Subgroups — #30868134

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569997	C₂	1	1	yes	maximal, characteristic
2	#22569139	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568853	C₂	1	1	yes	maximal, characteristic
4	#22561152	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560956	C₂	1	2	yes	maximal, complemented
7	#22560910	C₂	1	2	yes	maximal
8	#22560860	C₂	1	2	yes	maximal
9	#22560811	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic
12	#16863953	C₂²	1	1	yes	characteristic
13	#16863703	C₄	1	2	yes	complemented
14	#16863696	C₄	1	2	yes	complemented
15	#16863691	C₄	1	1	yes	characteristic, complemented
16	#16863686	C₄	1	1	yes	characteristic, complemented
17	#16863681	C₄	1	1	yes	characteristic, complemented
18	#16863676	C₄	1	1	yes	characteristic, complemented
19	#16863523	—	2	1	no	—
20	#16863523	—	2	1	no	—
21	#16863520	—	2	1	no	—
22	#16863520	—	2	1	no	—
23	#16863457	—	2	1	no	—
24	#16863457	—	2	1	no	—
25	#16863449	—	2	1	no	—
26	#16863449	—	2	1	no	—
27	C₂×#14523978	C₂²	1	2	yes	—
28	#16863432	C₂²	1	2	yes	—
29	#16863424	C₂²	1	2	yes	—
30	#16863423	C₂²	1	1	yes	characteristic
31	#16863423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
33	#16863392	C₂²	1	2	yes	—
34	#16863363	C₂²	1	1	yes	characteristic
35	#16863363	C₂²	1	1	yes	characteristic
36	#16863354	C₂²	1	2	yes	—
37	C₄×#11761612	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
39	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#16863160	C₂²	1	2	yes	—
41	#16863156	C₂²	1	2	yes	—
42	#16863156	C₂²	1	2	yes	—
43	#16863151	C₂²	1	2	yes	—
44	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#104×C₆₇	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	#339×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761612	C₂³	1	1	yes	characteristic
50	C₂×#11761612	C₂³	1	1	yes	characteristic
51	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
54	#14524043	C₂×C₄	1	1	yes	characteristic
55	#14524043	C₂×C₄	1	1	yes	characteristic
56	#14524035	C₂×C₄	1	2	yes	—
57	#14524022	—	2	1	no	—
58	#14524022	—	2	1	no	—
59	C₄×Dic₁₃₄	—	2	1	no	—
60	C₄×Dic₁₃₄	—	2	1	no	—
61	#14524002	C₂³	1	2	yes	—
62	#14523978	—	4	2	no	—
63	#14523976	C₂×C₄	1	2	yes	—
64	#14523976	C₂×C₄	1	2	yes	—
65	#14523974	C₂×C₄	1	2	yes	—
66	#14523974	C₂×C₄	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
71	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
72	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
74	Dic₆₇⋊C₈	—	2	2	no	—
75	Dic₆₇⋊C₈	—	2	2	no	—
76	#14523967	C₂×C₄	1	2	yes	—
77	#14523967	C₂×C₄	1	2	yes	—
78	#14523964	C₂×C₄	1	2	yes	—
79	#14523964	C₂×C₄	1	2	yes	—
80	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
81	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
82	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
83	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
84	#14523954	—	2	1	no	—
85	#14523953	—	2	2	no	—
86	C₄×C₆₇⋊C₈	—	2	1	no	—
87	C₄×C₆₇⋊C₈	—	4	2	no	—
88	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
89	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
90	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
91	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
92	C₈×Dic₆₇	—	2	2	no	—
93	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
