Subgroups — C₆₇×#33351 #30712602

Order 17152.

1212 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3487	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2889	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2884	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2122	C₂	1	6	yes	maximal, complemented
5	C₆₇×#2120	C₂	1	6	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic, complemented
8	#568×C₆₇	—	2	1	no	—
9	#550×C₆₇	—	2	3	no	—
10	#550×C₆₇	—	2	1	no	—
11	#548×C₆₇	—	2	3	no	—
12	#548×C₆₇	—	2	1	no	—
13	#535×C₆₇	C₄	1	3	yes	complemented
14	#535×C₆₇	C₂²	1	3	yes	—
15	#525×C₆₇	C₄	1	3	yes	complemented
16	#525×C₆₇	C₂²	1	3	yes	—
17	#519×C₆₇	—	2	1	no	—
18	#519×C₆₇	C₄	1	1	yes	characteristic, complemented
19	#413×C₆₇	C₂²	1	6	yes	—
20	C₂×#101×C₆₇	—	2	3	no	—
21	#410×C₆₇	—	2	3	no	—
22	#410×C₆₇	C₂²	1	6	yes	—
23	#352×C₆₇	C₂²	1	4	yes	—
24	AΣL(1,𝔽₁₆)×C₆₇	C₂²	1	12	yes	—
25	2₋¹⁺⁴×C₆₇	—	4	4	no	—
26	C₂×#41×C₆₇	—	2	3	no	—
27	C₂×#41×C₆₇	—	2	1	no	—
28	C₂×#41×C₆₇	—	4	4	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	3	yes	—
30	C₂×#41×C₆₇	D₄	1	3	yes	—
31	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
32	C₂×#41×C₆₇	C₂³	1	3	yes	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Q₈×C₆₇	D₄	1	3	yes	—
35	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₆₇	C₂×C₄	1	3	yes	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
38	#131×C₆₇	—	4	4	no	—
39	C₂×M₁₆×C₆₇	—	2	6	no	—
40	#127×C₆₇	—	2	3	no	—
41	#127×C₆₇	—	2	3	no	—
42	#124×C₆₇	—	2	3	no	—
43	#124×C₆₇	—	2	3	no	—
44	#123×C₆₇	C₂×C₄	1	3	yes	—
45	#123×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²⋊Q₈×C₆₇	—	2	3	no	—
47	C₂²⋊Q₈×C₆₇	—	2	3	no	—
48	#121×C₆₇	—	2	3	no	—
49	#121×C₆₇	—	2	3	no	—
50	#121×C₆₇	—	2	6	no	—
51	#121×C₆₇	—	2	3	no	—
52	#121×C₆₇	—	2	3	no	—
53	#120×C₆₇	C₂×C₄	1	6	yes	—
54	#120×C₆₇	C₂×C₄	1	3	yes	—
55	#120×C₆₇	C₂³	1	3	yes	—
56	C₄×Q₈×C₆₇	—	2	3	no	—
57	C₄×Q₈×C₆₇	—	2	1	no	—
58	C₄×Q₈×C₆₇	—	4	4	no	—
59	C₄×D₄×C₆₇	—	2	6	no	—
60	C₄×D₄×C₆₇	—	2	3	no	—
61	C₄×D₄×C₆₇	—	2	3	no	—
62	#117×C₆₇	—	2	6	no	—
63	#117×C₆₇	C₂×C₄	1	3	yes	—
64	#117×C₆₇	C₂×C₄	1	3	yes	—
65	#101×C₆₇	—	2	6	no	—
66	#100×C₆₇	C₂³	1	4	yes	—
67	#99×C₆₇	—	2	6	no	—
68	#99×C₆₇	C₂³	1	4	yes	—
69	C₂⁴×C₆₇	C₂²×C₄	1	3	yes	—
70	#41×C₆₇	—	2	4	no	—
71	#41×C₆₇	—	4	6	no	—
72	#41×C₆₇	—	4	6	no	—
73	#41×C₆₇	—	4	6	no	—
74	#41×C₆₇	—	4	12	no	—
75	#41×C₆₇	—	4	4	no	—
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	—	2	4	no	—
78	C₂×Q₈×C₆₇	—	4	6	no	—
79	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
80	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
81	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Q₈×C₆₇	C₂×D₄	1	3	yes	—
83	C₂×D₄×C₆₇	—	2	6	no	—
84	C₂×D₄×C₆₇	—	2	3	no	—
85	C₂×D₄×C₆₇	—	2	3	no	—
