Subgroups — #30875542

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569011	C₂	1	1	yes	maximal, characteristic
2	#22568874	C₂	1	2	yes	maximal, complemented
3	#22561102	C₂	1	1	yes	maximal, characteristic
4	#22561097	C₂	1	1	yes	maximal, characteristic
5	#22560858	C₂	1	2	yes	maximal, complemented
6	#22560858	C₂	1	2	yes	maximal, complemented
7	#22560384	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560230	C₂	1	2	yes	maximal, complemented
9	#22560093	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	2	yes	—
14	#16863509	C₄	1	2	yes	complemented
15	#16863504	C₄	1	2	yes	complemented
16	#16863498	C₄	1	2	yes	complemented
17	#16863498	C₄	1	2	yes	complemented
18	#16863456	—	2	2	no	—
19	#16863456	—	2	2	no	—
20	#16863452	—	2	2	no	—
21	#16863448	—	2	2	no	—
22	C₂×#14523979	C₂²	1	1	yes	characteristic
23	C₂×#14523979	C₂²	1	1	yes	characteristic
24	C₂×#14523979	C₂²	1	2	yes	—
25	#16863434	C₂²	1	1	yes	characteristic
26	#16863420	C₂²	1	2	yes	—
27	C₂×#14523971	C₂²	1	2	yes	—
28	C₂×#14523971	C₂²	1	2	yes	—
29	#16863393	C₂²	1	2	yes	—
30	#16863362	C₂²	1	2	yes	—
31	#16863362	C₂²	1	2	yes	—
32	#16863351	C₂²	1	2	yes	—
33	#16863351	C₂²	1	2	yes	—
34	#16863156	C₂²	1	2	yes	—
35	#16863156	C₂²	1	2	yes	—
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	#336×C₆₇	C₂²	1	1	yes	characteristic
42	#336×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#11761628	C₂³	1	2	yes	—
44	#14524065	—	2	2	no	—
45	#14524065	—	2	2	no	—
46	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
47	C₂×#11761617	C₂×C₄	1	2	yes	—
48	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
49	C₂×#11761612	C₂³	1	2	yes	—
50	C₂×#11761612	C₂³	1	2	yes	—
51	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
53	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
54	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
55	C₄×C₂²⋊D₆₇	—	2	2	no	—
56	C₄×C₂²⋊D₆₇	—	2	2	no	—
57	#14524002	C₂³	1	2	yes	—
58	#14524000	—	2	2	no	—
59	C₄²×D₆₇	—	2	2	no	—
60	#14523979	—	2	2	no	—
61	#14523979	—	2	2	no	—
62	#14523979	—	4	2	no	—
63	#14523977	C₂×C₄	1	1	yes	characteristic
64	#14523977	C₂×C₄	1	1	yes	characteristic
65	#14523975	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
67	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
68	#14523971	—	4	2	no	—
69	#14523971	—	4	2	no	—
70	#14523968	C₂×C₄	1	2	yes	—
71	#14523968	C₂×C₄	1	2	yes	—
72	#14523965	C₂×C₄	1	2	yes	—
73	#14523965	C₂×C₄	1	2	yes	—
74	(C₂×D₆₇)⋊C₈	—	2	2	no	—
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	#14523953	—	2	2	no	—
79	#14523953	—	2	2	no	—
80	C₈×Dic₆₇	—	2	2	no	—
81	C₈×Dic₆₇	—	2	2	no	—
82	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
91	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
92	#97×C₆₇	—	2	2	no	—
93	#97×C₆₇	C₂×C₄	1	2	yes	—
94	#97×C₆₇	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₆₇	—	2	2	no	—
96	#95×C₆₇	—	2	1	no	—
97	#11761628	—	2	2	no	—
98	#11761628	—	2	2	no	—
99	#11761628	—	2	2	no	—
100	#11761628	—	2	2	no	—
101	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
