Subgroups — C₆₇×#33967 #30713218

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3396	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3395	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2251	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2248	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2208	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2207	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#559×C₆₇	—	2	1	no	—
13	#557×C₆₇	—	2	1	no	—
14	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
17	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
19	#495×C₆₇	C₂²	1	2	yes	—
20	#494×C₆₇	C₂²	1	1	yes	characteristic
21	#494×C₆₇	C₂²	1	2	yes	—
22	#493×C₆₇	C₂²	1	1	yes	characteristic
23	#463×C₆₇	—	2	2	no	—
24	#462×C₆₇	—	2	1	no	—
25	#462×C₆₇	—	2	2	no	—
26	#460×C₆₇	—	2	1	no	—
27	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₆₇	C₂²	1	2	yes	—
32	#366×C₆₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
35	#358×C₆₇	C₂²	1	1	yes	characteristic
36	#358×C₆₇	C₂²	1	1	yes	characteristic
37	#358×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	2	yes	—
39	#346×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₆₇	—	2	2	no	—
42	C₂×Q₁₆×C₆₇	—	2	1	no	—
43	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₆₇	—	2	2	no	—
47	C₂×D₈×C₆₇	—	2	1	no	—
48	C₂×D₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	#127×C₆₇	—	2	1	no	—
52	#125×C₆₇	—	4	1	no	—
53	#125×C₆₇	—	4	2	no	—
54	#124×C₆₇	—	2	2	no	—
55	#124×C₆₇	—	4	1	no	—
56	#123×C₆₇	—	4	2	no	—
57	C₄×Q₈×C₆₇	—	2	1	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×Q₈×C₆₇	—	4	1	no	—
61	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	—	4	1	no	—
64	Q₃₂×C₆₇	—	2	2	no	—
65	Q₃₂×C₆₇	—	4	2	no	—
66	SD₃₂×C₆₇	—	2	4	no	—
67	SD₃₂×C₆₇	—	4	2	no	—
68	SD₃₂×C₆₇	—	4	2	no	—
69	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
72	#107×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
76	#103×C₆₇	—	2	2	no	—
77	#103×C₆₇	—	2	1	no	—
78	#103×C₆₇	—	2	1	no	—
79	#102×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂×Q₈×C₆₇	—	2	1	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	1	no	—
85	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	Q₁₆×C₆₇	—	2	2	no	—
91	Q₁₆×C₆₇	—	2	1	no	—
92	Q₁₆×C₆₇	—	2	1	no	—
93	Q₁₆×C₆₇	—	2	1	no	—
94	Q₁₆×C₆₇	—	2	2	no	—
95	Q₁₆×C₆₇	—	2	1	no	—
96	Q₁₆×C₆₇	—	8	1	no	—
97	Q₁₆×C₆₇	—	8	2	no	—
98	SD₁₆×C₆₇	—	8	2	no	—
99	D₈×C₆₇	—	2	2	no	—
100	D₈×C₆₇	—	2	2	no	—
101	D₈×C₆₇	—	8	1	no	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₆₇	—	2	1	no	—
106	C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₄⋊C₄×C₆₇	—	4	1	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
116	C₂²⋊C₄×C₆₇	—	4	2	no	—
117	C₂²⋊C₄×C₆₇	—	4	2	no	—
118	C₂²⋊C₄×C₆₇	—	4	2	no	—
119	C₄²×C₆₇	—	4	1	no	—
120	C₄²×C₆₇	—	4	2	no	—
121	C₄²×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	C₂×D₄	1	2	yes	—
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₆₇	—	2	1	no	—
125	C₁₆×C₆₇	—	2	2	no	—
126	C₁₆×C₆₇	#41	1	2	yes	—
127	C₂³×C₆₇	—	4	2	no	—
128	Q₈×C₆₇	—	2	2	no	—
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	4	1	no	—
132	Q₈×C₆₇	—	4	1	no	—
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	4	1	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	8	2	no	—
138	D₄×C₆₇	—	16	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	—	16	2	no	—
145	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₆₇	—	2	1	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	8	2	no	—
157	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	—	8	2	no	—
163	C₄×C₆₇	—	8	1	no	—
164	C₄×C₆₇	—	8	1	no	—
165	C₄×C₆₇	—	8	1	no	—
166	C₄×C₆₇	#510	1	2	yes	—
167	C₄×C₆₇	#550	1	1	yes	characteristic
168	C₄×C₆₇	#576	1	1	yes	characteristic
169	#33967	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₆₇	—	16	2	no	—
171	C₂×C₆₇	#3391	1	1	yes	characteristic
172	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
173	C₂×C₆₇	#3416	1	1	yes	characteristic
174	#3396	C₂×C₆₇	1	1	yes	characteristic
175	#3395	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#2251	C₂×C₆₇	1	2	yes	complemented
177	#2248	C₂×C₆₇	1	1	yes	characteristic
178	#2211	C₂×C₆₇	1	2	yes	complemented
179	#2209	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#2208	C₂×C₆₇	1	2	yes	complemented
181	#2207	C₂×C₆₇	1	1	yes	characteristic