86	C₂×D₄×C₆₇	—	2	3	no	—
87	C₂×D₄×C₆₇	—	2	3	no	—
88	C₂×D₄×C₆₇	—	4	6	no	—
89	C₂×D₄×C₆₇	C₂²⋊C₄	1	3	yes	—
90	C₂×D₄×C₆₇	C₂²×C₄	1	3	yes	—
91	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
92	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	—	2	3	no	—
94	C₂²×C₄×C₆₇	—	2	3	no	—
95	C₂²×C₄×C₆₇	—	4	6	no	—
96	C₂²×C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
101	M₁₆×C₆₇	—	2	4	no	—
102	M₁₆×C₆₇	—	4	6	no	—
103	C₂×C₈×C₆₇	—	4	6	no	—
104	C₄⋊C₄×C₆₇	—	2	3	no	—
105	C₄⋊C₄×C₆₇	—	2	3	no	—
106	C₄⋊C₄×C₆₇	—	2	3	no	—
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	4	6	no	—
109	C₂²⋊C₄×C₆₇	—	2	3	no	—
110	C₂²⋊C₄×C₆₇	—	2	1	no	—
111	C₂²⋊C₄×C₆₇	—	2	6	no	—
112	C₂²⋊C₄×C₆₇	—	2	3	no	—
113	C₂²⋊C₄×C₆₇	—	2	3	no	—
114	C₂²⋊C₄×C₆₇	—	2	4	no	—
115	C₂²⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²⋊C₄×C₆₇	C₂²×C₄	1	3	yes	—
117	C₄²×C₆₇	—	2	3	no	—
118	C₄²×C₆₇	—	2	3	no	—
119	C₄²×C₆₇	—	4	6	no	—
120	C₂³×C₆₇	—	2	3	no	—
121	C₂³×C₆₇	—	4	3	no	—
122	C₂³×C₆₇	—	8	3	no	—
123	C₂³×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
124	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₆₇	—	2	6	no	—
127	Q₈×C₆₇	—	2	6	no	—
128	Q₈×C₆₇	—	4	3	no	—
129	Q₈×C₆₇	#99	1	4	yes	—
130	D₄×C₆₇	—	2	6	no	—
131	D₄×C₆₇	—	4	3	no	—
132	D₄×C₆₇	—	4	3	no	—
133	D₄×C₆₇	—	4	1	no	—
134	D₄×C₆₇	—	4	6	no	—
135	D₄×C₆₇	—	4	6	no	—
136	D₄×C₆₇	—	8	6	no	—
137	D₄×C₆₇	—	8	3	no	—
138	D₄×C₆₇	—	8	3	no	—
139	C₂×C₄×C₆₇	—	2	6	no	—
140	C₂×C₄×C₆₇	—	2	6	no	—
141	C₂×C₄×C₆₇	—	2	6	no	—
142	C₂×C₄×C₆₇	—	2	3	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	4	6	no	—
146	C₂×C₄×C₆₇	—	4	6	no	—
147	C₂×C₄×C₆₇	—	4	6	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
151	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₆₇	—	4	4	no	—
155	C₂²×C₆₇	—	2	1	no	—
156	C₂²×C₆₇	—	2	1	no	—
157	C₂²×C₆₇	—	2	3	no	—
158	C₂²×C₆₇	—	4	3	no	—
159	C₂²×C₆₇	—	4	1	no	—
160	C₂²×C₆₇	—	4	1	no	—
161	C₂²×C₆₇	—	4	3	no	—
162	C₂²×C₆₇	—	4	3	no	—
163	C₂²×C₆₇	—	4	3	no	—
164	C₂²×C₆₇	—	8	6	no	—
165	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	3	no	—
168	C₄×C₆₇	—	2	3	no	—
169	C₄×C₆₇	—	4	3	no	—
170	C₄×C₆₇	—	4	1	no	—
171	C₄×C₆₇	—	4	1	no	—
172	C₄×C₆₇	—	8	4	no	—
173	C₄×C₆₇	C₂×#99	1	6	yes	—
174	#33351	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	—	4	1	no	—
177	C₂×C₆₇	—	4	3	no	—
178	C₂×C₆₇	—	4	1	no	—
179	C₂×C₆₇	—	8	3	no	—
180	C₂×C₆₇	C₂²×#99	1	1	yes	characteristic
181	#3487	C₂×C₆₇	1	1	yes	characteristic
182	#2889	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#2884	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#2122	C₂×C₆₇	1	6	yes	complemented
185	#2120	C₂×C₆₇	1	6	yes	complemented
186	C₆₇	#33351	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
