Subgroups — C₆₇×#33135 #30712386

Order 17152.

994 conjugacy classes of subgroups.

The subgroup lattice has 354 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	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2854	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×#1936	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1893	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	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#123×C₆₇	—	2	2	no	—
13	C₂×#121×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#466×C₆₇	—	2	4	no	—
16	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×#102×C₆₇	C₂²	1	2	yes	—
19	C₂×#102×C₆₇	C₂²	1	4	yes	—
20	C₂×#97×C₆₇	—	2	4	no	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	#390×C₆₇	C₂²	1	4	yes	—
24	#389×C₆₇	C₂²	1	4	yes	—
25	#388×C₆₇	—	2	4	no	—
26	#386×C₆₇	—	2	2	no	—
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	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×D₈×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	—	2	4	no	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	D₄	1	4	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	#123×C₆₇	—	4	8	no	—
45	#121×C₆₇	—	2	4	no	—
46	#121×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄²×C₆₇	—	2	2	no	—
55	C₂×C₄²×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	D₄	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
58	#107×C₆₇	—	2	2	no	—
59	#106×C₆₇	—	2	4	no	—
60	#102×C₆₇	—	2	4	no	—
61	#102×C₆₇	—	2	8	no	—
62	#97×C₆₇	—	2	4	no	—
63	#97×C₆₇	—	2	4	no	—
64	#95×C₆₇	—	2	2	no	—
65	#95×C₆₇	—	2	4	no	—
66	C₂⁴×C₆₇	—	2	2	no	—
67	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	—	2	4	no	—
70	C₂×D₄×C₆₇	—	2	2	no	—
71	C₂×D₄×C₆₇	—	2	4	no	—
72	C₂×D₄×C₆₇	—	4	4	no	—
73	C₂×D₄×C₆₇	—	4	4	no	—
74	C₂×D₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	4	2	no	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	D₈×C₆₇	—	4	4	no	—
90	C₂×C₈×C₆₇	—	2	8	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	8	no	—
95	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	#41	1	4	yes	—
98	C₂×C₈×C₆₇	#41	1	4	yes	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	—	2	4	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	4	8	no	—
104	C₂²⋊C₄×C₆₇	—	4	8	no	—
105	C₂²⋊C₄×C₆₇	—	4	8	no	—
106	C₂²⋊C₄×C₆₇	—	4	4	no	—
107	C₂²⋊C₄×C₆₇	—	4	4	no	—
108	C₄²×C₆₇	—	2	4	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	—	2	2	no	—
112	C₂³×C₆₇	—	2	2	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	—	4	2	no	—
117	C₂³×C₆₇	—	4	4	no	—
118	C₂³×C₆₇	#121	1	2	yes	—
119	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
120	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	D₄×C₆₇	—	4	2	no	—
122	D₄×C₆₇	—	4	2	no	—
123	D₄×C₆₇	—	4	4	no	—
124	D₄×C₆₇	—	8	8	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	4	2	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	2	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	#121	1	2	yes	—
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	—	2	4	no	—
148	C₈×C₆₇	#123	1	4	yes	complemented
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	8	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	—	8	4	no	—
154	C₂²×C₆₇	—	8	4	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	8	2	no	—
157	C₂²×C₆₇	#469	1	4	yes	—
158	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
159	C₂²×C₆₇	#532	1	2	yes	—
160	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
161	C₂²×C₆₇	#575	1	2	yes	—
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	4	2	no	—
164	C₄×C₆₇	—	4	2	no	—
165	C₄×C₆₇	—	8	4	no	—
166	C₄×C₆₇	—	8	2	no	—
167	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
168	C₄×C₆₇	#525	1	1	yes	characteristic
169	C₄×C₆₇	#529	1	2	yes	—
170	#33135	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₆₇	—	2	4	no	—
172	C₂×C₆₇	—	16	2	no	—
173	C₂×C₆₇	#2854	1	1	yes	characteristic
174	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
175	C₂×C₆₇	#3054	1	1	yes	characteristic
176	C₂×C₆₇	#3180	1	4	yes	—
177	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
178	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
179	#2854	C₂×C₆₇	1	2	yes	complemented
180	#1972	C₂×C₆₇	1	2	yes	complemented
181	#1944	C₂×C₆₇	1	2	yes	complemented