95	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₆₇	—	2	1	no	—
98	C₄×D₄×C₆₇	—	2	1	no	—
99	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#117×C₆₇	—	2	1	no	—
102	#117×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄²×C₆₇	—	2	1	no	—
105	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₆₇	—	2	1	no	—
108	C₄⋊C₈×C₆₇	—	2	1	no	—
109	#104×C₆₇	—	4	2	no	—
110	#95×C₆₇	—	2	1	no	—
111	C₂×Dic₁₃₄	—	2	1	no	—
112	C₂×Dic₁₃₄	—	2	1	no	—
113	#11761612	—	2	1	no	—
114	#11761612	—	2	1	no	—
115	#11761612	—	2	1	no	—
116	#11761612	—	2	1	no	—
117	C₂×C₆₇⋊C₈	—	2	1	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	—	2	1	no	—
120	C₂×C₆₇⋊C₈	—	2	1	no	—
121	C₂×C₆₇⋊C₈	—	2	2	no	—
122	C₂×C₆₇⋊C₈	—	2	1	no	—
123	C₂×C₆₇⋊C₈	—	2	1	no	—
124	C₂×C₆₇⋊C₈	—	2	1	no	—
125	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
127	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
128	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₄⋊Dic₆₇	—	2	1	no	—
132	C₄⋊Dic₆₇	—	2	1	no	—
133	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
134	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
135	Dic₆₇⋊C₄	—	2	2	no	—
136	Dic₆₇⋊C₄	—	2	2	no	—
137	C₂²⋊Dic₆₇	—	2	2	no	—
138	C₄×Dic₆₇	—	2	2	no	—
139	#41×C₆₇	—	2	1	no	—
140	#41×C₆₇	—	2	1	no	—
141	#41×C₆₇	—	2	1	no	—
142	#41×C₆₇	—	2	1	no	—
143	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
153	M₁₆×C₆₇	—	2	2	no	—
154	M₁₆×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	1	no	—
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₂²⋊C₄×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	1	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂³×C₆₇	—	2	1	no	—
174	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
175	Q₈×C₆₇	—	2	1	no	—
176	Q₈×C₆₇	—	2	1	no	—
177	D₄×C₆₇	—	2	1	no	—
178	D₄×C₆₇	—	2	1	no	—
179	D₄×C₆₇	—	4	1	no	—
180	D₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
201	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
202	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
203	C₂×Dic₆₇	—	2	2	no	—
204	C₂×Dic₆₇	—	2	2	no	—
205	Dic₁₃₄	—	4	2	no	—
206	Dic₁₃₄	—	4	2	no	—
207	C₈×C₆₇	—	4	2	no	—
208	C₆₇⋊C₈	—	2	2	no	—
209	C₆₇⋊C₈	—	2	2	no	—
210	C₆₇⋊C₈	—	2	1	no	—
211	C₆₇⋊C₈	—	2	1	no	—
212	C₆₇⋊C₈	—	2	1	no	—
213	C₆₇⋊C₈	—	2	1	no	—
214	C₆₇⋊C₈	—	4	1	no	—
215	C₆₇⋊C₈	—	4	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	1	no	—
221	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₆₇	#541	1	1	yes	characteristic
223	C₂²×C₆₇	#542	1	1	yes	characteristic
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	4	1	no	—
229	C₄×C₆₇	—	4	1	no	—
230	C₄×C₆₇	—	4	1	no	—
231	C₄×C₆₇	#443	1	2	yes	—
232	C₄×C₆₇	#519	1	1	yes	characteristic
233	C₄×C₆₇	#520	1	1	yes	characteristic
234	C₄×C₆₇	#536	1	1	yes	characteristic
235	C₄×C₆₇	#537	1	1	yes	characteristic
236	Dic₆₇	—	4	2	no	—
237	Dic₆₇	—	4	2	no	—
238	#22130	—	67	1	no	maximal, Sylow
239	C₂×C₆₇	—	2	1	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	—	8	1	no	—
242	C₂×C₆₇	#2323	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
244	C₂×C₆₇	#2955	1	1	yes	characteristic
245	#3076	—	67	1	no	—
246	#2960	—	67	1	no	—
247	#2922	—	67	1	no	—
248	#1986	—	67	1	no	—
249	#1950	—	67	1	no	—
250	#1945	—	67	2	no	—
251	#1937	—	67	2	no	—
252	#1929	—	67	2	no	—
253	#1922	—	67	2	no	—
254	#1806	D₆₇	1	1	yes	characteristic
255	C₈×C₄⋊C₄	—	67	1	no	—