188	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic, complemented
189	#568	—	2	1	no	—
190	#550	—	2	3	no	—
191	#550	—	2	1	no	—
192	#548	—	2	3	no	—
193	#548	—	2	1	no	—
194	#535	C₄×C₆₇	1	3	yes	complemented
195	#535	C₂²×C₆₇	1	3	yes	—
196	#525	C₄×C₆₇	1	3	yes	complemented
197	#525	C₂²×C₆₇	1	3	yes	—
198	#519	—	2	1	no	—
199	#519	C₄×C₆₇	1	1	yes	characteristic, complemented
200	#413	C₂²×C₆₇	1	6	yes	—
201	C₂×#101	—	2	3	no	—
202	#410	—	2	3	no	—
203	#410	C₂²×C₆₇	1	6	yes	—
204	#352	C₂²×C₆₇	1	4	yes	—
205	AΣL(1,𝔽₁₆)	C₂²×C₆₇	1	12	yes	—
206	2₋¹⁺⁴	—	4	4	no	—
207	C₂×#41	—	2	3	no	—
208	C₂×#41	—	2	1	no	—
209	C₂×#41	—	4	4	no	—
210	C₂×#41	C₂×C₄×C₆₇	1	3	yes	—
211	C₂×#41	D₄×C₆₇	1	3	yes	—
212	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
213	C₂×#41	C₂³×C₆₇	1	3	yes	—
214	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
215	C₂²×Q₈	D₄×C₆₇	1	3	yes	—
216	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
217	C₂²×D₄	C₂×C₄×C₆₇	1	3	yes	—
218	C₂³×C₄	C₂×C₄×C₆₇	1	3	yes	—
219	#131	—	4	4	no	—
220	C₂×M₁₆	—	2	6	no	—
221	#127	—	2	3	no	—
222	#127	—	2	3	no	—
223	#124	—	2	3	no	—
224	#124	—	2	3	no	—
225	#123	C₂×C₄×C₆₇	1	3	yes	—
226	#123	C₂³×C₆₇	1	1	yes	characteristic
227	C₂²⋊Q₈	—	2	3	no	—
228	C₂²⋊Q₈	—	2	3	no	—
229	#121	—	2	3	no	—
230	#121	—	2	3	no	—
231	#121	—	2	6	no	—
232	#121	—	2	3	no	—
233	#121	—	2	3	no	—
234	#120	C₂×C₄×C₆₇	1	6	yes	—
235	#120	C₂×C₄×C₆₇	1	3	yes	—
236	#120	C₂³×C₆₇	1	3	yes	—
237	C₄×Q₈	—	2	3	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	4	4	no	—
240	C₄×D₄	—	2	6	no	—
241	C₄×D₄	—	2	3	no	—
242	C₄×D₄	—	2	3	no	—
243	#117	—	2	6	no	—
244	#117	C₂×C₄×C₆₇	1	3	yes	—
245	#117	C₂×C₄×C₆₇	1	3	yes	—
246	#101	—	2	6	no	—
247	#100	C₂³×C₆₇	1	4	yes	—
248	#99	—	2	6	no	—
249	#99	C₂³×C₆₇	1	4	yes	—
250	C₂⁴	C₂²×C₄×C₆₇	1	3	yes	—
251	#41	—	2	4	no	—
252	#41	—	4	6	no	—
253	#41	—	4	6	no	—
254	#41	—	4	6	no	—
255	#41	—	4	12	no	—
256	#41	—	4	4	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	4	6	no	—
260	C₂×Q₈	C₂²⋊C₄×C₆₇	1	3	yes	—
261	C₂×Q₈	C₂²⋊C₄×C₆₇	1	3	yes	—
262	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
263	C₂×Q₈	C₂×D₄×C₆₇	1	3	yes	—
264	C₂×D₄	—	2	6	no	—
265	C₂×D₄	—	2	3	no	—
266	C₂×D₄	—	2	3	no	—
267	C₂×D₄	—	2	3	no	—
268	C₂×D₄	—	2	3	no	—
269	C₂×D₄	—	4	6	no	—
270	C₂×D₄	C₂²⋊C₄×C₆₇	1	3	yes	—
271	C₂×D₄	C₂²×C₄×C₆₇	1	3	yes	—
272	C₂×D₄	C₂×D₄×C₆₇	1	3	yes	—
273	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	—	2	3	no	—
275	C₂²×C₄	—	2	3	no	—
276	C₂²×C₄	—	4	6	no	—
277	C₂²×C₄	C₂²⋊C₄×C₆₇	1	6	yes	—
278	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
282	M₁₆	—	2	4	no	—
283	M₁₆	—	4	6	no	—
284	C₂×C₈	—	4	6	no	—
285	C₄⋊C₄	—	2	3	no	—
286	C₄⋊C₄	—	2	3	no	—
287	C₄⋊C₄	—	2	3	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	6	no	—