182	#2183	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#2175	C₂×C₆₇	1	2	yes	complemented
184	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
185	C₆₇	#33967	1	1	yes	characteristic, direct product factor, Sylow
186	#559	—	2	1	no	—
187	#557	—	2	1	no	—
188	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
189	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
190	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
191	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
192	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
193	#495	C₂²×C₆₇	1	2	yes	—
194	#494	C₂²×C₆₇	1	2	yes	—
195	#494	C₂²×C₆₇	1	1	yes	characteristic
196	#493	C₂²×C₆₇	1	1	yes	characteristic
197	#463	—	2	2	no	—
198	#462	—	2	1	no	—
199	#462	—	2	2	no	—
200	#460	—	2	1	no	—
201	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
202	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
203	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
204	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
205	C₄×D₈	C₂²×C₆₇	1	2	yes	—
206	#366	C₂²×C₆₇	1	1	yes	characteristic
207	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
208	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
209	#358	C₂²×C₆₇	1	1	yes	characteristic
210	#358	C₂²×C₆₇	1	1	yes	characteristic
211	#358	C₂²×C₆₇	1	2	yes	—
212	#357	C₂²×C₆₇	1	2	yes	—
213	#346	C₂²×C₆₇	1	2	yes	—
214	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
218	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
219	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	C₂³×C₆₇	1	2	yes	—
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	#127	—	2	1	no	—
226	#125	—	4	1	no	—
227	#125	—	4	2	no	—
228	#124	—	2	2	no	—
229	#124	—	4	1	no	—
230	#123	—	4	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	4	1	no	—
235	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	4	1	no	—
238	Q₃₂	—	2	2	no	—
239	Q₃₂	—	4	2	no	—
240	SD₃₂	—	2	4	no	—
241	SD₃₂	—	4	2	no	—
242	SD₃₂	—	4	2	no	—
243	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
244	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
245	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
246	#107	C₂³×C₆₇	1	1	yes	characteristic
247	#107	C₂³×C₆₇	1	1	yes	characteristic
248	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
249	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
250	#103	—	2	2	no	—
251	#103	—	2	1	no	—
252	#103	—	2	1	no	—
253	#102	—	2	2	no	—
254	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₆₇	1	2	yes	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	8	1	no	—
271	Q₁₆	—	8	2	no	—
272	SD₁₆	—	8	2	no	—
273	D₈	—	2	2	no	—
274	D₈	—	2	2	no	—
275	D₈	—	8	1	no	—
276	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
289	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	C₂×D₄×C₆₇	1	2	yes	—
297	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	#41×C₆₇	1	2	yes	—
301	C₂³	—	4	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	16	2	no	—
319	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
320	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
321	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
327	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	#510×C₆₇	1	2	yes	—
341	C₄	#550×C₆₇	1	1	yes	characteristic
342	C₄	#576×C₆₇	1	1	yes	characteristic
343	C₂	—	16	2	no	—
344	C₂	C₆₇×#3391	1	1	yes	characteristic
345	C₂	C₂×#510×C₆₇	1	1	yes	characteristic
346	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	#493×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3395	C₂	semidirect
3. C₆₇×#2251	C₂	semidirect
5. C₆₇×#2211	C₂	semidirect
6. C₆₇×#2209	C₂	semidirect
7. C₆₇×#2208	C₂	semidirect
9. C₆₇×#2183	C₂	semidirect
10. C₆₇×#2175	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
169. #33967	C₆₇	direct (N × H)
175. #3395	C₂×C₆₇	semidirect
176. #2251	C₂×C₆₇	semidirect
178. #2211	C₂×C₆₇	semidirect
179. #2209	C₂×C₆₇	semidirect
180. #2208	C₂×C₆₇	semidirect
182. #2183	C₂×C₆₇	semidirect
183. #2175	C₂×C₆₇	semidirect
184. C₄×Q₃₂	C₂×C₆₇	semidirect
185. C₆₇	#33967	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3396	2
C₆₇×#3395	2
C₆₇×#2251	2
C₆₇×#2248	2
C₆₇×#2211	2
C₆₇×#2209	2
C₆₇×#2208	2
C₆₇×#2207	2
C₆₇×#2183	2
C₆₇×#2175	2
C₄×Q₃₂×C₆₇	2
#33967	67

Subgroups — C67×#33967 #30713218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33967 #30713218