Subgroups — #31736507

Order 17152.

1778 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574028	C₂	1	2	yes	maximal, complemented
2	#22573128	C₂	1	4	yes	maximal, complemented
3	#22573117	C₂	1	4	yes	maximal, complemented
4	#22573098	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
6	#22573074	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
8	#22570749	C₂	1	2	yes	maximal, complemented
9	C₂×#16863801	C₂	1	1	yes	maximal, characteristic, complemented
10	#22570058	C₂	1	4	yes	maximal, complemented
11	#22570024	C₂	1	2	yes	maximal, complemented
12	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
13	#22569085	C₂	1	2	yes	maximal
14	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
15	Q₈×Dic₁₃₄	C₂²	1	4	yes	—
16	#16864125	C₂²	1	4	yes	—
17	#16864103	—	2	4	no	—
18	#16864103	C₂²	1	4	yes	—
19	D₄×Dic₁₃₄	—	2	4	no	—
20	D₄×Dic₁₃₄	C₂²	1	4	yes	—
21	#16864024	—	2	4	no	—
22	#16864024	C₂²	1	4	yes	—
23	#16863953	—	2	2	no	—
24	#16863953	C₂²	1	1	yes	characteristic
25	C₂×#14524043	—	2	2	no	—
26	C₂×#14524043	C₂²	1	2	yes	—
27	#16863879	C₂²	1	4	yes	—
28	#16863803	C₂²	1	8	yes	—
29	#16863801	C₂²	1	2	yes	—
30	#16863801	C₂²	1	2	yes	—
31	#16863801	C₂²	1	2	yes	—
32	#16863801	C₂²	1	2	yes	—
33	#16863800	C₂²	1	4	yes	—
34	#16863797	C₂²	1	4	yes	—
35	#16863705	C₂²	1	8	yes	—
36	#16863694	C₂²	1	8	yes	—
37	#16863691	C₂²	1	4	yes	—
38	#16863689	C₂²	1	8	yes	—
39	#16863686	C₂²	1	4	yes	—
40	#16863681	C₂²	1	4	yes	—
41	#16863676	C₂²	1	4	yes	—
42	#16863674	C₂²	1	8	yes	—
43	#16863526	C₂²	1	2	yes	—
44	#16863526	C₂²	1	2	yes	—
45	C₈×Dic₁₃₄	C₂²	1	2	yes	—
46	C₈×Dic₁₃₄	C₂²	1	2	yes	—
47	#16863497	C₂²	1	8	yes	—
48	Q₁₆×Dic₆₇	C₂²	1	4	yes	—
49	D₈×Dic₆₇	C₂²	1	4	yes	—
50	#16863439	C₂²	1	4	yes	—
51	C₂×#14523977	C₂²	1	2	yes	—
52	C₂×#14523977	C₂²	1	2	yes	—
53	#16863422	C₂²	1	2	yes	—
54	#16863420	C₂²	1	1	yes	characteristic
55	#16863397	C₂²	1	4	yes	—
56	#16863351	C₂²	1	2	yes	—
57	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×#41×C₆₇	C₂²	1	2	yes	—
59	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
60	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
61	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
62	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
63	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
64	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
65	#427×C₆₇	C₂²	1	1	yes	characteristic
66	#416×C₆₇	C₂²	1	2	yes	—
67	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
68	C₂²×Dic₁₃₄	—	2	4	no	—
69	C₂²×Dic₁₃₄	—	2	2	no	—
70	C₂×#11761612	C₂³	1	2	yes	—
71	#14524043	—	2	4	no	—
72	#14524043	—	2	4	no	—
73	#14524043	—	2	4	no	—
74	#14524043	—	2	2	no	—
75	#14524043	—	2	2	no	—
76	#14524043	D₄	1	1	yes	characteristic
77	#14524043	D₄	1	1	yes	characteristic
78	#14524043	C₂³	1	4	yes	—
79	#14524043	C₂³	1	4	yes	—
80	#14524042	—	2	8	no	—
81	#14524042	C₂³	1	8	yes	—
