Subgroups — #30878701

Order 17152.

820 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569126	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569119	C₂	1	2	yes	maximal, complemented
3	#22568820	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561103	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561000	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560880	C₂	1	2	yes	maximal, complemented
7	#22560573	C₂	1	2	yes	maximal, complemented
8	#22560431	C₂	1	2	yes	maximal, complemented
9	#22560117	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560051	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3067	C₂	1	1	yes	maximal, characteristic
12	#16864163	C₂²	1	2	yes	—
13	#16864163	C₂²	1	2	yes	—
14	#16863518	—	2	2	no	—
15	#16863518	—	2	4	no	—
16	#16863513	—	2	2	no	—
17	#16863513	—	2	4	no	—
18	#16863445	—	2	2	no	—
19	#16863445	C₂²	1	2	yes	—
20	#16863445	C₂²	1	2	yes	—
21	C₂×#14523979	—	2	2	no	—
22	C₂×#14523979	C₂²	1	1	yes	characteristic
23	C₂×#14523979	C₂²	1	1	yes	characteristic
24	#16863413	C₂²	1	2	yes	—
25	C₂×#14523970	C₂²	1	2	yes	—
26	C₂×#14523970	C₂²	1	1	yes	characteristic
27	#16863392	C₂²	1	1	yes	characteristic
28	#16863363	C₂²	1	2	yes	—
29	#16863351	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₆₇	C₂²	1	2	yes	—
31	C₂×#14523953	C₂²	1	1	yes	characteristic
32	#16863168	C₂²	1	2	yes	—
33	#16863160	C₂²	1	1	yes	characteristic
34	#16863160	C₂²	1	2	yes	—
35	#16863157	C₂²	1	1	yes	characteristic
36	#575×C₆₇	C₂²	1	2	yes	—
37	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
39	#473×C₆₇	C₄	1	2	yes	complemented
40	#472×C₆₇	C₄	1	2	yes	complemented
41	#471×C₆₇	C₄	1	4	yes	complemented
42	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
43	#413×C₆₇	C₂²	1	2	yes	—
44	#14524050	—	2	4	no	—
45	#14524050	—	2	4	no	—
46	#14524050	—	2	4	no	—
47	#14524050	—	4	2	no	—
48	C₂×#11761612	—	2	4	no	—
49	C₂×#11761612	—	4	2	no	—
50	C₂×#11761612	C₂³	1	2	yes	—
51	C₂×#11761612	C₂³	1	2	yes	—
52	C₂×#11761612	C₂³	1	1	yes	characteristic
53	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
54	#14524002	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
56	#14523979	—	2	4	no	—
57	#14523979	—	2	4	no	—
58	#14523979	—	2	1	no	—
59	#14523979	—	2	1	no	—
60	#14523979	D₄	1	2	yes	—
61	#14523979	D₄	1	2	yes	—
62	#14523970	—	2	4	no	—
63	#14523970	—	2	2	no	—
64	#14523970	—	2	2	no	—
65	#14523970	D₄	1	4	yes	—
66	#14523967	—	2	1	no	—
67	#14523964	—	2	1	no	—
68	C₂²⋊(C₆₇⋊C₈)	—	4	2	no	—
69	#14523953	—	2	2	no	—
70	#14523953	—	2	1	no	—
71	#14523953	—	2	1	no	—
72	#14523953	—	2	1	no	—
73	C₈×Dic₆₇	—	2	2	no	—
74	C₈×Dic₆₇	—	2	1	no	—
75	C₂×#41×C₆₇	C₂³	1	2	yes	—
76	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
77	#137×C₆₇	—	2	2	no	—
78	#137×C₆₇	C₂×C₄	1	4	yes	—
79	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
80	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	4	yes	—
81	#135×C₆₇	—	2	4	no	—
82	#135×C₆₇	—	2	1	no	—
83	#135×C₆₇	—	2	1	no	—
84	#135×C₆₇	—	2	1	no	—
85	#135×C₆₇	—	2	1	no	—
86	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
89	#131×C₆₇	—	2	1	no	—
90	#131×C₆₇	—	2	1	no	—
91	#131×C₆₇	C₂×C₄	1	2	yes	—
92	#131×C₆₇	C₂×C₄	1	2	yes	—
93	C₂×M₁₆×C₆₇	—	2	1	no	—
94	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
95	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	—	2	1	no	—