256	C₆₇	#22130	1	1	yes	characteristic, complemented, Sylow
257	#533	—	67	1	no	—
258	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
259	#479	—	67	2	no	—
260	#478	—	67	2	no	—
261	#477	—	67	1	no	—
262	#476	—	67	1	no	—
263	#475	—	67	1	no	—
264	#474	—	67	1	no	—
265	#446	—	134	1	no	—
266	#446	—	134	1	no	—
267	C₈×Q₈	—	134	1	no	—
268	C₈×Q₈	—	134	1	no	—
269	#433	—	134	1	no	—
270	#433	—	134	1	no	—
271	#431	—	134	1	no	—
272	#431	—	134	1	no	—
273	C₂×#108	—	67	2	no	—
274	#427	—	67	2	no	—
275	#424	—	67	2	no	—
276	#424	—	67	1	no	—
277	#424	—	67	1	no	—
278	#423	C₂×D₆₇	1	1	yes	characteristic
279	C₂×C₄⋊C₈	—	67	1	no	—
280	C₂×#104	C₂×D₆₇	1	2	yes	—
281	#416	—	67	2	no	—
282	#408	—	67	1	no	—
283	#408	—	67	1	no	—
284	#405	—	67	2	no	—
285	C₄×M₁₆	—	67	1	no	—
286	C₂×C₄×C₈	—	67	2	no	—
287	C₂×C₄×C₈	—	67	1	no	—
288	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
289	#341	—	67	2	no	—
290	#339	—	67	2	no	—
291	#339	—	67	2	no	—
292	#339	C₂×D₆₇	1	2	yes	—
293	#336	—	67	2	no	—
294	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
295	C₂×M₁₆	—	67	1	no	—
296	C₂×M₁₆	—	67	1	no	—
297	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
298	C₂²×C₈	—	67	1	no	—
299	C₂²×C₈	—	67	2	no	—
300	C₂²×C₈	—	67	1	no	—
301	C₄⋊Q₈	—	67	1	no	—
302	C₄⋊Q₈	—	67	1	no	—
303	#125	—	67	2	no	—
304	C₂²⋊Q₈	—	134	1	no	—
305	C₂²⋊Q₈	—	134	1	no	—
306	C₄×Q₈	—	134	1	no	—
307	C₄×Q₈	—	134	1	no	—
308	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
309	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
313	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
314	#117	—	2	1	no	—
315	#117	—	67	2	no	—
316	#117	C₂²×D₆₇	1	1	yes	characteristic
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
320	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
321	#108	—	268	2	no	—
322	#107	—	67	2	no	—
323	#107	—	67	2	no	—
324	#106	—	67	2	no	—
325	#106	—	67	2	no	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	—	67	1	no	—
329	C₄⋊C₈	—	67	1	no	—
330	C₄⋊C₈	—	67	2	no	—
331	C₄⋊C₈	—	134	1	no	—
332	C₄⋊C₈	—	134	1	no	—
333	C₄⋊C₈	—	134	1	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	2	no	—
337	#104	—	4	2	no	—
338	#103	—	67	2	no	—
339	#103	—	67	2	no	—
340	#102	—	67	2	no	—
341	#102	—	67	2	no	—
342	C₂²⋊C₈	—	134	1	no	—
343	C₂²⋊C₈	—	134	1	no	—
344	C₂²⋊C₈	—	134	1	no	—
345	C₂²⋊C₈	—	134	1	no	—
346	#97	—	134	2	no	—
347	#97	—	134	1	no	—
348	C₄×C₈	—	268	2	no	—
349	C₄×C₈	—	67	1	no	—
350	C₄×C₈	—	67	1	no	—
351	C₄×C₈	—	67	1	no	—
352	C₄×C₈	—	67	1	no	—
353	C₄×C₈	—	134	2	no	—
354	C₄×C₈	—	134	1	no	—
355	#95	—	2	1	no	—
356	#41	—	2	1	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	2	1	no	—
360	C₂×Q₈	—	134	1	no	—
361	C₂×Q₈	—	134	1	no	—
362	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	#11761627	1	1	yes	characteristic
370	C₂²×C₄	#11761627	1	1	yes	characteristic
371	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
372	M₁₆	—	2	2	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	134	1	no	—
375	M₁₆	—	134	1	no	—
376	M₁₆	—	134	1	no	—
377	M₁₆	—	134	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	67	2	no	—
380	C₂×C₈	—	67	2	no	—
381	C₂×C₈	—	67	1	no	—
382	C₂×C₈	—	67	1	no	—
383	C₂×C₈	—	67	1	no	—
384	C₂×C₈	—	67	1	no	—
385	C₂×C₈	—	134	1	no	—
386	C₂×C₈	—	134	1	no	—
387	C₂×C₈	—	134	1	no	—
388	C₂×C₈	—	134	2	no	—