82	#14524035	—	2	4	no	—
83	#14524035	D₄	1	2	yes	—
84	#14524034	—	2	8	no	—
85	#14524034	C₂³	1	8	yes	—
86	#14524022	—	2	4	no	—
87	#14524022	—	2	4	no	—
88	#14524022	D₄	1	2	yes	—
89	#14524022	D₄	1	2	yes	—
90	#14524020	—	2	8	no	—
91	#14524020	—	2	8	no	—
92	#14524020	—	2	8	no	—
93	Q₈×Dic₆₇	C₂³	1	8	yes	—
94	C₄×Dic₁₃₄	—	2	4	no	—
95	C₄×Dic₁₃₄	—	2	4	no	—
96	C₄×Dic₁₃₄	C₂³	1	2	yes	—
97	C₄×Dic₁₃₄	C₂³	1	2	yes	—
98	D₄×Dic₆₇	—	2	8	no	—
99	D₄×Dic₆₇	C₂³	1	8	yes	—
100	#14524002	—	2	4	no	—
101	#14524002	C₂³	1	2	yes	—
102	C₂×C₄⋊Dic₆₇	—	2	4	no	—
103	C₂×C₄⋊Dic₆₇	—	2	4	no	—
104	C₂×C₄⋊Dic₆₇	—	2	2	no	—
105	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
106	#14523977	C₂³	1	4	yes	—
107	#14523977	C₂³	1	4	yes	—
108	#14523977	C₂³	1	4	yes	—
109	#14523977	C₂³	1	4	yes	—
110	#14523976	C₂³	1	8	yes	—
111	#14523974	C₂³	1	8	yes	—
112	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
113	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
114	Dic₆₇⋊C₈	C₂³	1	4	yes	—
115	Dic₆₇⋊C₈	C₂³	1	4	yes	—
116	#14523967	C₂³	1	8	yes	—
117	#14523964	C₂³	1	8	yes	—
118	#14523953	C₂³	1	4	yes	—
119	C₈×Dic₆₇	C₂³	1	4	yes	—
120	C₂×#41×C₆₇	C₂³	1	2	yes	—
121	#135×C₆₇	Q₈	1	8	yes	complemented
122	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
124	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
125	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
126	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
127	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
128	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
129	C₄×D₄×C₆₇	—	2	2	no	—
130	C₄×D₄×C₆₇	—	2	2	no	—
131	C₄×D₄×C₆₇	C₂³	1	2	yes	—
132	C₄×D₄×C₆₇	C₂³	1	2	yes	—
133	#117×C₆₇	—	2	2	no	—
134	#117×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	—	2	2	no	—
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	#107×C₆₇	C₂³	1	1	yes	characteristic
138	#107×C₆₇	C₂³	1	1	yes	characteristic
139	#106×C₆₇	C₂³	1	2	yes	—
140	#103×C₆₇	C₂³	1	2	yes	—
141	#103×C₆₇	C₂³	1	2	yes	—
142	#102×C₆₇	C₂³	1	2	yes	—
143	#102×C₆₇	C₂³	1	2	yes	—
144	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	C₂×Dic₁₃₄	—	2	4	no	—
149	C₂×Dic₁₃₄	—	2	4	no	—
150	C₂×Dic₁₃₄	—	2	8	no	—
151	C₂×Dic₁₃₄	—	8	8	no	—
152	C₂×Dic₁₃₄	—	8	4	no	—
153	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
154	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
155	C₂²×Dic₆₇	—	2	4	no	—
156	C₂²×Dic₆₇	—	2	8	no	—
157	#11761612	—	2	4	no	—
158	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
159	C₄⋊Dic₆₇	—	2	4	no	—
160	C₄⋊Dic₆₇	—	2	4	no	—
161	C₄⋊Dic₆₇	—	2	4	no	—
162	C₄⋊Dic₆₇	—	2	8	no	—
163	C₄⋊Dic₆₇	—	2	8	no	—
164	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
165	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
166	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
167	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