97	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	#108×C₆₇	C₂×C₄	1	2	yes	—
99	#108×C₆₇	C₂×C₄	1	2	yes	—
100	#101×C₆₇	C₂×C₄	1	4	yes	—
101	#100×C₆₇	C₂×C₄	1	4	yes	—
102	C₂²×Dic₆₇	—	2	1	no	—
103	#11761612	—	2	1	no	—
104	#11761612	—	2	2	no	—
105	#11761612	—	2	2	no	—
106	#11761612	—	2	2	no	—
107	#11761612	—	4	4	no	—
108	#11761612	—	4	4	no	—
109	#11761612	C₂×D₄	1	4	yes	—
110	#11761612	C₂×D₄	1	2	yes	—
111	C₂×C₆₇⋊C₈	—	2	2	no	—
112	C₂×C₆₇⋊C₈	—	2	1	no	—
113	C₂×C₆₇⋊C₈	—	2	1	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	—	4	4	no	—
118	C₂×C₆₇⋊C₈	—	4	4	no	—
119	C₄⋊Dic₆₇	—	2	1	no	—
120	C₂²⋊Dic₆₇	—	2	1	no	—
121	C₄×Dic₆₇	—	2	1	no	—
122	C₄×Dic₆₇	—	2	1	no	—
123	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
124	#41×C₆₇	—	2	1	no	—
125	#41×C₆₇	—	2	1	no	—
126	#41×C₆₇	—	2	2	no	—
127	#41×C₆₇	—	2	2	no	—
128	#41×C₆₇	—	2	2	no	—
129	#41×C₆₇	—	2	2	no	—
130	#41×C₆₇	C₂²×C₄	1	2	yes	—
131	#41×C₆₇	C₂×D₄	1	2	yes	—
132	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
134	C₂×D₄×C₆₇	—	2	1	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₆₇	—	2	4	no	—
142	Q₁₆×C₆₇	—	2	1	no	—
143	Q₁₆×C₆₇	—	2	1	no	—
144	SD₁₆×C₆₇	—	2	4	no	—
145	SD₁₆×C₆₇	—	2	4	no	—
146	SD₁₆×C₆₇	—	2	2	no	—
147	SD₁₆×C₆₇	—	2	2	no	—
148	D₈×C₆₇	—	2	4	no	—
149	D₈×C₆₇	—	2	1	no	—
150	D₈×C₆₇	—	2	1	no	—
151	M₁₆×C₆₇	—	2	1	no	—
152	M₁₆×C₆₇	—	2	2	no	—
153	M₁₆×C₆₇	—	2	2	no	—
154	M₁₆×C₆₇	—	2	2	no	—
155	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
156	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
166	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
170	Q₈×C₆₇	—	2	1	no	—
171	Q₈×C₆₇	—	2	2	no	—
172	Q₈×C₆₇	—	2	2	no	—
173	Q₈×C₆₇	C₄×D₄	1	2	yes	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	—	2	2	no	—
177	D₄×C₆₇	—	2	2	no	—
178	D₄×C₆₇	—	2	1	no	—
179	D₄×C₆₇	—	4	2	no	—
180	D₄×C₆₇	—	4	2	no	—
181	D₄×C₆₇	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×Dic₆₇	—	2	2	no	—
195	C₂×Dic₆₇	—	4	1	no	—
196	C₂×Dic₆₇	—	8	1	no	—
197	C₈×C₆₇	—	2	2	no	—
198	C₈×C₆₇	—	2	4	no	—
199	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
202	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
203	C₆₇⋊C₈	—	2	2	no	—
204	C₆₇⋊C₈	—	2	4	no	—
205	C₆₇⋊C₈	—	4	2	no	—
206	C₆₇⋊C₈	—	4	1	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	4	1	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	8	1	no	—
211	C₂²×C₆₇	—	8	2	no	—
212	C₂²×C₆₇	#520	1	1	yes	characteristic
213	C₂²×C₆₇	#547	1	1	yes	characteristic
214	C₂²×C₆₇	#548	1	1	yes	characteristic
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	4	2	no	—
217	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
219	C₄×C₆₇	D₄²	1	1	yes	characteristic
220	C₄×C₆₇	#553	1	1	yes	characteristic
221	Dic₆₇	—	4	2	no	—
222	Dic₆₇	—	8	1	no	—
223	#23043	—	67	1	no	maximal, Sylow
224	C₂×C₆₇	—	2	1	no	—
225	C₂×C₆₇	—	2	1	no	—
226	C₂×C₆₇	—	2	1	no	—
227	C₂×C₆₇	—	4	2	no	—
228	C₂×C₆₇	—	8	2	no	—
229	C₂×C₆₇	#2361	1	1	yes	characteristic
230	#3067	D₆₇	1	1	yes	characteristic
231	#2957	—	67	1	no	—
232	#2956	—	67	2	no	—
233	#2916	—	67	1	no	—
234	#1975	—	67	1	no	—
235	#1953	—	67	1	no	—
236	#1932	—	67	2	no	—
237	#1881	—	67	2	no	—
238	#1858	—	67	2	no	—
239	#1778	—	67	1	no	—
240	#1765	—	67	1	no	—
241	C₆₇	#23043	1	1	yes	characteristic, complemented, Sylow
242	#575	C₂×D₆₇	1	2	yes	—
243	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
244	#568	—	67	2	no	—
245	#568	—	67	2	no	—