290	C₂²⋊C₄	—	2	3	no	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	2	6	no	—
293	C₂²⋊C₄	—	2	3	no	—
294	C₂²⋊C₄	—	2	3	no	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₂²⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
297	C₂²⋊C₄	C₂²×C₄×C₆₇	1	3	yes	—
298	C₄²	—	2	3	no	—
299	C₄²	—	2	3	no	—
300	C₄²	—	4	6	no	—
301	C₂³	—	2	3	no	—
302	C₂³	—	4	3	no	—
303	C₂³	—	8	3	no	—
304	C₂³	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
305	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
306	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
307	Q₈	—	2	6	no	—
308	Q₈	—	2	6	no	—
309	Q₈	—	4	3	no	—
310	Q₈	#99×C₆₇	1	4	yes	—
311	D₄	—	2	6	no	—
312	D₄	—	4	3	no	—
313	D₄	—	4	3	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	6	no	—
316	D₄	—	4	6	no	—
317	D₄	—	8	6	no	—
318	D₄	—	8	3	no	—
319	D₄	—	8	3	no	—
320	C₂×C₄	—	2	6	no	—
321	C₂×C₄	—	2	6	no	—
322	C₂×C₄	—	2	6	no	—
323	C₂×C₄	—	2	3	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	6	no	—
327	C₂×C₄	—	4	6	no	—
328	C₂×C₄	—	4	6	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
333	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
334	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
335	C₈	—	4	4	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	3	no	—
339	C₂²	—	4	3	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	3	no	—
343	C₂²	—	4	3	no	—
344	C₂²	—	4	3	no	—
345	C₂²	—	8	6	no	—
346	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	3	no	—
349	C₄	—	2	3	no	—
350	C₄	—	4	3	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	8	4	no	—
354	C₄	C₂×#99×C₆₇	1	6	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	4	1	no	—
357	C₂	—	4	1	no	—
358	C₂	—	4	3	no	—
359	C₂	—	8	3	no	—
360	C₂	C₂²×#99×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂²×#99
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₇
prime radical	C₆₇×#3487	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×Q₈×C₆₇ ◁ C₂×2₋¹⁺⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2889	C₂	semidirect
3. C₆₇×#2884	C₂	semidirect
4. C₆₇×#2122	C₂	semidirect
5. C₆₇×#2120	C₂	semidirect
7. C₂²×#41×C₆₇	C₄	semidirect
13. #535×C₆₇	C₄	semidirect
15. #525×C₆₇	C₄	semidirect
18. #519×C₆₇	C₄	semidirect
174. #33351	C₆₇	direct (N × H)
182. #2889	C₂×C₆₇	semidirect
183. #2884	C₂×C₆₇	semidirect
184. #2122	C₂×C₆₇	semidirect
185. #2120	C₂×C₆₇	semidirect
186. C₆₇	#33351	direct (N × H)
188. C₂²×#41	C₄×C₆₇	semidirect
194. #535	C₄×C₆₇	semidirect
196. #525	C₄×C₆₇	semidirect
199. #519	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×#3487	2
C₆₇×#2889	2
C₆₇×#2884	2
C₆₇×#2122	2
C₆₇×#2120	2
#33351	67

Subgroups — C67×#33351 #30712602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33351 #30712602