168	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
169	Dic₆₇⋊C₄	—	2	8	no	—
170	Dic₆₇⋊C₄	—	2	8	no	—
171	Dic₆₇⋊C₄	C₂×D₄	1	4	yes	—
172	Dic₆₇⋊C₄	C₂×D₄	1	4	yes	—
173	C₂²⋊Dic₆₇	—	2	8	no	—
174	C₂²⋊Dic₆₇	C₂×D₄	1	4	yes	—
175	C₄×Dic₆₇	—	2	8	no	—
176	C₄×Dic₆₇	C₂⁴	1	4	yes	—
177	#41×C₆₇	—	2	4	no	—
178	#41×C₆₇	C₂×Q₈	1	8	yes	complemented
179	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
180	C₂×D₄×C₆₇	—	2	2	no	—
181	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	—	2	2	no	—
183	C₂²×C₄×C₆₇	—	2	2	no	—
184	C₂²×C₄×C₆₇	—	2	2	no	—
185	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	Q₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
189	SD₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
190	D₈×C₆₇	C₂×Q₈	1	4	yes	complemented
191	C₂×C₈×C₆₇	—	2	2	no	—
192	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
193	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	2	2	no	—
199	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₂²⋊C₄×C₆₇	—	2	2	no	—
203	C₂²⋊C₄×C₆₇	—	2	2	no	—
204	C₄²×C₆₇	—	2	2	no	—
205	C₄²×C₆₇	—	2	2	no	—
206	C₄²×C₆₇	—	2	2	no	—
207	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₆₇	—	2	2	no	—
214	Q₈×C₆₇	C₂²×Q₈	1	4	yes	—
215	D₄×C₆₇	—	2	8	no	—
216	D₄×C₆₇	—	2	2	no	—
217	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	8	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	4	4	no	—
225	C₂×C₄×C₆₇	—	4	4	no	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
232	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
233	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
234	C₂×Dic₆₇	—	2	8	no	—
235	C₂×Dic₆₇	—	8	8	no	—
236	C₂×Dic₆₇	—	8	4	no	—
237	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
238	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
239	Dic₁₃₄	—	2	4	no	—
240	Dic₁₃₄	—	2	4	no	—
241	Dic₁₃₄	—	4	8	no	—
242	Dic₁₃₄	—	8	8	no	—
243	Dic₁₃₄	—	8	4	no	—
244	C₈×C₆₇	—	2	1	no	—
245	C₈×C₆₇	—	2	1	no	—
246	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
247	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
248	C₆₇⋊C₈	—	2	4	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
255	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
256	C₄×C₆₇	—	2	4	no	—
257	C₄×C₆₇	—	2	1	no	—
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	4	2	no	—
260	C₄×C₆₇	—	4	2	no	—
261	C₄×C₆₇	D₄×Q₈	1	2	yes	—
262	C₄×C₆₇	#576	1	2	yes	—
263	C₄×C₆₇	#578	1	2	yes	—
264	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
265	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
266	Dic₆₇	—	2	4	no	—
267	Dic₆₇	—	2	4	no	—
268	Dic₆₇	—	4	8	no	—
269	#61373	—	67	1	no	maximal, Sylow
270	C₂×C₆₇	—	2	2	no	—
271	C₂×C₆₇	—	4	4	no	—
272	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
273	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
274	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
275	#3503	—	67	2	no	—
