Subgroups — C₆₇×#20143 #30699394

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3354	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3303	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1703	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1671	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1654	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1634	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1634	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1586	C₂	1	2	yes	maximal, complemented
10	#555×C₆₇	C₂²	1	1	yes	characteristic
11	#555×C₆₇	C₂²	1	1	yes	characteristic
12	#547×C₆₇	C₂²	1	2	yes	—
13	#498×C₆₇	C₂²	1	2	yes	—
14	#498×C₆₇	C₂²	1	1	yes	characteristic
15	#494×C₆₇	C₂²	1	1	yes	characteristic
16	#487×C₆₇	C₂²	1	2	yes	—
17	#485×C₆₇	—	2	2	no	—
18	#482×C₆₇	—	2	2	no	—
19	#481×C₆₇	—	2	2	no	—
20	#476×C₆₇	C₂²	1	2	yes	—
21	#476×C₆₇	C₂²	1	2	yes	—
22	#474×C₆₇	C₂²	1	2	yes	—
23	#469×C₆₇	—	2	2	no	—
24	#465×C₆₇	—	2	2	no	—
25	#461×C₆₇	C₂²	1	2	yes	—
26	#449×C₆₇	—	2	2	no	—
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	#436×C₆₇	C₂²	1	2	yes	—
29	C₈×D₄×C₆₇	C₂²	1	2	yes	—
30	#428×C₆₇	—	2	2	no	—
31	#416×C₆₇	—	2	2	no	—
32	#334×C₆₇	C₂²	1	2	yes	—
33	#333×C₆₇	C₂²	1	1	yes	characteristic
34	#333×C₆₇	C₂²	1	1	yes	characteristic
35	#332×C₆₇	C₂²	1	2	yes	—
36	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×D₈×C₆₇	D₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	#124×C₆₇	—	4	1	no	—
49	#124×C₆₇	—	4	1	no	—
50	#123×C₆₇	—	4	2	no	—
51	C₂²⋊Q₈×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	2	no	—
55	C₄×D₄×C₆₇	C₂³	1	2	yes	—
56	C₄×D₄×C₆₇	C₂³	1	2	yes	—
57	#117×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	#107×C₆₇	—	2	2	no	—
61	#107×C₆₇	D₄	1	2	yes	—
62	#106×C₆₇	—	2	2	no	—
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	2	2	no	—
65	#106×C₆₇	—	2	2	no	—
66	#106×C₆₇	—	4	1	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#103×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	2	no	—
72	#102×C₆₇	—	2	2	no	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	D₄	1	2	yes	—
76	#102×C₆₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	#41×C₆₇	—	4	4	no	—
82	#41×C₆₇	—	8	2	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	1	no	—
85	C₂×Q₈×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	4	2	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	Q₁₆×C₆₇	—	8	2	no	—
95	SD₁₆×C₆₇	—	8	2	no	—
96	D₈×C₆₇	—	4	2	no	—
97	M₁₆×C₆₇	—	4	4	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	8	2	no	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂²⋊C₄×C₆₇	—	4	2	no	—
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂³×C₆₇	—	2	2	no	—
120	C₂³×C₆₇	—	2	2	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	Q₈×C₆₇	—	4	2	no	—
123	Q₈×C₆₇	—	8	1	no	—
124	Q₈×C₆₇	—	8	1	no	—
125	D₄×C₆₇	—	2	2	no	—
126	D₄×C₆₇	—	2	2	no	—
127	D₄×C₆₇	—	2	2	no	—
128	D₄×C₆₇	—	2	2	no	—
129	D₄×C₆₇	—	8	4	no	—
130	D₄×C₆₇	—	16	1	no	—
131	D₄×C₆₇	—	16	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	8	4	no	—
142	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	—	4	2	no	—
148	C₈×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	—	4	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×C₆₇	—	4	2	no	—
154	C₂²×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	D₄²	1	1	yes	characteristic
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	—	8	2	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	#575	1	2	yes	—
163	C₄×C₆₇	#577	1	2	yes	—
164	#20143	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₆₇	—	8	2	no	—
166	C₂×C₆₇	—	8	2	no	—
167	C₂×C₆₇	#3267	1	2	yes	—
168	C₂×C₆₇	#3291	1	1	yes	characteristic
169	#3354	C₂×C₆₇	1	2	yes	complemented
170	#3303	C₂×C₆₇	1	2	yes	complemented
171	#1716	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#1703	C₂×C₆₇	1	2	yes	complemented
173	#1671	C₂×C₆₇	1	2	yes	complemented
