Subgroups — #30876710

Order 17152.

1016 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569119	C₂	1	2	yes	maximal, complemented
2	#22568756	C₂	1	2	yes	maximal, complemented
3	#22561038	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560880	C₂	1	2	yes	maximal, complemented
5	#22560712	C₂	1	2	yes	maximal, complemented
6	#22560593	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560451	C₂	1	2	yes	maximal, complemented
8	#104×Dic₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3018	C₂	1	1	yes	maximal, characteristic
10	#16864163	C₂²	1	2	yes	—
11	#16864163	C₂²	1	1	yes	characteristic, complemented
12	#41×Dic₆₇	C₂²	1	2	yes	—
13	#16863518	—	2	4	no	—
14	#16863513	—	2	4	no	—
15	#16863445	C₂²	1	2	yes	complemented
16	#16863420	C₂²	1	1	yes	characteristic, complemented
17	#16863413	—	2	4	no	—
18	#16863413	C₂²	1	2	yes	—
19	#16863413	C₂²	1	2	yes	—
20	C₂×#14523970	—	2	4	no	—
21	C₂×#14523970	C₂²	1	2	yes	—
22	C₂×#14523970	C₂²	1	2	yes	—
23	C₂×#14523970	C₂²	1	2	yes	complemented
24	#16863370	C₂²	1	1	yes	characteristic, complemented
25	M₁₆×Dic₆₇	C₂²	1	2	yes	complemented
26	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
27	#16863168	C₂²	1	2	yes	complemented
28	#16863157	C₂²	1	2	yes	complemented
29	#16863156	C₂²	1	2	yes	complemented
30	#585×C₆₇	C₂²	1	1	yes	characteristic
31	#575×C₆₇	C₂²	1	2	yes	—
32	#532×C₆₇	C₂²	1	1	yes	characteristic
33	#456×C₆₇	C₄	1	2	yes	complemented
34	#455×C₆₇	C₄	1	2	yes	complemented
35	#454×C₆₇	C₄	1	2	yes	complemented
36	#453×C₆₇	C₄	1	2	yes	complemented
37	C₂×#104×C₆₇	C₂²	1	2	yes	—
38	#413×C₆₇	C₂²	1	1	yes	characteristic
39	#14524050	—	2	4	no	—
40	#14524050	—	2	4	no	—
41	#14524050	—	4	2	no	—
42	C₂×#11761612	—	2	4	no	—
43	C₂×#11761612	—	4	1	no	—
44	C₂×#11761612	C₂³	1	2	yes	—
45	C₂×#11761612	C₂³	1	2	yes	—
46	Q₈×Dic₆₇	—	2	2	no	—
47	D₄×Dic₆₇	—	2	2	no	—
48	D₄×Dic₆₇	—	4	2	no	—
49	#14524002	—	2	2	no	—
50	#14524002	C₂³	1	2	yes	—
51	C₂×C₄×Dic₆₇	—	2	2	no	—
52	C₂×C₄×Dic₆₇	—	4	1	no	—
53	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
54	#14523979	—	2	4	no	—
55	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
56	#14523970	—	2	4	no	—
57	#14523970	—	2	2	no	—
58	#14523970	—	2	2	no	—
59	#14523970	—	2	2	no	—
60	#14523970	—	2	2	no	—
61	#14523970	—	2	4	no	—
62	#14523970	D₄	1	4	yes	complemented
63	#14523970	D₄	1	4	yes	complemented
64	#14523958	—	4	1	no	—
65	#14523953	—	2	2	no	—
66	C₈×Dic₆₇	—	2	2	no	—
67	2₋¹⁺⁴×C₆₇	—	2	2	no	—
68	2₋¹⁺⁴×C₆₇	C₂×C₄	1	2	yes	—
69	H(5,𝔽₂)×C₆₇	—	2	2	no	—
70	H(5,𝔽₂)×C₆₇	—	4	1	no	—
71	H(5,𝔽₂)×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×#41×C₆₇	—	2	4	no	—
73	C₂×#41×C₆₇	—	4	1	no	—
74	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×#41×C₆₇	C₂³	1	2	yes	—
77	#137×C₆₇	—	2	2	no	—
78	#137×C₆₇	C₂×C₄	1	4	yes	complemented
79	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
80	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	4	yes	complemented
81	#135×C₆₇	—	2	4	no	—