276	#3386	—	67	4	no	—
277	#3385	—	67	4	no	—
278	#3383	—	67	2	no	—
279	Q₈×Q₁₆	—	67	2	no	—
280	#3380	—	67	2	no	—
281	Q₈×D₈	—	67	2	no	—
282	#3164	—	67	2	no	—
283	C₂×#500	—	67	1	no	—
284	#3083	—	67	4	no	—
285	#3079	—	67	2	no	—
286	#2960	—	67	1	no	—
287	#2951	—	67	2	no	—
288	C₄×#135	D₆₇	1	1	yes	characteristic
289	C₆₇	#61373	1	1	yes	characteristic, complemented, Sylow
290	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
291	Q₈²	—	67	4	no	—
292	#557	—	67	4	no	—
293	#554	—	67	4	no	—
294	#554	—	134	4	no	—
295	D₄×Q₈	—	67	4	no	—
296	D₄×Q₈	—	134	4	no	—
297	#544	—	67	4	no	—
298	#544	—	134	4	no	—
299	#533	—	67	1	no	—
300	#533	—	134	2	no	—
301	C₂×C₄⋊Q₈	—	67	2	no	—
302	C₂×C₄⋊Q₈	—	134	2	no	—
303	#520	—	67	4	no	—
304	C₄×#41	C₂×D₆₇	1	2	yes	—
305	#501	—	67	8	no	—
306	#500	—	67	4	no	—
307	#500	—	67	2	no	—
308	#500	—	67	2	no	—
309	#500	—	67	2	no	—
310	#500	—	67	2	no	—
311	#499	—	67	4	no	—
312	#479	—	67	8	no	—
313	#478	—	67	8	no	—
314	#477	—	67	4	no	—
315	#477	—	67	8	no	—
316	#476	—	67	4	no	—
317	#475	—	67	4	no	—
318	#474	—	67	4	no	—
319	#474	—	67	8	no	—
320	#446	—	67	2	no	—
321	#446	—	67	2	no	—
322	C₈×Q₈	—	67	2	no	—
323	C₈×Q₈	—	67	2	no	—
324	#440	—	67	8	no	—
325	C₄×Q₁₆	—	67	4	no	—
326	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
327	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
328	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
329	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
330	C₄×D₈	—	67	4	no	—
331	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
332	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
333	#428	—	67	4	no	—
334	#427	C₂×D₆₇	1	1	yes	characteristic
335	C₂×#107	—	67	2	no	—
336	C₂×#107	—	67	2	no	—
337	#424	—	67	2	no	—
338	#423	—	67	1	no	—
339	#417	—	67	4	no	—
340	#416	C₂×D₆₇	1	2	yes	—
341	#405	—	67	2	no	—
342	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
343	C₂×#41	C₂²×D₆₇	1	2	yes	—
344	C₂²×Q₈	—	134	4	no	—
345	C₂²×Q₈	—	134	2	no	—
346	#135	Dic₁₃₄	1	8	yes	complemented
347	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
348	C₂×SD₁₆	C₂²×D₆₇	1	2	yes	—
349	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
350	C₂×M₁₆	—	67	2	no	—
351	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
352	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
353	C₄⋊Q₈	—	67	8	no	—
354	C₄⋊Q₈	—	67	1	no	—
355	C₄⋊Q₈	—	67	1	no	—
356	C₄⋊Q₈	—	67	4	no	—
357	C₄⋊Q₈	—	67	4	no	—
358	C₄⋊Q₈	—	134	4	no	—
359	C₄⋊Q₈	—	134	4	no	—
360	C₄⋊Q₈	—	134	4	no	—
361	C₄⋊Q₈	—	134	8	no	—
362	C₄⋊Q₈	—	134	2	no	—
363	C₄⋊Q₈	—	134	2	no	—
364	#125	—	67	8	no	—
365	#125	—	67	2	no	—
366	#125	—	134	4	no	—
367	#125	—	134	8	no	—
368	C₂²⋊Q₈	—	67	2	no	—
369	C₂²⋊Q₈	—	67	2	no	—
370	C₂²⋊Q₈	—	134	8	no	—
371	C₂²⋊Q₈	—	134	8	no	—
372	C₂²⋊Q₈	—	134	8	no	—
373	C₂²⋊Q₈	—	134	4	no	—
374	C₂²⋊Q₈	—	134	4	no	—
375	C₄×Q₈	—	67	2	no	—
376	C₄×Q₈	—	67	2	no	—
377	C₄×Q₈	—	67	8	no	—
