Subgroups — C₆₇×#33900 #30713151

Order 17152.

558 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×#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3287	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2232	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2220	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2139	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₇	C₂²	1	2	yes	—
13	#554×C₆₇	—	2	1	no	—
14	#550×C₆₇	—	2	1	no	—
15	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	#500×C₆₇	C₂²	1	1	yes	characteristic
19	#493×C₆₇	C₂²	1	1	yes	characteristic
20	#484×C₆₇	—	2	2	no	—
21	#477×C₆₇	—	2	1	no	—
22	#467×C₆₇	C₂²	1	2	yes	—
23	#466×C₆₇	C₂²	1	2	yes	—
24	#460×C₆₇	—	2	1	no	—
25	#449×C₆₇	—	2	2	no	—
26	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#107×C₆₇	C₂²	1	2	yes	—
30	#366×C₆₇	C₂²	1	1	yes	characteristic
31	#366×C₆₇	C₂²	1	2	yes	—
32	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#358×C₆₇	C₂²	1	1	yes	characteristic
34	#358×C₆₇	C₂²	1	2	yes	—
35	#358×C₆₇	C₂²	1	1	yes	characteristic
36	#357×C₆₇	C₂²	1	2	yes	—
37	#357×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₇	—	2	2	no	—
42	#135×C₆₇	—	2	4	no	—
43	#135×C₆₇	—	4	2	no	—
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	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	#127×C₆₇	—	2	1	no	—
52	#127×C₆₇	—	4	1	no	—
53	#125×C₆₇	—	4	1	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	4	2	no	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	4	2	no	—
58	C₄×Q₈×C₆₇	—	2	1	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	4	1	no	—
61	C₄×D₄×C₆₇	—	4	1	no	—
62	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	Q₃₂×C₆₇	—	2	2	no	—
65	SD₃₂×C₆₇	—	4	2	no	—
66	M₃₂×C₆₇	—	4	2	no	—
67	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
76	#103×C₆₇	—	2	1	no	—
77	#103×C₆₇	—	2	2	no	—
78	#103×C₆₇	—	2	1	no	—
79	#102×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#41×C₆₇	—	4	4	no	—
83	#41×C₆₇	—	8	2	no	—
84	C₂×Q₈×C₆₇	—	2	1	no	—
85	C₂×Q₈×C₆₇	—	2	1	no	—
86	C₂×Q₈×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	1	no	—
91	C₂×D₄×C₆₇	—	4	1	no	—
92	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
96	Q₁₆×C₆₇	—	2	1	no	—
97	Q₁₆×C₆₇	—	2	1	no	—
98	Q₁₆×C₆₇	—	2	1	no	—
99	Q₁₆×C₆₇	#41	1	2	yes	—
100	SD₁₆×C₆₇	—	4	4	no	—
101	D₈×C₆₇	—	2	2	no	—
102	D₈×C₆₇	—	2	2	no	—
103	D₈×C₆₇	—	8	1	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	2	no	—
117	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
118	C₂²⋊C₄×C₆₇	—	4	2	no	—
119	C₂²⋊C₄×C₆₇	—	4	2	no	—
120	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
121	C₄²×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₆₇	—	2	1	no	—
125	C₁₆×C₆₇	—	2	1	no	—
126	C₁₆×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	C₂×D₈	1	2	yes	—
129	Q₈×C₆₇	—	2	2	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	4	1	no	—
132	Q₈×C₆₇	—	4	1	no	—
133	Q₈×C₆₇	—	16	1	no	—
134	D₄×C₆₇	—	2	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	8	4	no	—
138	D₄×C₆₇	—	8	2	no	—
139	D₄×C₆₇	—	16	1	no	—
140	D₄×C₆₇	—	16	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	8	4	no	—
149	C₂×C₄×C₆₇	—	16	2	no	—
150	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₆₇	—	2	2	no	—
155	C₈×C₆₇	—	2	1	no	—
156	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
157	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₆₇	—	2	1	no	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	—	8	1	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	—	8	1	no	—
171	C₄×C₆₇	—	8	1	no	—
172	C₄×C₆₇	#510	1	2	yes	—
173	C₄×C₆₇	#546	1	1	yes	characteristic
174	C₄×C₆₇	#577	1	1	yes	characteristic
175	#33900	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	4	2	no	—
177	C₂×C₆₇	—	16	2	no	—
178	C₂×C₆₇	#3294	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
180	C₂×C₆₇	#3415	1	1	yes	characteristic
181	#3338	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#3287	C₂×C₆₇	1	1	yes	characteristic
183	#2240	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#2232	C₂×C₆₇	1	2	yes	complemented
185	#2227	C₂×C₆₇	1	1	yes	characteristic, complemented
186	#2220	C₂×C₆₇	1	2	yes	complemented
187	#2207	C₂×C₆₇	1	1	yes	characteristic, complemented
188	#2189	C₂×C₆₇	1	2	yes	complemented
189	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
190	#2169	C₂×C₆₇	1	1	yes	characteristic, complemented
191	#2139	C₂×C₆₇	1	2	yes	complemented