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
84	#127×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
85	#124×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
86	#124×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
87	#121×C₆₇	—	4	1	no	—
88	C₄×D₄×C₆₇	—	4	1	no	—
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	#104×C₆₇	C₂×C₄	1	4	yes	complemented
91	#104×C₆₇	C₂×C₄	1	4	yes	complemented
92	#101×C₆₇	C₂×C₄	1	2	yes	complemented
93	#100×C₆₇	C₂×C₄	1	2	yes	complemented
94	C₂²×Dic₆₇	—	2	2	no	—
95	C₂²×Dic₆₇	—	4	2	no	—
96	#11761612	—	2	2	no	—
97	#11761612	—	2	2	no	—
98	#11761612	—	2	2	no	—
99	#11761612	—	4	2	no	—
100	#11761612	—	4	4	no	—
101	#11761612	C₂×D₄	1	4	yes	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	4	2	no	—
105	C₂×C₆₇⋊C₈	—	4	4	no	—
106	C₄⋊Dic₆₇	—	2	2	no	—
107	C₄⋊Dic₆₇	—	4	2	no	—
108	C₂²⋊Dic₆₇	—	2	2	no	—
109	C₂²⋊Dic₆₇	—	4	2	no	—
110	C₄×Dic₆₇	—	2	2	no	—
111	C₄×Dic₆₇	—	2	2	no	—
112	C₄×Dic₆₇	—	4	2	no	—
113	C₄×Dic₆₇	—	4	4	no	—
114	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
115	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
116	#41×C₆₇	—	2	2	no	—
117	#41×C₆₇	—	2	2	no	—
118	#41×C₆₇	—	2	4	no	—
119	#41×C₆₇	—	2	4	no	—
120	#41×C₆₇	—	2	2	no	—
121	#41×C₆₇	—	2	2	no	—
122	#41×C₆₇	—	2	2	no	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	4	2	no	—
125	#41×C₆₇	—	4	2	no	—
126	#41×C₆₇	—	4	4	no	—
127	#41×C₆₇	—	4	2	no	—
128	#41×C₆₇	C₂²×C₄	1	4	yes	—
129	#41×C₆₇	C₂×D₄	1	4	yes	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	4	1	no	—
132	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×Q₈×C₆₇	#41	1	2	yes	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	4	no	—
137	C₂×D₄×C₆₇	—	2	1	no	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	4	1	no	—
140	C₂×D₄×C₆₇	—	4	2	no	—
141	C₂×D₄×C₆₇	—	4	1	no	—
142	C₂×D₄×C₆₇	—	4	4	no	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂×D₄×C₆₇	#41	1	2	yes	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	4	2	no	—
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	Q₁₆×C₆₇	—	2	4	no	—
153	SD₁₆×C₆₇	—	2	4	no	—
154	SD₁₆×C₆₇	—	2	4	no	—
155	D₈×C₆₇	—	2	4	no	—
156	M₁₆×C₆₇	—	2	2	no	—
157	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	4	1	no	—
160	C₂²⋊C₄×C₆₇	—	4	1	no	—
161	C₂²⋊C₄×C₆₇	—	4	1	no	—
162	C₄²×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	4	2	no	—
169	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
170	Q₈×C₆₇	—	2	2	no	—
171	Q₈×C₆₇	—	2	2	no	—
172	Q₈×C₆₇	—	2	2	no	—
173	Q₈×C₆₇	—	4	2	no	—
174	Q₈×C₆₇	C₄×D₄	1	4	yes	—
175	D₄×C₆₇	—	2	4	no	—
176	D₄×C₆₇	—	2	4	no	—
177	D₄×C₆₇	—	2	2	no	—
178	D₄×C₆₇	—	2	2	no	—
179	D₄×C₆₇	—	2	2	no	—
180	D₄×C₆₇	—	4	4	no	—
181	D₄×C₆₇	—	4	2	no	—
182	D₄×C₆₇	—	4	2	no	—
183	D₄×C₆₇	—	4	2	no	—
184	D₄×C₆₇	—	8	2	no	—
185	D₄×C₆₇	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	4	1	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	—	4	1	no	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