102	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
103	C₂×C₂²⋊D₆₇	—	2	2	no	—
104	C₂×C₄×D₆₇	—	2	2	no	—
105	C₂×C₄×D₆₇	—	2	2	no	—
106	C₂×C₄×D₆₇	—	2	2	no	—
107	Dic₂₆₈	—	2	2	no	—
108	#11761617	—	2	4	no	—
109	D₅₃₆	—	2	2	no	—
110	#11761612	—	2	2	no	—
111	#11761612	—	2	2	no	—
112	#11761612	—	2	2	no	—
113	#11761612	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	—	2	2	no	—
116	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
117	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
118	Dic₆₇⋊C₄	—	2	2	no	—
119	Dic₆₇⋊C₄	—	2	2	no	—
120	C₂²⋊Dic₆₇	—	2	2	no	—
121	(C₂×D₆₇)⋊C₄	—	2	2	no	—
122	(C₂×D₆₇)⋊C₄	—	2	2	no	—
123	C₄×Dic₆₇	—	2	2	no	—
124	C₄×Dic₆₇	—	2	2	no	—
125	C₄×Dic₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	#41	1	2	yes	—
144	C₂×C₈×C₆₇	#41	1	2	yes	—
145	C₄⋊C₄×C₆₇	—	4	2	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
151	C₄²×C₆₇	C₂×D₄	1	2	yes	—
152	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₂²×D₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	—	4	1	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
164	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
170	C₂²⋊D₆₇	—	4	2	no	—
171	C₂²⋊D₆₇	—	4	2	no	—
172	C₂×Dic₆₇	—	2	2	no	—
173	C₂×Dic₆₇	—	2	2	no	—
174	C₂×Dic₆₇	—	2	2	no	—
175	D₂₆₈	—	2	2	no	—
176	D₂₆₈	—	2	2	no	—
177	C₄×D₆₇	—	4	4	no	—
178	C₄×D₆₇	—	4	2	no	—
179	C₄×D₆₇	—	4	2	no	—
180	C₄×D₆₇	—	8	2	no	—
181	Dic₁₃₄	—	2	2	no	—
182	Dic₁₃₄	—	2	2	no	—
183	C₈×C₆₇	—	2	2	no	—
184	C₈×C₆₇	—	2	1	no	—
185	C₈×C₆₇	—	2	1	no	—
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	#117	1	2	yes	—
188	C₈×C₆₇	#117	1	2	yes	—
189	C₆₇⋊C₈	—	4	2	no	—
190	C₆₇⋊C₈	—	4	2	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂²×C₆₇	#555	1	1	yes	characteristic
195	C₂²×C₆₇	#556	1	1	yes	characteristic
196	C₂×D₆₇	—	4	2	no	—
197	C₂×D₆₇	—	4	2	no	—
198	C₂×D₆₇	—	4	2	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	8	1	no	—
203	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
204	C₄×C₆₇	#520	1	1	yes	characteristic
205	C₄×C₆₇	#547	1	1	yes	characteristic
206	C₄×C₆₇	#550	1	1	yes	characteristic
207	Dic₆₇	—	4	2	no	—
208	Dic₆₇	—	4	2	no	—
209	Dic₆₇	—	4	2	no	—
210	#22804	—	67	1	no	maximal, Sylow
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	2	1	no	—
213	C₂×C₆₇	#2354	1	1	yes	characteristic
214	C₂×C₆₇	#2955	1	1	yes	characteristic
215	C₂×C₆₇	#2956	1	1	yes	characteristic
216	D₆₇	—	8	2	no	—
217	#2943	—	67	1	no	—
218	#2925	—	67	2	no	—
219	#1975	—	67	1	no	—
220	#1974	—	67	1	no	—
221	#1929	—	67	2	no	—
222	#1929	—	67	2	no	—
223	#1849	—	67	1	no	—
224	#1806	—	67	2	no	—
225	#1773	—	67	1	no	—
226	#1772	—	67	1	no	—
227	#1771	D₆₇	1	1	yes	characteristic, complemented
228	C₆₇	#22804	1	1	yes	characteristic, complemented, Sylow
229	C₂×#135	—	67	1	no	—
230	C₄×#41	—	67	2	no	—
231	#442	—	67	2	no	—
232	#441	—	67	2	no	—
233	#440	—	67	2	no	—
234	#440	—	67	2	no	—
235	#433	—	134	2	no	—
236	#433	—	134	2	no	—
237	#432	—	134	2	no	—
238	#431	—	134	2	no	—
239	C₂×#108	—	67	2	no	—