192	C₆₇	#33900	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×#135	C₂²×C₆₇	1	2	yes	—
194	#554	—	2	1	no	—
195	#550	—	2	1	no	—
196	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
198	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
199	#500	C₂²×C₆₇	1	1	yes	characteristic
200	#493	C₂²×C₆₇	1	1	yes	characteristic
201	#484	—	2	2	no	—
202	#477	—	2	1	no	—
203	#467	C₂²×C₆₇	1	2	yes	—
204	#466	C₂²×C₆₇	1	2	yes	—
205	#460	—	2	1	no	—
206	#449	—	2	2	no	—
207	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
208	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
209	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×#107	C₂²×C₆₇	1	2	yes	—
211	#366	C₂²×C₆₇	1	1	yes	characteristic
212	#366	C₂²×C₆₇	1	2	yes	—
213	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
214	#358	C₂²×C₆₇	1	1	yes	characteristic
215	#358	C₂²×C₆₇	1	2	yes	—
216	#358	C₂²×C₆₇	1	1	yes	characteristic
217	#357	C₂²×C₆₇	1	2	yes	—
218	#357	C₂²×C₆₇	1	2	yes	—
219	#357	C₂²×C₆₇	1	2	yes	—
220	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
221	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#41	—	2	2	no	—
223	#135	—	2	4	no	—
224	#135	—	4	2	no	—
225	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
226	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×D₈	—	2	1	no	—
229	C₂×D₈	C₂³×C₆₇	1	2	yes	—
230	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
231	C₄⋊Q₈	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	—	4	1	no	—
234	#125	—	4	1	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	4	2	no	—
237	#121	—	2	2	no	—
238	#121	—	4	2	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	—	4	1	no	—
243	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	2	2	no	—
245	Q₃₂	—	2	2	no	—
246	SD₃₂	—	4	2	no	—
247	M₃₂	—	4	2	no	—
248	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
249	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
250	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
251	#107	—	2	2	no	—
252	#107	—	2	1	no	—
253	#107	C₂³×C₆₇	1	1	yes	characteristic
254	#107	C₂³×C₆₇	1	2	yes	—
255	#107	C₂³×C₆₇	1	1	yes	characteristic
256	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
257	#103	—	2	1	no	—
258	#103	—	2	1	no	—
259	#103	—	2	2	no	—
260	#102	—	2	2	no	—
261	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
262	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
263	#41	—	4	4	no	—
264	#41	—	8	2	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	2	1	no	—
279	Q₁₆	—	2	1	no	—
280	Q₁₆	#41×C₆₇	1	2	yes	—
281	SD₁₆	—	4	4	no	—
282	D₈	—	2	2	no	—
283	D₈	—	2	2	no	—
284	D₈	—	8	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
289	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	C₂×D₈×C₆₇	1	2	yes	—
310	Q₈	—	2	2	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	16	1	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	4	no	—
319	D₄	—	8	2	no	—
320	D₄	—	16	1	no	—
321	D₄	—	16	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	4	no	—
330	C₂×C₄	—	16	2	no	—
331	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
333	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
338	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	1	no	—
353	C₄	#510×C₆₇	1	2	yes	—
354	C₄	#546×C₆₇	1	1	yes	characteristic
355	C₄	#577×C₆₇	1	1	yes	characteristic
356	C₂	—	4	2	no	—
357	C₂	—	16	2	no	—
358	C₂	C₆₇×#3294	1	1	yes	characteristic
359	C₂	C₂×#510×C₆₇	1	1	yes	characteristic
360	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3287	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3338	C₂	semidirect
3. C₆₇×#2240	C₂	semidirect
4. C₆₇×#2232	C₂	semidirect
5. C₆₇×#2227	C₂	semidirect
6. C₆₇×#2220	C₂	semidirect
7. C₆₇×#2207	C₂	semidirect
8. C₆₇×#2189	C₂	semidirect
9. C₄×Q₃₂×C₆₇	C₂	semidirect
10. C₆₇×#2169	C₂	semidirect
11. C₆₇×#2139	C₂	semidirect
175. #33900	C₆₇	direct (N × H)
181. #3338	C₂×C₆₇	semidirect
183. #2240	C₂×C₆₇	semidirect
184. #2232	C₂×C₆₇	semidirect
185. #2227	C₂×C₆₇	semidirect
186. #2220	C₂×C₆₇	semidirect
187. #2207	C₂×C₆₇	semidirect
188. #2189	C₂×C₆₇	semidirect
189. C₄×Q₃₂	C₂×C₆₇	semidirect
190. #2169	C₂×C₆₇	semidirect
191. #2139	C₂×C₆₇	semidirect
192. C₆₇	#33900	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×#3338	2
C₆₇×#3287	2
C₆₇×#2240	2
C₆₇×#2232	2
C₆₇×#2227	2
C₆₇×#2220	2
C₆₇×#2207	2
C₆₇×#2189	2
C₄×Q₃₂×C₆₇	2
C₆₇×#2169	2
C₆₇×#2139	2
#33900	67

Subgroups — C67×#33900 #30713151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33900 #30713151