246	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
247	#473	Dic₆₇	1	2	yes	complemented
248	#472	Dic₆₇	1	2	yes	complemented
249	#471	Dic₆₇	1	4	yes	complemented
250	#444	—	134	4	no	—
251	#444	—	134	2	no	—
252	#443	—	134	4	no	—
253	#443	—	134	2	no	—
254	#430	—	67	2	no	—
255	#430	—	67	2	no	—
256	#430	—	134	2	no	—
257	C₂×#108	—	67	1	no	—
258	C₂×#108	—	67	1	no	—
259	C₂×#108	—	134	2	no	—
260	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
261	#421	—	67	2	no	—
262	C₂×#104	—	67	1	no	—
263	C₂×#104	—	67	2	no	—
264	#416	—	67	1	no	—
265	#413	C₂×D₆₇	1	2	yes	—
266	#408	—	67	2	no	—
267	#405	—	67	1	no	—
268	C₄×M₁₆	—	67	2	no	—
269	C₂×#97	—	67	1	no	—
270	#344	—	67	2	no	—
271	#341	—	67	1	no	—
272	#341	—	67	2	no	—
273	#339	—	67	1	no	—
274	C₂×#41	C₂²×D₆₇	1	2	yes	—
275	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
276	#137	—	2	2	no	—
277	#137	C₂×Dic₆₇	1	4	yes	—
278	AGL(1,ℤ/8)	—	2	2	no	—
279	AGL(1,ℤ/8)	C₂×Dic₆₇	1	4	yes	—
280	#135	—	2	4	no	—
281	#135	—	2	1	no	—
282	#135	—	2	1	no	—
283	#135	—	2	1	no	—
284	#135	—	2	1	no	—
285	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
286	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
287	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
288	#131	—	2	1	no	—
289	#131	—	2	1	no	—
290	#131	—	268	2	no	—
291	#131	—	134	4	no	—
292	#131	—	134	4	no	—
293	#131	—	134	4	no	—
294	#131	C₂×Dic₆₇	1	2	yes	—
295	#131	C₂×Dic₆₇	1	2	yes	—
296	C₂×M₁₆	—	2	1	no	—
297	C₂×M₁₆	—	268	2	no	—
298	C₂×M₁₆	—	67	2	no	—
299	C₂×M₁₆	—	67	1	no	—
300	C₂×M₁₆	—	67	2	no	—
301	C₂×M₁₆	—	134	4	no	—
302	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
303	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	67	1	no	—
306	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
307	#117	—	67	1	no	—
308	C₂×C₄²	—	67	1	no	—
309	#108	—	67	2	no	—
310	#108	—	67	2	no	—
311	#108	—	134	4	no	—
312	#108	—	134	4	no	—
313	#108	—	134	1	no	—
314	#108	—	134	1	no	—
315	#108	C₂×Dic₆₇	1	2	yes	—
316	#108	C₂×Dic₆₇	1	2	yes	—
317	#104	—	67	4	no	—
318	#104	—	134	2	no	—
319	#104	—	134	2	no	—
320	#104	—	134	4	no	—
321	#103	—	134	1	no	—
322	#102	—	134	1	no	—
323	#101	C₂×Dic₆₇	1	4	yes	—
324	#100	C₂×Dic₆₇	1	4	yes	—
325	C₂²⋊C₈	—	268	2	no	—
326	#97	—	134	2	no	—
327	#97	—	134	1	no	—
328	#97	—	134	1	no	—
329	#97	—	134	1	no	—
330	C₄×C₈	—	134	2	no	—
331	C₄×C₈	—	134	1	no	—
332	#41	—	2	1	no	—
333	#41	—	2	1	no	—
334	#41	—	2	2	no	—
335	#41	—	2	2	no	—
336	#41	—	2	2	no	—
337	#41	—	2	2	no	—
338	#41	C₂²×Dic₆₇	1	2	yes	—
339	#41	D₄×D₆₇	1	2	yes	—
340	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
341	C₂×Q₈	#11761627	1	1	yes	characteristic
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
345	C₂×D₄	#11761627	1	1	yes	characteristic
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	134	1	no	—
349	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
350	Q₁₆	—	2	4	no	—
351	Q₁₆	—	2	1	no	—
352	Q₁₆	—	2	1	no	—
353	SD₁₆	—	2	4	no	—
354	SD₁₆	—	2	4	no	—
355	SD₁₆	—	2	2	no	—
356	SD₁₆	—	2	2	no	—
357	D₈	—	2	4	no	—
358	D₈	—	2	1	no	—
359	D₈	—	2	1	no	—
360	M₁₆	—	2	1	no	—
361	M₁₆	—	2	2	no	—
362	M₁₆	—	2	2	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	268	4	no	—
365	M₁₆	—	268	4	no	—
366	M₁₆	—	67	2	no	—
367	M₁₆	—	67	4	no	—
368	M₁₆	—	134	2	no	—
369	M₁₆	—	134	2	no	—
370	M₁₆	—	134	1	no	—
371	M₁₆	—	134	2	no	—
372	M₁₆	C₂²×Dic₆₇	1	2	yes	—