240	C₂×#108	—	67	1	no	—
241	C₂×#108	—	67	1	no	—
242	#427	—	67	1	no	—
243	#423	—	67	2	no	—
244	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
245	C₂×#104	—	67	2	no	—
246	C₂×#104	—	67	2	no	—
247	#416	—	67	2	no	—
248	#408	—	67	2	no	—
249	#408	—	67	2	no	—
250	#405	—	67	2	no	—
251	#405	—	67	2	no	—
252	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
253	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
255	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
256	#339	—	67	2	no	—
257	#339	—	67	2	no	—
258	#336	C₂×D₆₇	1	1	yes	characteristic
259	#336	C₂×D₆₇	1	1	yes	characteristic
260	C₂×#41	—	67	2	no	—
261	#135	—	134	2	no	—
262	#135	—	134	2	no	—
263	C₂×Q₁₆	—	67	1	no	—
264	C₂×SD₁₆	—	67	2	no	—
265	C₂×D₈	—	67	1	no	—
266	C₂×M₁₆	—	67	2	no	—
267	C₂×M₁₆	—	67	2	no	—
268	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
269	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
270	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
272	C₄×Q₈	—	67	2	no	—
273	C₄×Q₈	—	67	2	no	—
274	C₄×D₄	—	67	2	no	—
275	C₄×D₄	—	67	2	no	—
276	C₄×D₄	—	134	2	no	—
277	C₄×D₄	—	134	2	no	—
278	#117	—	67	2	no	—
279	#117	—	134	2	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄²	—	134	2	no	—
282	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
283	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
284	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
285	#108	—	268	2	no	—
286	#108	—	134	2	no	—
287	#108	—	134	2	no	—
288	#107	—	67	1	no	—
289	#107	—	67	1	no	—
290	#106	—	67	2	no	—
291	C₄⋊C₈	—	134	2	no	—
292	C₄⋊C₈	—	134	2	no	—
293	C₄⋊C₈	D₂₆₈	1	2	yes	—
294	C₄⋊C₈	D₂₆₈	1	2	yes	—
295	#104	—	268	2	no	—
296	#104	—	268	2	no	—
297	#103	—	67	2	no	—
298	#103	—	67	2	no	—
299	#102	—	67	2	no	—
300	#102	—	67	2	no	—
301	C₂²⋊C₈	—	134	2	no	—
302	C₂²⋊C₈	—	134	2	no	—
303	C₂²⋊C₈	—	134	2	no	—
304	C₂²⋊C₈	—	134	2	no	—
305	#97	—	2	2	no	—
306	#97	—	134	2	no	—
307	#97	—	134	2	no	—
308	#97	C₄×D₆₇	1	2	yes	—
309	#97	C₄×D₆₇	1	2	yes	—
310	C₄×C₈	—	2	2	no	—
311	C₄×C₈	—	134	2	no	—
312	C₄×C₈	—	134	2	no	—
313	#95	—	2	1	no	—
314	#41	—	134	2	no	—
315	#41	—	134	2	no	—
316	#41	—	134	2	no	—
317	#41	—	134	2	no	—
318	C₂×Q₈	—	67	2	no	—
319	C₂×D₄	—	67	2	no	—
320	C₂×D₄	—	134	2	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	134	2	no	—
324	C₂²×C₄	—	134	2	no	—
325	C₂²×C₄	—	134	2	no	—
326	C₂²×C₄	#11761628	1	1	yes	characteristic
327	C₂²×C₄	#11761628	1	1	yes	characteristic
328	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
329	Q₁₆	—	134	2	no	—
330	SD₁₆	—	134	4	no	—
331	D₈	—	134	2	no	—
332	M₁₆	—	134	2	no	—
333	M₁₆	—	134	2	no	—
334	M₁₆	—	134	2	no	—
335	M₁₆	—	134	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	134	2	no	—
343	C₂×C₈	—	134	2	no	—
344	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
345	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
346	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
347	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
348	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