203	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×Dic₆₇	—	2	2	no	—
205	C₂×Dic₆₇	—	2	2	no	—
206	C₂×Dic₆₇	—	4	4	no	—
207	C₂×Dic₆₇	—	4	2	no	—
208	C₂×Dic₆₇	—	8	4	no	—
209	C₈×C₆₇	—	2	4	no	—
210	C₆₇⋊C₈	—	2	4	no	—
211	C₆₇⋊C₈	—	4	2	no	—
212	C₂²×C₆₇	—	2	4	no	—
213	C₂²×C₆₇	—	4	1	no	—
214	C₂²×C₆₇	—	4	1	no	—
215	C₂²×C₆₇	—	4	1	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	—	8	1	no	—
218	C₂²×C₆₇	—	8	1	no	—
219	C₂²×C₆₇	—	16	2	no	—
220	C₂²×C₆₇	#518	1	1	yes	characteristic
221	C₂²×C₆₇	#546	1	2	yes	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	4	1	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	—	4	1	no	—
227	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
228	C₄×C₆₇	D₄²	1	1	yes	characteristic
229	C₄×C₆₇	#551	1	1	yes	characteristic
230	Dic₆₇	—	2	4	no	—
231	Dic₆₇	—	4	2	no	—
232	Dic₆₇	—	8	2	no	—
233	#22899	—	67	1	no	maximal, Sylow
234	C₂×C₆₇	—	2	2	no	—
235	C₂×C₆₇	—	2	1	no	—
236	C₂×C₆₇	—	4	4	no	—
237	C₂×C₆₇	—	8	1	no	—
238	C₂×C₆₇	—	8	1	no	—
239	C₂×C₆₇	#2358	1	1	yes	characteristic
240	#3018	D₆₇	1	1	yes	characteristic
241	#2956	—	67	2	no	—
242	#2901	—	67	2	no	—
243	#1962	—	67	1	no	—
244	#1932	—	67	2	no	—
245	#1906	—	67	2	no	—
246	#1884	—	67	1	no	—
247	#1861	—	67	2	no	—
248	C₄×#104	—	67	2	no	—
249	C₆₇	#22899	1	1	yes	characteristic, complemented, Sylow
250	#585	C₂×D₆₇	1	1	yes	characteristic
251	#575	C₂×D₆₇	1	2	yes	—
252	#568	—	67	1	no	—
253	#568	—	67	2	no	—
254	#532	C₂×D₆₇	1	1	yes	characteristic
255	C₄×#41	—	67	2	no	—
256	#456	Dic₆₇	1	2	yes	complemented
257	#455	Dic₆₇	1	2	yes	complemented
258	#454	Dic₆₇	1	2	yes	complemented
259	#453	Dic₆₇	1	2	yes	complemented
260	#444	—	134	4	no	—
261	#443	—	134	4	no	—
262	#430	—	67	2	no	—
263	#423	—	67	1	no	—
264	#421	—	67	2	no	—
265	#421	—	67	2	no	—
266	#421	—	134	4	no	—
267	C₂×#104	—	67	2	no	—
268	C₂×#104	—	67	2	no	—
269	C₂×#104	—	67	2	no	—
270	C₂×#104	—	134	4	no	—
271	C₂×#104	C₂×D₆₇	1	2	yes	—
272	#413	C₂×D₆₇	1	1	yes	characteristic
273	#410	—	67	1	no	—
274	C₄×M₁₆	—	67	2	no	—
275	C₄³	—	67	1	no	—
276	#344	—	67	2	no	—
277	#339	—	67	2	no	—
278	#339	—	67	2	no	—
279	2₋¹⁺⁴	—	2	2	no	—
280	2₋¹⁺⁴	C₂×Dic₆₇	1	2	yes	—
281	H(5,𝔽₂)	—	2	2	no	—
282	H(5,𝔽₂)	—	4	1	no	—
283	H(5,𝔽₂)	C₂×Dic₆₇	1	2	yes	—
284	C₂×#41	—	2	4	no	—
285	C₂×#41	—	4	1	no	—
286	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
287	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
288	C₂×#41	C₂²×D₆₇	1	2	yes	—
289	#137	—	2	2	no	—
290	#137	C₂×Dic₆₇	1	4	yes	complemented
291	AGL(1,ℤ/8)	—	2	2	no	—
292	AGL(1,ℤ/8)	C₂×Dic₆₇	1	4	yes	complemented
293	#135	—	2	4	no	—
294	#131	—	268	2	no	—
295	#131	—	134	4	no	—
296	#131	—	134	4	no	—
297	C₂×M₁₆	—	268	1	no	—
298	C₂×M₁₆	—	67	2	no	—
299	C₂×M₁₆	—	67	2	no	—
300	C₂×M₁₆	—	134	4	no	—
301	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
302	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
303	#127	C₂×Dic₆₇	1	1	yes	characteristic, complemented