174	#1654	C₂×C₆₇	1	2	yes	complemented
175	#1634	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#1634	C₂×C₆₇	1	1	yes	characteristic, complemented
177	#1586	C₂×C₆₇	1	2	yes	complemented
178	C₆₇	#20143	1	1	yes	characteristic, direct product factor, Sylow
179	#555	C₂²×C₆₇	1	1	yes	characteristic
180	#555	C₂²×C₆₇	1	1	yes	characteristic
181	#547	C₂²×C₆₇	1	2	yes	—
182	#498	C₂²×C₆₇	1	2	yes	—
183	#498	C₂²×C₆₇	1	1	yes	characteristic
184	#494	C₂²×C₆₇	1	1	yes	characteristic
185	#487	C₂²×C₆₇	1	2	yes	—
186	#485	—	2	2	no	—
187	#482	—	2	2	no	—
188	#481	—	2	2	no	—
189	#476	C₂²×C₆₇	1	2	yes	—
190	#476	C₂²×C₆₇	1	2	yes	—
191	#474	C₂²×C₆₇	1	2	yes	—
192	#469	—	2	2	no	—
193	#465	—	2	2	no	—
194	#461	C₂²×C₆₇	1	2	yes	—
195	#449	—	2	2	no	—
196	C₄×D₈	C₂²×C₆₇	1	2	yes	—
197	#436	C₂²×C₆₇	1	2	yes	—
198	C₈×D₄	C₂²×C₆₇	1	2	yes	—
199	#428	—	2	2	no	—
200	#416	—	2	2	no	—
201	#334	C₂²×C₆₇	1	2	yes	—
202	#333	C₂²×C₆₇	1	1	yes	characteristic
203	#333	C₂²×C₆₇	1	1	yes	characteristic
204	#332	C₂²×C₆₇	1	2	yes	—
205	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
206	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
207	#322	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×#41	—	2	2	no	—
209	C₂×Q₁₆	—	4	1	no	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×D₈	D₄×C₆₇	1	2	yes	—
212	C₂×M₁₆	—	2	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
216	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
217	#124	—	4	1	no	—
218	#124	—	4	1	no	—
219	#123	—	4	2	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	C₄×D₄	C₂³×C₆₇	1	2	yes	—
225	C₄×D₄	C₂³×C₆₇	1	2	yes	—
226	#117	—	2	2	no	—
227	#117	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	D₄×C₆₇	1	2	yes	—
231	#106	—	2	2	no	—
232	#106	—	2	2	no	—
233	#106	—	2	2	no	—
234	#106	—	2	2	no	—
235	#106	—	4	1	no	—
236	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
237	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	#103	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	D₄×C₆₇	1	2	yes	—
245	#102	D₄×C₆₇	1	2	yes	—
246	C₂²⋊C₈	—	2	2	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₆₇	1	2	yes	—
250	#41	—	4	4	no	—
251	#41	—	8	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
259	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	Q₁₆	—	8	2	no	—
264	SD₁₆	—	8	2	no	—
265	D₈	—	4	2	no	—
266	M₁₆	—	4	4	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	8	2	no	—
271	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	1	no	—
293	Q₈	—	8	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	2	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	8	4	no	—
299	D₄	—	16	1	no	—
300	D₄	—	16	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	4	no	—
311	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
312	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	D₄²×C₆₇	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₄	#575×C₆₇	1	2	yes	—
332	C₄	#577×C₆₇	1	2	yes	—
333	C₂	—	8	2	no	—
334	C₂	—	8	2	no	—
335	C₂	C₆₇×#3267	1	2	yes	—
336	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3354	C₂	semidirect
2. C₆₇×#3303	C₂	semidirect
3. C₆₇×#1716	C₂	semidirect
4. C₆₇×#1703	C₂	semidirect
5. C₆₇×#1671	C₂	semidirect
6. C₆₇×#1654	C₂	semidirect
7. C₆₇×#1634	C₂	semidirect
8. C₆₇×#1634	C₂	semidirect
9. C₆₇×#1586	C₂	semidirect
164. #20143	C₆₇	direct (N × H)
169. #3354	C₂×C₆₇	semidirect
170. #3303	C₂×C₆₇	semidirect
171. #1716	C₂×C₆₇	semidirect
172. #1703	C₂×C₆₇	semidirect
173. #1671	C₂×C₆₇	semidirect
174. #1654	C₂×C₆₇	semidirect
175. #1634	C₂×C₆₇	semidirect
176. #1634	C₂×C₆₇	semidirect
177. #1586	C₂×C₆₇	semidirect
178. C₆₇	#20143	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×#3354	2
C₆₇×#3303	2
C₆₇×#1716	2
C₆₇×#1703	2
C₆₇×#1671	2
C₆₇×#1654	2
C₆₇×#1634	2
C₆₇×#1586	2
#20143	67

Subgroups — C67×#20143 #30699394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20143 #30699394