C₂²×C₆₇	1	2	yes	—
199	#340	C₂²×C₆₇	1	4	yes	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
202	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
203	C₂×D₈	—	2	2	no	—
204	C₂×D₈	—	2	4	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	4	no	—
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	D₄×C₆₇	1	4	yes	—
209	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
211	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
212	#121	—	2	4	no	—
213	#121	—	2	4	no	—
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	—	2	4	no	—
224	C₂×C₂²⋊C₄	—	2	4	no	—
225	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
226	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
227	#107	—	2	2	no	—
228	#106	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	8	no	—
231	#97	—	2	4	no	—
232	C₄×C₈	—	2	4	no	—
233	#95	—	2	2	no	—
234	#95	—	2	4	no	—
235	C₂⁴	—	2	2	no	—
236	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	8	4	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
251	C₂²×C₄	#41×C₆₇	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
253	D₈	—	4	4	no	—
254	M₁₆	—	2	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
261	C₂×C₈	#41×C₆₇	1	4	yes	—
262	C₂×C₈	#41×C₆₇	1	4	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	8	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	8	4	no	—
273	C₂²⋊C₄	—	8	4	no	—
274	C₂²⋊C₄	—	8	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	#121×C₆₇	1	2	yes	—
286	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	4	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	#121×C₆₇	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
307	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
308	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	#123×C₆₇	1	4	yes	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	8	4	no	—
321	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
322	C₂²	#532×C₆₇	1	2	yes	—
323	C₂²	#576×C₆₇	1	2	yes	—
324	C₂²	#577×C₆₇	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	4	no	—
330	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
331	C₄	#525×C₆₇	1	1	yes	characteristic
332	C₄	#529×C₆₇	1	2	yes	—
333	C₂	—	4	2	no	—
334	C₂	—	16	2	no	—
335	C₂	C₆₇×#2854	1	1	yes	characteristic
336	C₂	C₆₇×#3058	1	1	yes	characteristic
337	C₂	C₆₇×#3060	1	1	yes	characteristic
338	C₂	C₆₇×#3179	1	4	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×#1972	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1934	C₂	semidirect
7. C₆₇×#1895	C₂	semidirect
8. C₆₇×#1807	C₂	semidirect
163. #33132	C₆₇	direct (N × H)
171. C₂×#405	C₂×C₆₇	semidirect
172. #2624	C₂×C₆₇	semidirect
173. #1972	C₂×C₆₇	semidirect
174. #1944	C₂×C₆₇	semidirect
175. #1934	C₂×C₆₇	semidirect
176. #1895	C₂×C₆₇	semidirect
177. #1807	C₂×C₆₇	semidirect
178. C₆₇	#33132	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×#1972	2
C₆₇×#1944	2
C₆₇×#1934	2
C₆₇×#1895	2
C₆₇×#1807	2
#33132	67

Subgroups — C67×#33132 #30712383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33132 #30712383