304	#124	C₂×Dic₆₇	1	1	yes	characteristic, complemented
305	#124	C₂×Dic₆₇	1	1	yes	characteristic, complemented
306	#121	—	4	1	no	—
307	C₄×Q₈	—	134	2	no	—
308	C₄×D₄	—	4	1	no	—
309	C₄×D₄	—	268	2	no	—
310	C₄×D₄	—	134	2	no	—
311	#117	—	67	2	no	—
312	#117	—	134	2	no	—
313	C₂×C₄²	—	268	1	no	—
314	C₂×C₄²	—	67	2	no	—
315	C₂×C₄²	—	134	2	no	—
316	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
317	#108	—	134	4	no	—
318	C₄⋊C₈	—	268	1	no	—
319	#104	—	67	4	no	—
320	#104	—	67	4	no	—
321	#104	—	134	2	no	—
322	#104	—	134	2	no	—
323	#104	—	134	2	no	—
324	#104	—	134	2	no	—
325	#104	—	134	4	no	—
326	#104	—	134	4	no	—
327	#104	C₂×Dic₆₇	1	4	yes	complemented
328	#104	C₂×Dic₆₇	1	4	yes	complemented
329	#101	C₂×Dic₆₇	1	2	yes	complemented
330	#100	C₂×Dic₆₇	1	2	yes	complemented
331	#99	—	268	1	no	—
332	#97	—	134	2	no	—
333	C₄×C₈	—	134	2	no	—
334	#41	—	2	2	no	—
335	#41	—	2	2	no	—
336	#41	—	2	4	no	—
337	#41	—	2	4	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	4	2	no	—
343	#41	—	4	2	no	—
344	#41	—	4	4	no	—
345	#41	—	4	2	no	—
346	#41	C₂²×Dic₆₇	1	4	yes	—
347	#41	D₄×D₆₇	1	4	yes	—
348	C₂×Q₈	—	2	4	no	—
349	C₂×Q₈	—	4	1	no	—
350	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
351	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
352	C₂×Q₈	#11761627	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	4	no	—
361	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
362	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
363	C₂×D₄	#11761627	1	2	yes	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	268	2	no	—
370	C₂²×C₄	—	134	2	no	—
371	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
372	Q₁₆	—	2	4	no	—
373	SD₁₆	—	2	4	no	—
374	SD₁₆	—	2	4	no	—
375	D₈	—	2	4	no	—
376	M₁₆	—	2	2	no	—
377	M₁₆	—	268	4	no	—
378	M₁₆	—	268	2	no	—
379	M₁₆	—	67	4	no	—
380	M₁₆	—	134	2	no	—
381	M₁₆	—	134	2	no	—
382	M₁₆	—	134	2	no	—
383	M₁₆	C₂²×Dic₆₇	1	4	yes	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	268	4	no	—
386	C₂×C₈	—	268	2	no	—
387	C₂×C₈	—	134	2	no	—
388	C₂×C₈	—	134	2	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	—	268	2	no	—
391	C₄⋊C₄	—	134	2	no	—
392	C₂²⋊C₄	—	4	1	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	268	2	no	—
395	C₂²⋊C₄	—	134	2	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	268	2	no	—
398	C₄²	—	268	4	no	—
399	C₄²	—	67	2	no	—
400	C₄²	—	67	2	no	—
401	C₄²	—	134	2	no	—
402	C₄²	—	134	2	no	—
403	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
404	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
405	C₂³	—	2	2	no	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	4	2	no	—
409	C₂³	#14524088	1	1	yes	characteristic
410	Q₈	—	2	2	no	—
411	Q₈	—	2	2	no	—
412	Q₈	—	2	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	D₄×Dic₆₇	1	4	yes	—
415	D₄	—	2	4	no	—
416	D₄	—	2	4	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	4	no	—
421	D₄	—	4	2	no	—
422	D₄	—	4	2	no	—
423	D₄	—	4	2	no	—
424	D₄	—	8	2	no	—