349	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
350	C₂×C₈	#11761627	1	2	yes	—
351	C₂×C₈	#11761629	1	2	yes	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	67	2	no	—
354	C₄⋊C₄	—	67	2	no	—
355	C₄⋊C₄	—	134	2	no	—
356	C₄⋊C₄	—	134	2	no	—
357	C₂²⋊C₄	—	134	2	no	—
358	C₂²⋊C₄	—	134	2	no	—
359	C₂²⋊C₄	—	134	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	134	2	no	—
365	C₄²	—	134	2	no	—
366	C₄²	—	134	2	no	—
367	C₄²	C₂×C₄×D₆₇	1	2	yes	—
368	C₄²	C₂×D₂₆₈	1	2	yes	—
369	C₂³	—	134	2	no	—
370	C₂³	C₂×#11761628	1	1	yes	characteristic
371	Q₈	—	134	2	no	—
372	Q₈	—	134	2	no	—
373	D₄	—	268	2	no	—
374	D₄	—	268	2	no	—
375	D₄	—	134	2	no	—
376	D₄	—	134	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	268	4	no	—
386	C₂×C₄	—	268	2	no	—
387	C₂×C₄	—	268	2	no	—
388	C₂×C₄	—	536	2	no	—
389	C₂×C₄	—	134	2	no	—
390	C₂×C₄	—	134	2	no	—
391	C₂×C₄	—	134	2	no	—
392	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
393	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
394	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
396	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
397	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
398	C₂×C₄	#14524089	1	1	yes	characteristic
399	C₂×C₄	#14524090	1	1	yes	characteristic
400	C₈	—	2	1	no	—
401	C₈	—	2	1	no	—
402	C₈	—	2	2	no	—
403	C₈	—	2	2	no	—
404	C₈	—	268	2	no	—
405	C₈	—	268	2	no	—
406	C₈	#14524001	1	2	yes	—
407	C₈	#14524001	1	2	yes	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	268	2	no	—
411	C₂²	—	268	2	no	—
412	C₂²	—	268	2	no	—
413	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
414	C₂²	#16864111	1	1	yes	characteristic
415	C₂²	#16864119	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	8	1	no	—
420	C₄	—	268	2	no	—
421	C₄	—	268	2	no	—
422	C₄	—	268	2	no	—
423	C₄	C₂×#14524001	1	1	yes	characteristic
424	C₄	#16863876	1	1	yes	characteristic
425	C₄	#16864050	1	1	yes	characteristic
426	C₄	#16864068	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	536	2	no	—
430	C₂	#22563156	1	1	yes	characteristic
431	C₂	#22569113	1	1	yes	characteristic
432	C₂	#22569120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1771	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1771	D₆₇
hypocenter	C₆₇	#22804
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1771

Complements

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

N	H ≅ G/N	type
2. #22568874	C₂	semidirect
5. #22560858	C₂	semidirect
6. #22560858	C₂	semidirect
7. #22560384	C₂	semidirect
8. #22560230	C₂	semidirect
9. #22560093	C₂	semidirect
10. #22560088	C₂	semidirect
11. C₆₇×#1771	C₂	semidirect
14. #16863509	C₄	semidirect
15. #16863504	C₄	semidirect
16. #16863498	C₄	semidirect
17. #16863498	C₄	semidirect
227. #1771	D₆₇	semidirect
228. C₆₇	#22804	semidirect

Maximal subgroups

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

subgroup	index
#22569011	2
#22568874	2
#22561102	2
#22561097	2
#22560858	2
#22560384	2
#22560230	2
#22560093	2
#22560088	2
C₆₇×#1771	2
#22804	67