378	C₄×Q₈	—	134	4	no	—
379	C₄×Q₈	—	134	4	no	—
380	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
381	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
382	C₄×D₄	—	2	2	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	67	8	no	—
385	C₄×D₄	—	134	8	no	—
386	C₄×D₄	C₂²×D₆₇	1	2	yes	—
387	C₄×D₄	C₂²×D₆₇	1	2	yes	—
388	#117	—	2	2	no	—
389	#117	—	67	2	no	—
390	#117	—	134	4	no	—
391	#117	C₂²×D₆₇	1	2	yes	—
392	C₂×C₄⋊C₄	—	67	4	no	—
393	C₂×C₄⋊C₄	—	134	4	no	—
394	C₂×C₄⋊C₄	—	134	4	no	—
395	C₂×C₄⋊C₄	—	134	2	no	—
396	C₂×C₄²	—	2	2	no	—
397	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
398	#107	—	67	4	no	—
399	#107	—	67	4	no	—
400	#107	—	67	4	no	—
401	#107	—	67	4	no	—
402	#107	—	67	8	no	—
403	#107	C₂²×D₆₇	1	1	yes	characteristic
404	#107	C₂²×D₆₇	1	1	yes	characteristic
405	#106	—	67	8	no	—
406	#106	C₂²×D₆₇	1	2	yes	—
407	C₄⋊C₈	—	67	4	no	—
408	C₄⋊C₈	—	67	4	no	—
409	C₄⋊C₈	—	67	2	no	—
410	C₄⋊C₈	—	67	2	no	—
411	#103	—	67	8	no	—
412	#103	C₂²×D₆₇	1	2	yes	—
413	#103	C₂²×D₆₇	1	2	yes	—
414	#102	—	67	8	no	—
415	#102	C₂²×D₆₇	1	2	yes	—
416	#102	C₂²×D₆₇	1	2	yes	—
417	#97	—	67	4	no	—
418	C₄×C₈	—	67	4	no	—
419	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
420	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
421	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
422	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
423	#41	—	2	4	no	—
424	#41	C₂×Dic₁₃₄	1	8	yes	complemented
425	C₂×Q₈	—	536	4	no	—
426	C₂×Q₈	—	536	8	no	—
427	C₂×Q₈	—	67	2	no	—
428	C₂×Q₈	—	67	2	no	—
429	C₂×Q₈	—	134	8	no	—
430	C₂×Q₈	—	134	4	no	—
431	C₂×Q₈	—	134	4	no	—
432	C₂×Q₈	C₂³×D₆₇	1	2	yes	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	C₂³×D₆₇	1	2	yes	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	134	8	no	—
439	C₂²×C₄	—	134	4	no	—
440	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
441	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
443	Q₁₆	C₂×Dic₁₃₄	1	4	yes	complemented
444	SD₁₆	C₂×Dic₁₃₄	1	8	yes	complemented
445	D₈	C₂×Dic₁₃₄	1	4	yes	complemented
446	M₁₆	—	134	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	67	4	no	—
449	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	complemented
450	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
451	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
452	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
453	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	67	2	no	—
457	C₄⋊C₄	—	67	2	no	—
458	C₄⋊C₄	—	67	2	no	—
459	C₄⋊C₄	—	67	2	no	—
460	C₄⋊C₄	—	67	4	no	—
461	C₄⋊C₄	—	67	4	no	—
462	C₄⋊C₄	—	67	8	no	—
463	C₄⋊C₄	—	134	8	no	—
464	C₄⋊C₄	—	134	8	no	—
465	C₄⋊C₄	—	134	8	no	—
466	C₄⋊C₄	—	134	8	no	—
467	C₄⋊C₄	—	134	4	no	—
468	C₄⋊C₄	—	134	4	no	—
469	C₄⋊C₄	—	134	4	no	—
470	C₄⋊C₄	C₂³×D₆₇	1	2	yes	—
471	C₄⋊C₄	C₂³×D₆₇	1	2	yes	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	67	4	no	—