389	C₂×C₈	—	134	1	no	—
390	C₂×C₈	—	134	1	no	—
391	C₂×C₈	—	134	2	no	—
392	C₂×C₈	—	134	1	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	67	2	no	—
397	C₄⋊C₄	—	67	2	no	—
398	C₄⋊C₄	—	134	2	no	—
399	C₄⋊C₄	—	134	2	no	—
400	C₄⋊C₄	—	134	1	no	—
401	C₄⋊C₄	—	134	1	no	—
402	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
403	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
404	C₂²⋊C₄	—	2	1	no	—
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	134	2	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	134	2	no	—
414	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
415	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
416	C₄²	D₄×D₆₇	1	1	yes	characteristic
417	C₄²	D₄×D₆₇	1	1	yes	characteristic
418	C₂³	—	2	1	no	—
419	C₂³	C₂×#11761627	1	1	yes	characteristic
420	Q₈	—	2	1	no	—
421	Q₈	—	2	1	no	—
422	Q₈	—	268	2	no	—
423	Q₈	—	268	2	no	—
424	D₄	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	C₂×C₄	—	2	2	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	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	134	2	no	—
441	C₂×C₄	—	134	2	no	—
442	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
443	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
444	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
445	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
446	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
448	C₂×C₄	#14524088	1	1	yes	characteristic
449	C₂×C₄	#14524090	1	1	yes	characteristic
450	C₂×C₄	#14524090	1	1	yes	characteristic
451	C₂×C₄	#14524091	1	1	yes	characteristic
452	C₈	—	4	2	no	—
453	C₈	—	268	1	no	—
454	C₈	—	268	1	no	—
455	C₈	—	134	2	no	—
456	C₈	—	134	2	no	—
457	C₈	—	134	1	no	—
458	C₈	—	134	1	no	—
459	C₈	—	134	1	no	—
460	C₈	—	134	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
467	C₂²	#16864006	1	1	yes	characteristic
468	C₂²	#16864013	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	4	no	—
473	C₄	—	4	1	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	268	2	no	—
477	C₄	—	268	2	no	—
478	C₄	#16863510	1	2	yes	—
479	C₄	#16863873	1	1	yes	characteristic
480	C₄	#16863877	1	1	yes	characteristic
481	C₄	#16863969	1	1	yes	characteristic
482	C₄	#16863975	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	8	1	no	—
486	C₂	#22562887	1	1	yes	characteristic
487	C₂	C₂×#16863510	1	1	yes	characteristic
488	C₂	#22569108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1806	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1806	D₆₇
hypocenter	C₆₇	#22130
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1806

Complements

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

N	H ≅ G/N	type
2. #22569139	C₂	semidirect
4. #22561152	C₂	semidirect
5. #22560982	C₂	semidirect
6. #22560956	C₂	semidirect
9. #22560811	C₂	semidirect
10. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
13. #16863703	C₄	semidirect
14. #16863696	C₄	semidirect
15. #16863691	C₄	semidirect
16. #16863686	C₄	semidirect
17. #16863681	C₄	semidirect
18. #16863676	C₄	semidirect
256. C₆₇	#22130	semidirect

Maximal subgroups

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

subgroup	index
#22569997	2
#22569139	2
#22568853	2
#22561152	2
#22560982	2
#22560956	2
#22560910	2
#22560860	2
#22560811	2
C₄⋊C₄×C₆₇⋊C₈	2
C₆₇×#1806	2
#22130	67