425	D₄	D₄×Dic₆₇	1	4	yes	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	268	4	no	—
441	C₂×C₄	—	268	2	no	—
442	C₂×C₄	—	536	4	no	—
443	C₂×C₄	—	134	2	no	—
444	C₂×C₄	—	134	2	no	—
445	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
447	C₂×C₄	C₂×#11761627	1	2	yes	—
448	C₂×C₄	#14524088	1	1	yes	characteristic
449	C₈	—	2	4	no	—
450	C₈	—	268	2	no	—
451	C₈	—	134	4	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	8	2	no	—
457	C₂²	—	8	1	no	—
458	C₂²	—	8	1	no	—
459	C₂²	—	16	2	no	—
460	C₂²	#16863871	1	1	yes	characteristic
461	C₂²	#16864033	1	2	yes	—
462	C₄	—	2	2	no	—
463	C₄	—	2	4	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	268	2	no	—
468	C₄	—	536	2	no	—
469	C₄	—	134	4	no	—
470	C₄	C₂×D₄×Dic₆₇	1	2	yes	—
471	C₄	#16864028	1	1	yes	characteristic
472	C₄	#16864077	1	1	yes	characteristic
473	C₂	—	2	1	no	—
474	C₂	—	2	2	no	—
475	C₂	—	4	4	no	—
476	C₂	—	8	1	no	—
477	C₂	—	8	1	no	—
478	C₂	#22563193	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16864028
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3018	C₂
socle	C₂×C₆₇	#2358
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3018	D₆₇
hypocenter	C₆₇	#22899
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3018

Complements

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

N	H ≅ G/N	type
1. #22569119	C₂	semidirect
2. #22568756	C₂	semidirect
3. #22561038	C₂	semidirect
4. #22560880	C₂	semidirect
5. #22560712	C₂	semidirect
6. #22560593	C₂	semidirect
7. #22560451	C₂	semidirect
8. #104×Dic₆₇	C₂	semidirect
11. #16864163	C₂²	semidirect
15. #16863445	C₂²	semidirect
16. #16863420	C₂²	semidirect
23. C₂×#14523970	C₂²	semidirect
24. #16863370	C₂²	semidirect
25. M₁₆×Dic₆₇	C₂²	semidirect
26. C₄²×Dic₆₇	C₂²	semidirect
27. #16863168	C₂²	semidirect
28. #16863157	C₂²	semidirect
29. #16863156	C₂²	semidirect
33. #456×C₆₇	C₄	semidirect
34. #455×C₆₇	C₄	semidirect
35. #454×C₆₇	C₄	semidirect
36. #453×C₆₇	C₄	semidirect
62. #14523970	D₄	semidirect
63. #14523970	D₄	semidirect
78. #137×C₆₇	C₂×C₄	semidirect
80. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
83. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
84. #127×C₆₇	C₂×C₄	semidirect
85. #124×C₆₇	C₂×C₄	semidirect
86. #124×C₆₇	C₂×C₄	semidirect
90. #104×C₆₇	C₂×C₄	semidirect
91. #104×C₆₇	C₂×C₄	semidirect
92. #101×C₆₇	C₂×C₄	semidirect
93. #100×C₆₇	C₂×C₄	semidirect
249. C₆₇	#22899	semidirect
256. #456	Dic₆₇	semidirect
257. #455	Dic₆₇	semidirect
258. #454	Dic₆₇	semidirect
259. #453	Dic₆₇	semidirect
290. #137	C₂×Dic₆₇	semidirect
292. AGL(1,ℤ/8)	C₂×Dic₆₇	semidirect
302. C₄⋊Q₈	C₂×Dic₆₇	semidirect
303. #127	C₂×Dic₆₇	semidirect
304. #124	C₂×Dic₆₇	semidirect
305. #124	C₂×Dic₆₇	semidirect
327. #104	C₂×Dic₆₇	semidirect
328. #104	C₂×Dic₆₇	semidirect
329. #101	C₂×Dic₆₇	semidirect
330. #100	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569119	2
#22568756	2
#22561038	2
#22560880	2
#22560712	2
#22560593	2
#22560451	2
#104×Dic₆₇	2
C₆₇×#3018	2
#22899	67