373	M₁₆	C₂²×Dic₆₇	1	4	yes	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	268	4	no	—
379	C₂×C₈	—	268	4	no	—
380	C₂×C₈	—	134	1	no	—
381	C₂×C₈	—	134	1	no	—
382	C₂×C₈	—	134	1	no	—
383	C₂×C₈	—	134	2	no	—
384	C₂×C₈	—	134	2	no	—
385	C₂×C₈	—	134	1	no	—
386	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
387	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
388	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
389	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
390	C₂×C₈	#11761627	1	1	yes	characteristic
391	C₂×C₈	#11761627	1	1	yes	characteristic
392	C₄⋊C₄	—	134	1	no	—
393	C₂²⋊C₄	—	134	1	no	—
394	C₄²	—	67	2	no	—
395	C₄²	—	134	1	no	—
396	C₄²	—	134	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	2	no	—
399	C₂³	#14524090	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	2	no	—
402	Q₈	—	2	2	no	—
403	Q₈	D₄×Dic₆₇	1	2	yes	—
404	D₄	—	2	2	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	1	no	—
407	D₄	—	2	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	2	no	—
411	D₄	D₄×Dic₆₇	1	2	yes	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	268	1	no	—
419	C₂×C₄	—	536	1	no	—
420	C₂×C₄	—	134	2	no	—
421	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
422	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
424	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
425	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
426	C₂×C₄	#14524089	1	1	yes	characteristic
427	C₈	—	2	2	no	—
428	C₈	—	2	4	no	—
429	C₈	—	268	2	no	—
430	C₈	—	268	1	no	—
431	C₈	—	134	4	no	—
432	C₈	—	134	2	no	—
433	C₈	D₄×Dic₆₇	1	1	yes	characteristic
434	C₈	D₄×Dic₆₇	1	1	yes	characteristic
435	C₈	D₄×Dic₆₇	1	1	yes	characteristic
436	C₈	D₄×Dic₆₇	1	1	yes	characteristic
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	8	1	no	—
441	C₂²	—	8	2	no	—
442	C₂²	#16863879	1	1	yes	characteristic
443	C₂²	#16864046	1	1	yes	characteristic
444	C₂²	#16864052	1	1	yes	characteristic
445	C₄	—	2	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	268	2	no	—
448	C₄	—	536	1	no	—
449	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
450	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
451	C₄	#16864030	1	1	yes	characteristic
452	C₄	#16864098	1	1	yes	characteristic
453	C₂	—	2	1	no	—
454	C₂	—	2	1	no	—
455	C₂	—	2	1	no	—
456	C₂	—	4	2	no	—
457	C₂	—	8	2	no	—
458	C₂	#22563229	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16864030
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3067	C₂
socle	C₂×C₆₇	#2361
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3067	D₆₇
hypocenter	C₆₇	#23043
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3067

Complements

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

N	H ≅ G/N	type
1. #22569126	C₂	semidirect
2. #22569119	C₂	semidirect
3. #22568820	C₂	semidirect
4. #22561103	C₂	semidirect
5. #22561000	C₂	semidirect
6. #22560880	C₂	semidirect
7. #22560573	C₂	semidirect
8. #22560431	C₂	semidirect
9. #22560117	C₂	semidirect
10. #22560051	C₂	semidirect
39. #473×C₆₇	C₄	semidirect
40. #472×C₆₇	C₄	semidirect
41. #471×C₆₇	C₄	semidirect
241. C₆₇	#23043	semidirect
247. #473	Dic₆₇	semidirect
248. #472	Dic₆₇	semidirect
249. #471	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569126	2
#22569119	2
#22568820	2
#22561103	2
#22561000	2
#22560880	2
#22560573	2
#22560431	2
#22560117	2
#22560051	2
C₆₇×#3067	2
#23043	67