182	#1936	C₂×C₆₇	1	4	yes	complemented
183	#1893	C₂×C₆₇	1	2	yes	complemented
184	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
185	C₆₇	#33135	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
187	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
188	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
189	C₂×#123	—	2	2	no	—
190	C₂×#121	C₂²×C₆₇	1	2	yes	—
191	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
192	#466	—	2	4	no	—
193	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×#106	C₂²×C₆₇	1	2	yes	—
195	C₂×#102	C₂²×C₆₇	1	4	yes	—
196	C₂×#102	C₂²×C₆₇	1	2	yes	—
197	C₂×#97	—	2	4	no	—
198	C₂×#97	C₂²×C₆₇	1	2	yes	—
199	C₂×#97	C₂²×C₆₇	1	2	yes	—
200	#390	C₂²×C₆₇	1	4	yes	—
201	#389	C₂²×C₆₇	1	4	yes	—
202	#388	—	2	4	no	—
203	#386	—	2	2	no	—
204	#380	C₂²×C₆₇	1	2	yes	—
205	#340	C₂²×C₆₇	1	2	yes	—
206	#340	C₂²×C₆₇	1	4	yes	—
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
209	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
210	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
211	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	2	4	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	2	4	no	—
217	C₂²×C₈	D₄×C₆₇	1	4	yes	—
218	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
220	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
221	#123	—	4	8	no	—
222	#121	—	2	4	no	—
223	#121	—	2	4	no	—
224	C₂×C₄⋊C₄	—	2	4	no	—
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×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	D₄×C₆₇	1	2	yes	—
234	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
235	#107	—	2	2	no	—
236	#106	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	8	no	—
239	#97	—	2	4	no	—
240	#97	—	2	4	no	—
241	#95	—	2	2	no	—
242	#95	—	2	4	no	—
243	C₂⁴	—	2	2	no	—
244	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
264	C₂²×C₄	#41×C₆₇	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
266	D₈	—	4	4	no	—
267	C₂×C₈	—	2	8	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	8	no	—
272	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
274	C₂×C₈	#41×C₆₇	1	4	yes	—
275	C₂×C₈	#41×C₆₇	1	4	yes	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	8	no	—
281	C₂²⋊C₄	—	4	8	no	—
282	C₂²⋊C₄	—	4	8	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	4	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	#121×C₆₇	1	2	yes	—
296	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
297	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	#121×C₆₇	1	2	yes	—
319	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
321	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
322	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	4	no	—
325	C₈	#123×C₆₇	1	4	yes	complemented
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	4	no	—
333	C₂²	—	8	4	no	—
334	C₂²	#469×C₆₇	1	4	yes	—
335	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
336	C₂²	#532×C₆₇	1	2	yes	—
337	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
338	C₂²	#575×C₆₇	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	4	no	—
344	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
345	C₄	#525×C₆₇	1	1	yes	characteristic
346	C₄	#529×C₆₇	1	2	yes	—
347	C₂	—	2	4	no	—
348	C₂	—	16	2	no	—
349	C₂	C₆₇×#2854	1	1	yes	characteristic
350	C₂	C₂×#469×C₆₇	1	1	yes	characteristic
351	C₂	C₆₇×#3054	1	1	yes	characteristic
352	C₂	C₆₇×#3180	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 ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#97×C₆₇	C₂	semidirect
3. C₆₇×#2854	C₂	semidirect
4. C₆₇×#1972	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1936	C₂	semidirect
7. C₆₇×#1893	C₂	semidirect
8. C₆₇×#1807	C₂	semidirect
148. C₈×C₆₇	#123	semidirect
170. #33135	C₆₇	direct (N × H)
178. C₂²×#97	C₂×C₆₇	semidirect
179. #2854	C₂×C₆₇	semidirect
180. #1972	C₂×C₆₇	semidirect
181. #1944	C₂×C₆₇	semidirect
182. #1936	C₂×C₆₇	semidirect
183. #1893	C₂×C₆₇	semidirect
184. #1807	C₂×C₆₇	semidirect
185. C₆₇	#33135	direct (N × H)
325. C₈	#123×C₆₇	semidirect

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
C₂²×#97×C₆₇	2
C₆₇×#2854	2
C₆₇×#1972	2
C₆₇×#1944	2
C₆₇×#1936	2
C₆₇×#1893	2
C₆₇×#1807	2
#33135	67

Subgroups — C67×#33135 #30712386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33135 #30712386