476	C₂²⋊C₄	—	134	8	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	67	4	no	—
481	C₄²	—	134	8	no	—
482	C₄²	D₄×D₆₇	1	1	yes	characteristic
483	C₄²	D₄×D₆₇	1	1	yes	characteristic
484	C₄²	C₂³×D₆₇	1	1	yes	characteristic
485	C₄²	C₂³×D₆₇	1	1	yes	characteristic
486	C₂³	—	2	2	no	—
487	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
488	Q₈	—	2	2	no	—
489	Q₈	—	268	8	no	—
490	Q₈	—	536	4	no	—
491	Q₈	—	536	8	no	—
492	Q₈	—	134	4	no	—
493	Q₈	—	134	4	no	—
494	Q₈	C₂²×Dic₁₃₄	1	4	yes	—
495	D₄	—	2	8	no	—
496	D₄	—	2	2	no	—
497	D₄	C₂²×Dic₁₃₄	1	4	yes	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	536	4	no	—
507	C₂×C₄	—	536	8	no	—
508	C₂×C₄	—	67	4	no	—
509	C₂×C₄	—	67	4	no	—
510	C₂×C₄	—	134	8	no	—
511	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
513	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
517	C₂×C₄	#14524090	1	2	yes	—
518	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	134	4	no	—
522	C₈	C₂²×Dic₁₃₄	1	2	yes	—
523	C₈	C₂²×Dic₁₃₄	1	2	yes	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	D₄×Dic₁₃₄	1	2	yes	—
530	C₂²	C₂²×D₄×D₆₇	1	1	yes	characteristic
531	C₄	—	2	1	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	1	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	2	no	—
536	C₄	—	268	8	no	—
537	C₄	—	134	4	no	—
538	C₄	—	134	4	no	—
539	C₄	D₄×Dic₁₃₄	1	2	yes	—
540	C₄	#16864198	1	2	yes	—
541	C₄	#16864210	1	2	yes	—
542	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
543	C₄	C₂×#14524090	1	1	yes	characteristic
544	C₂	—	2	2	no	—
545	C₂	—	4	4	no	—
546	C₂	C₂×D₄×Dic₁₃₄	1	1	yes	characteristic
547	C₂	C₂×#16864198	1	1	yes	characteristic
548	C₂	C₂×#16864210	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄×D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂×#135×C₆₇	C₂²
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	#61373
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22574028	C₂	semidirect
2. #22573128	C₂	semidirect
3. #22573117	C₂	semidirect
4. #22573098	C₂	semidirect
5. Q₁₆×Dic₁₃₄	C₂	semidirect
6. #22573074	C₂	semidirect
7. D₈×Dic₁₃₄	C₂	semidirect
8. #22570749	C₂	semidirect
9. C₂×#16863801	C₂	semidirect
10. #22570058	C₂	semidirect
11. #22570024	C₂	semidirect
12. #22569143	C₂	semidirect
121. #135×C₆₇	Q₈	semidirect
178. #41×C₆₇	C₂×Q₈	semidirect
188. Q₁₆×C₆₇	C₂×Q₈	semidirect
189. SD₁₆×C₆₇	C₂×Q₈	semidirect
190. D₈×C₆₇	C₂×Q₈	semidirect
192. C₂×C₈×C₆₇	C₂×Q₈	semidirect
289. C₆₇	#61373	semidirect
346. #135	Dic₁₃₄	semidirect
424. #41	C₂×Dic₁₃₄	semidirect
443. Q₁₆	C₂×Dic₁₃₄	semidirect
444. SD₁₆	C₂×Dic₁₃₄	semidirect
445. D₈	C₂×Dic₁₃₄	semidirect
449. C₂×C₈	C₂×Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22574028	2
#22573128	2
#22573117	2
#22573098	2
Q₁₆×Dic₁₃₄	2
#22573074	2
D₈×Dic₁₃₄	2
#22570749	2
C₂×#16863801	2
#22570058	2
#22570024	2
#22569143	2
#22569085	2
C₄×#135×C₆₇	2
#61373	67